@article{hong_yang_2024, title={Parametric "non-nested" discriminants for multiplicities of univariate polynomials}, volume={3}, ISSN={["1869-1862"]}, DOI={10.1007/s11425-023-2143-3}, journal={SCIENCE CHINA-MATHEMATICS}, author={Hong, Hoon and Yang, Jing}, year={2024}, month={Mar} }
 @article{hong_ovchinnikov_pogudin_yap_2023, title={Global Identifiability of Differential Models (vol 73, pg 1831, 2020)}, volume={9}, ISSN={["1097-0312"]}, DOI={10.1002/cpa.22163}, abstractNote={We are grateful to Peter Thompson for pointing out an error in [1, Lemma 3.5, p. 1848]. The original proof worked only under the assumption that θ ̂ $\hat{\theta }$ is a vector of constants. However, some of the components of θ ̂ $\hat{\bm{\theta }}$ could be the states of the dynamic under consideration, and the lemma was used in such a setup (i.e., with θ ̂ $\hat{\bm{\theta }}$ involving states) later in [1, Proposition 3.4]. We give a more explicit version of the statement and provide a correct proof. The desired statement will be deduced from the following: Lemma 1.Consider a system of differential equations Proof.Consider the following differential ideal Now we will prove the claim. Consider the ring R : = C [ x , μ ] { u } [ 1 / Q ] $R := \mathbb {C}[\bm{x}, \bm{\mu }]\lbrace \bm{u}\rbrace [1/Q]$ . Let J be the ideal generated by I ∩ C [ x , μ ] { u } $I \cap \mathbb {C}[\bm{x}, \bm{\mu }]\lbrace \bm{u}\rbrace$ in R. The definition of I via the saturation at Q implies that Let R ∼ $\widetilde{R}$ be the localization of R with respect to C { u } $\mathbb {C}\lbrace \bm{u}\rbrace$ and J ∼ $\widetilde{J}$ be the ideal generated by J in this localization. The derivation L $\mathcal {L}$ can be naturally extended to R ∼ $\widetilde{R}$ , and J ∼ $\widetilde{J}$ is also L $\mathcal {L}$ -invariant. It is sufficient to prove that J ∼ ∩ C [ x , μ ] ≠ { 0 } $\widetilde{J}\cap \mathbb {C}[\bm{x}, \bm{\mu }] \ne \lbrace 0\rbrace$ . Consider a nonzero element of J ∼ ∩ C [ x , μ ] { u } $\widetilde{J} \cap \mathbb {C}[\bm{x}, \bm{\mu }]\lbrace \bm{u}\rbrace$ with the smallest number of monomials and, among such elements, an element of the smallest total degree. We will call it S. If S ∈ C [ x , μ ] $S\in \mathbb {C}[\bm{x}, \bm{\mu }]$ , we are done. Otherwise, one of u appears in S, say u1. Let h = ord u 1 S $h = \operatorname{ord}_{u_1}S$ . Since R ∼ $\widetilde{R}$ is a Noetherian ring, there exists N > 0 $N > 0$ such that The following corollary is equivalent to [1, Lemma 3.5, p. 1848] but explicitly highlights that some of the entries of θ ̂ $\hat{\bm{\theta }}$ may be initial conditions, not only system parameters. Corollary 1. (Clarified version of [[1], Lemma 3.5, p. 1848])In the notation of [1, Section 2.2], let P ( μ , x , u , … , u ( N ) ) ∈ C [ μ , x ] { u } $P(\bm{\mu }, \bm{x}, u, \ldots , u^{(N)})\in \mathbb {C}[\bm{\mu }, \bm{x}] \lbrace u \rbrace$ be nonzero. Then there exist nonempty Zariski open subsets Θ ⊂ C s $\Theta {\subset }\mathbb {C}^{s}$ and U ⊂ C ∞ ( 0 ) $U\subset \mathbb {C}^{\infty }(0)$ such that, for every θ ̂ = ( μ ̂ , x ̂ * ) ∈ Θ $\hat{\bm{\theta }} = (\hat{\bm{\mu }}, \hat{\bm{x}}^\ast )\in \Theta$ , u ̂ ∈ U $\hat{u}\in U$ , and the corresponding x ̂ = X ( θ ̂ , u ̂ ) $\hat{\bm{x}} = X(\hat{\bm{\theta }}, \hat{u})$ , the function P ( μ ̂ , x ̂ , u ̂ , … , ( u ̂ ) ( N ) ) $P(\hat{\bm{\mu }}, \hat{\bm{x}}, \hat{u},\ldots ,(\hat{u})^{(N)})$ is a nonzero element of C ∞ ( 0 ) $\mathbb {C}^{\infty }(0)$ . Proof.We apply Lemma 1 to the model Σ and the polynomial P as in the statement, and obtain polynomials P 1 ( x , μ ) $P_1(\bm{x}, \bm{\mu })$ and P2(u). We define Zariski open sets Θ and U by P 1 ≠ 0 $P_1 \ne 0$ and P 2 ( u ) | t = 0 ≠ 0 $P_2(\bm{u})|_{t = 0} \ne 0$ , respectively. Then the lemma implies that, for ( μ ̂ , x ̂ ∗ ) ∈ Θ $(\hat{\bm{\mu }}, \hat{\bm{x}}^*) \in \Theta$ and u ̂ ∈ U $\hat{u} \in U$ , P ( μ ̂ , x ̂ , u ̂ , … , ( u ̂ ) ( N ) ) $P(\hat{\bm{\mu }}, \hat{\bm{x}}, \hat{u},\ldots ,(\hat{u})^{(N)})$ will be a nonzero function. □ $\Box$}, journal={COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS}, author={Hong, Hoon and Ovchinnikov, Alexey and Pogudin, Gleb and Yap, Chee}, year={2023}, month={Sep} }
 @article{hong_wang_yang_2023, title={Improving Angular Speed Uniformity by Piecewise Radical Reparameterization}, volume={398}, ISSN={["2075-2180"]}, DOI={10.4204/EPTCS.398.19}, abstractNote={For a rational parameterization of a curve, it is desirable that its angular speed is as uniform as possible. Hence, given a rational parameterization, one wants to find re-parameterization with better uniformity. One natural way is to use piecewise rational reparameterization. However, it turns out that the piecewise rational reparameterization does not help when the angular speed of the given rational parameterization is zero at some points on the curve. In this paper, we show how to overcome the challenge by using piecewise radical reparameterization.}, journal={ELECTRONIC PROCEEDINGS IN THEORETICAL COMPUTER SCIENCE}, author={Hong, Hoon and Wang, Dongming and Yang, Jing}, year={2023}, pages={165–178} }
 @article{hong_yang_2021, title={A condition for multiplicity structure of univariate polynomials}, volume={5}, DOI={10.1016/j.jsc.2020.08.007}, abstractNote={We consider the problem of finding a condition for a univariate polynomial having a given multiplicity structure when the number of distinct roots is given. It is well known that such conditions can be written as conjunctions of several polynomial equations and one inequation in the coefficients, by using repeated parametric gcd's. In this paper, we give a novel condition which is not based on repeated gcd's. Furthermore, it is shown that the number of polynomials in the condition is optimal and the degree of polynomials is smaller than that in the previous condition based on repeated gcd's.}, journal={Journal of Symbolic Computation}, author={Hong, Hoon and Yang, Jing}, year={2021}, month={May}, pages={15} }
 @article{hong_yang_2021, title={A condition for multiplicity structure of univariate polynomials}, volume={104}, url={https://www.sciencedirect.com/science/article/pii/S0747717120300882}, DOI={https://doi.org/10.1016/j.jsc.2020.08.007}, abstractNote={We consider the problem of finding a condition for a univariate polynomial having a given multiplicity structure when the number of distinct roots is given. It is well known that such conditions can be written as conjunctions of several polynomial equations and one inequation in the coefficients, by using repeated parametric gcd's. In this paper, we give a novel condition which is not based on repeated gcd's. Furthermore, it is shown that the number of polynomials in the condition is optimal and the degree of polynomials is smaller than that in the previous condition based on repeated gcd's.}, journal={Journal of Symbolic Computation}, author={Hong, Hoon and Yang, Jing}, year={2021}, pages={523–538} }
 @article{al-kateeb_ambrosino_hong_lee_2021, title={Maximum gap in cyclotomic polynomials}, volume={229}, ISSN={["1096-1658"]}, DOI={10.1016/j.jnt.2021.04.013}, abstractNote={We study the maximum gap g (maximum of the differences between any two consecutive exponents) of cyclotomic polynomials. In 2012, Hong, Lee, Lee and Park showed that g(Φp1p2)=p1−1 for primes p2>p1. In 2017, based on numerous calculations, the following generalization was conjectured: g(Φmp)=φ(m) for square free odd m and prime p>m. The main contribution of this paper is a proof of this conjecture. The proof is based on the discovery of an elegant structure among certain sub-polynomials of Φmp, which are divisible by the m-th inverse cyclotomic polynomial Ψm=xm−1Φm.}, journal={JOURNAL OF NUMBER THEORY}, author={Al-Kateeb, Ala'a and Ambrosino, Mary and Hong, Hoon and Lee, Eunjeong}, year={2021}, month={Dec}, pages={1–15} }
 @article{hong_ovchinnikov_pogudin_yap_2020, title={Global Identifiability of Differential Models}, volume={73}, ISSN={["1097-0312"]}, url={http://dx.doi.org/10.1002/cpa.21921}, DOI={10.1002/cpa.21921}, abstractNote={Many real‐world processes and phenomena are modeled using systems of ordinary differential equations with parameters. Given such a system, we say that a parameter is globally identifiable if it can be uniquely recovered from input and output data. The main contribution of this paper is to provide theory, an algorithm, and software for deciding global identifiability. First, we rigorously derive an algebraic criterion for global identifiability (this is an analytic property), which yields a deterministic algorithm. Second, we improve the efficiency by randomizing the algorithm while guaranteeing the probability of correctness. With our new algorithm, we can tackle problems that could not be tackled before. A software based on the algorithm (called SIAN) is available at https://github.com/pogudingleb/SIAN. © 2020 Wiley Periodicals LLC}, number={9}, journal={COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS}, publisher={Wiley}, author={Hong, Hoon and Ovchinnikov, Alexey and Pogudin, Gleb and Yap, Chee}, year={2020}, month={Sep}, pages={1831–1879} }
 @article{hong_ovchinnikov_pogudin_yap_2019, title={SIAN: a tool for assessing structural identifiability of parametric ODEs}, volume={53}, ISSN={["1932-2240"]}, url={https://doi.org/10.1145/3371991.3371993}, DOI={10.1145/3371991.3371993}, abstractNote={Many important real-world processes are modeled using systems of ordinary differential equations (ODEs) involving unknown parameters. The values of these parameters are usually inferred from experimental data. However, due to the structure of the model, there might be multiple parameter values that yield the same observed behavior even in the case of continuous noise-free data. It is important to detect such situations a priori, before collecting actual data. In this case, the only input is the model itself, so it is natural to tackle this question by methods of symbolic computation.}, number={2}, journal={ACM COMMUNICATIONS IN COMPUTER ALGEBRA}, author={Hong, Hoon and Ovchinnikov, Alexey and Pogudin, Gleb and Yap, Chee}, year={2019}, month={Jun}, pages={37–40} }
 @article{hong_ovchinnikov_pogudin_yap_2019, title={SIAN: software for structural identifiability analysis of ODE models}, volume={35}, ISSN={["1460-2059"]}, url={http://dx.doi.org/10.1093/bioinformatics/bty1069}, DOI={10.1093/bioinformatics/bty1069}, abstractNote={Abstract}, number={16}, journal={BIOINFORMATICS}, publisher={Oxford University Press (OUP)}, author={Hong, Hoon and Ovchinnikov, Alexey and Pogudin, Gleb and Yap, Chee}, editor={Wren, JonathanEditor}, year={2019}, month={Aug}, pages={2873–2874} }
 @article{herman_hong_tsigaridas_2018, title={Improving root separation bounds}, volume={84}, ISSN={["0747-7171"]}, url={http://dx.doi.org/10.1016/j.jsc.2017.03.001}, DOI={10.1016/j.jsc.2017.03.001}, abstractNote={Let f be a polynomial (or polynomial system) with all simple roots. The root separation of f is the minimum of the pair-wise distances between the complex roots. A root separation bound is a lower bound on the root separation. Finding a root separation bound is a fundamental problem, arising in numerous disciplines. We present two new root separation bounds: one univariate bound, and one multivariate bound. The new bounds improve on the old bounds in two ways: The new bounds are usually significantly bigger (hence better) than the previous bounds. The new bounds scale correctly, unlike the previous bounds.}, journal={JOURNAL OF SYMBOLIC COMPUTATION}, publisher={Elsevier BV}, author={Herman, Aaron and Hong, Hoon and Tsigaridas, Elias}, year={2018}, pages={25–56} }
 @article{coss_hauenstein_hong_molzahn_2018, title={Locating and Counting Equilibria of the Kuramoto Model with Rank-One Coupling}, volume={2}, ISSN={["2470-6566"]}, url={http://dx.doi.org/10.1137/17m1128198}, DOI={10.1137/17m1128198}, abstractNote={The Kuramoto model describes synchronization behavior among coupled oscillators and enjoys successful application in a wide variety of fields. Many of these applications seek phase-coherent solutions, i.e., equilibria of the model. Historically, research has focused on situations where the number of oscillators, $n$, is extremely large and can be treated as being infinite. More recently, however, applications have arisen in areas such as electrical engineering with more modest values of $n$. For these, the equilibria can be located by finding the real solutions of a system of polynomial equations utilizing techniques from algebraic geometry. However, typical methods for solving such systems locate all complex solutions even though only the real solutions give equilibria. In this paper, we present an algorithm to locate only the real solutions of the model, thereby shortening computation time by several orders of magnitude in certain situations. This is accomplished by choosing specific equilibria representatives and the consequent algebraic decoupling of the system. The correctness of the algorithm (that it finds only and all the equilibria) is proved rigorously. Additionally, the algorithm can be implemented using interval methods so that the equilibria can be approximated up to any given precision without significantly more computational effort. We also compare this solving approach to other computational algebraic geometric methods. Furthermore, analyzing this approach allows us to prove, asymptotically, that the maximum number of equilibria grows at the same rate as the number of complex solutions of a corresponding polynomial system. Finally, we conjecture an upper bound on the maximum number of equilibria for any number of oscillators which generalizes the known cases and is obtained on a range of explicitly provided natural frequencies.}, number={1}, journal={SIAM JOURNAL ON APPLIED ALGEBRA AND GEOMETRY}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Coss, Owen and Hauenstein, Jonathan D. and Hong, Hoon and Molzahn, Daniel K.}, year={2018}, pages={45–71} }
 @article{hong_rafael sendra_2018, title={Number of common roots and resultant of two tropical univariate polynomials}, volume={511}, ISSN={["1090-266X"]}, url={http://dx.doi.org/10.1016/j.jalgebra.2018.06.027}, DOI={10.1016/j.jalgebra.2018.06.027}, abstractNote={It is well known that for two univariate polynomials over the complex number field the number of their common roots is equal to the order of their resultant. In this paper, we show that this fundamental relationship still holds for the tropical polynomials under suitable adaptation of the notion of order, if the roots are simple and non-zero.}, journal={JOURNAL OF ALGEBRA}, publisher={Elsevier BV}, author={Hong, Hoon and Rafael Sendra, J.}, year={2018}, month={Oct}, pages={420–439} }
 @inproceedings{hong_sturm_2018, title={Positive Solutions of Systems of Signed Parametric Polynomial Inequalities}, volume={11077}, url={https://doi.org/10.1007/978-3-319-99639-4\_17}, DOI={10.1007/978-3-319-99639-4\_17}, booktitle={Computer Algebra in Scientific Computing - 20th International Workshop,
               CASC 2018, Lille, France, September 17-21, 2018, Proceedings}, publisher={Springer}, author={Hong, Hoon and Sturm, Thomas}, editor={Gerdt, Vladimir P. and Koepf, Wolfram and Seiler, Werner M. and Vorozhtsov, Evgenii V.Editors}, year={2018}, pages={238–253} }
 @article{hong_hough_kogan_2017, title={Algorithm for computing mu-bases of univariate polynomials}, volume={80}, ISSN={["0747-7171"]}, url={http://dx.doi.org/10.1016/j.jsc.2016.08.013}, DOI={10.1016/j.jsc.2016.08.013}, abstractNote={We present a new algorithm for computing a μ-basis of the syzygy module of n polynomials in one variable over an arbitrary field K. The algorithm is conceptually different from the previously-developed algorithms by Cox, Sederberg, Chen, Zheng, and Wang for n=3, and by Song and Goldman for an arbitrary n. The algorithm involves computing a “partial” reduced row-echelon form of a (2d+1)×n(d+1) matrix over K, where d is the maximum degree of the input polynomials. The proof of the algorithm is based on standard linear algebra and is completely self-contained. The proof includes a proof of the existence of the μ-basis and as a consequence provides an alternative proof of the freeness of the syzygy module. The theoretical (worst case asymptotic) computational complexity of the algorithm is O(d2n+d3+n2). We have implemented this algorithm (HHK) and the one developed by Song and Goldman (SG). Experiments on random inputs indicate that SG is faster than HHK when d is sufficiently large for a fixed n, and that HHK is faster than SG when n is sufficiently large for a fixed d.}, journal={JOURNAL OF SYMBOLIC COMPUTATION}, publisher={Elsevier BV}, author={Hong, Hoon and Hough, Zachary and Kogan, Irina A.}, year={2017}, pages={844–874} }
 @article{han_dai_hong_xia_2017, title={Open weak CAD and its applications}, volume={80}, ISSN={["0747-7171"]}, url={http://dx.doi.org/10.1016/j.jsc.2016.07.032}, DOI={10.1016/j.jsc.2016.07.032}, abstractNote={The concept of open weak CAD is introduced. Every open CAD is an open weak CAD. On the contrary, an open weak CAD is not necessarily an open CAD. An algorithm for computing projection polynomials of open weak CADs is proposed. The key idea is to compute the intersection of projection factor sets produced by different projection orders. The resulting open weak CAD often has smaller number of sample points than open CADs. The algorithm can be used for computing sample points for all open connected components of f≠0 for a given polynomial f. It can also be used for many other applications, such as testing semi-definiteness of polynomials and copositive problems. In fact, we solved several difficult semi-definiteness problems efficiently by using the algorithm. Furthermore, applying the algorithm to copositive problems, we find an explicit expression of the polynomials producing open weak CADs under some conditions, which significantly improves the efficiency of solving copositive problems.}, journal={JOURNAL OF SYMBOLIC COMPUTATION}, publisher={Elsevier BV}, author={Han, Jingjun and Dai, Liyun and Hong, Hoon and Xia, Bican}, year={2017}, pages={785–816} }
 @article{hong_kim_scholten_sendra_2017, title={Resultants over commutative idempotent semirings I: Algebraic aspect}, volume={79}, ISSN={["1095-855X"]}, url={http://dx.doi.org/10.1016/j.jsc.2016.02.009}, DOI={10.1016/j.jsc.2016.02.009}, abstractNote={The resultant theory plays a crucial role in computational algebra and algebraic geometry. The theory has two aspects: algebraic and geometric. In this paper, we focus on the algebraic aspect. One of the most important and well known algebraic properties of the resultant is that it is equal to the determinant of the Sylvester matrix. In 2008, Odagiri proved that a similar property holds over the tropical semiring if one replaces subtraction with addition. The tropical semiring belongs to a large family of algebraic structures called commutative idempotent semiring. In this paper, we prove that the same property (with subtraction replaced with addition) holds over an arbitrary commutative idempotent semiring.}, journal={JOURNAL OF SYMBOLIC COMPUTATION}, publisher={Elsevier BV}, author={Hong, Hoon and Kim, Yonggu and Scholten, Georgy and Sendra, J. Rafael}, year={2017}, pages={285–308} }
 @article{mccallum_hong_2016, title={On using Lazard's projection in CAD construction}, volume={72}, ISSN={["0747-7171"]}, url={http://dx.doi.org/10.1016/j.jsc.2015.02.001}, DOI={10.1016/j.jsc.2015.02.001}, abstractNote={In 1994 Lazard proposed an improved projection operation for cylindrical algebraic decomposition (CAD). For the proof he introduced a certain notion of valuation of a multivariate Puiseux series at a point. However a gap in one of the key supporting results for the improved projection was subsequently noticed. In this paper we show that Lazard's projection is valid for CAD construction for so-called well-oriented polynomial sets. Our proof does not make use of Lazard's notion of valuation, however.}, journal={JOURNAL OF SYMBOLIC COMPUTATION}, publisher={Elsevier BV}, author={McCallum, Scott and Hong, Hoon}, year={2016}, pages={65–81} }
 @article{herman_hong_2016, title={Quality of positive root bounds}, volume={74}, ISSN={["0747-7171"]}, url={http://dx.doi.org/10.1016/j.jsc.2015.09.006}, DOI={10.1016/j.jsc.2015.09.006}, abstractNote={In this paper, we study the quality of positive root bounds. A positive root bound of a polynomial is an upper bound on the largest positive root. Higher quality means that the relative over-estimation (the ratio of the bound and the largest positive root) is smaller. We report three findings. Most known positive root bounds can be arbitrarily bad; that is, the relative over-estimation can approach infinity, even when the degree and the coefficient size are fixed. When the number of sign variations is the same as the number of positive roots, the relative over-estimation of a positive root bound due to Hong (BH) is at most linear in the degree, no matter what the coefficient size is. When the number of sign variations is one, the relative over-estimation of BH is at most constant, in particular 4, no matter what the degree and the coefficient size are.}, journal={JOURNAL OF SYMBOLIC COMPUTATION}, publisher={Elsevier BV}, author={Herman, Aaron and Hong, Hoon}, year={2016}, pages={592–602} }
 @article{erascu_hong_2016, title={Real quantifier elimination for the synthesis of optimal numerical algorithms (Case study: Square root computation)}, volume={75}, ISSN={["0747-7171"]}, url={http://dx.doi.org/10.1016/j.jsc.2015.11.010}, DOI={10.1016/j.jsc.2015.11.010}, abstractNote={We report on our on-going efforts to apply real quantifier elimination to the synthesis of optimal numerical algorithms. In particular, we describe a case study on the square root problem: given a real number x and an error bound ε, find a real interval such that it contains x and its width is less than or equal to ε. A typical numerical algorithm starts with an initial interval and repeatedly updates it by applying a “refinement map” on it until it becomes narrow enough. Thus the synthesis amounts to finding a refinement map that ensures the correctness and optimality of the resulting algorithm. This problem can be formulated as a real quantifier elimination. Hence, in principle, the synthesis can be carried out automatically. However, the computational requirement is huge, making the automatic synthesis practically impossible with the current general real quantifier elimination software. We overcame the difficulty by (1) carefully reducing a complicated quantified formula into several simpler ones and (2) automatically eliminating the quantifiers from the resulting ones using the state-of-the-art quantifier elimination software. As the result, we were able to synthesize semi-automatically an optimal quadratically1 convergent map, which is better than the well known hand-crafted Secant-Newton map. Interestingly, the optimal synthesized map is not contracting as one would naturally expect.}, journal={JOURNAL OF SYMBOLIC COMPUTATION}, publisher={Elsevier BV}, author={Erascu, Madalina and Hong, Hoon}, year={2016}, pages={110–126} }
 @article{harlim_hong_robbins_2015, title={An algebraic method for constructing stable and consistent autoregressive filters}, volume={283}, ISSN={["1090-2716"]}, url={http://dx.doi.org/10.1016/j.jcp.2014.12.004}, DOI={10.1016/j.jcp.2014.12.004}, abstractNote={In this paper, we introduce an algebraic method to construct stable and consistent univariate autoregressive (AR) models of low order for filtering and predicting nonlinear turbulent signals with memory depth. By stable, we refer to the classical stability condition for the AR model. By consistent, we refer to the classical consistency constraints of Adams–Bashforth methods of order-two. One attractive feature of this algebraic method is that the model parameters can be obtained without directly knowing any training data set as opposed to many standard, regression-based parameterization methods. It takes only long-time average statistics as inputs. The proposed method provides a discretization time step interval which guarantees the existence of stable and consistent AR model and simultaneously produces the parameters for the AR models. In our numerical examples with two chaotic time series with different characteristics of decaying time scales, we find that the proposed AR models produce significantly more accurate short-term predictive skill and comparable filtering skill relative to the linear regression-based AR models. These encouraging results are robust across wide ranges of discretization times, observation times, and observation noise variances. Finally, we also find that the proposed model produces an improved short-time prediction relative to the linear regression-based AR-models in forecasting a data set that characterizes the variability of the Madden–Julian Oscillation, a dominant tropical atmospheric wave pattern.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, publisher={Elsevier BV}, author={Harlim, John and Hong, Hoon and Robbins, Jacob L.}, year={2015}, month={Feb}, pages={241–257} }
 @article{hong_lee_lee_2015, title={Explicit formula for optimal ate pairing over cyclotomic family of elliptic curves}, volume={34}, ISSN={["1090-2465"]}, url={http://dx.doi.org/10.1016/j.ffa.2014.12.007}, DOI={10.1016/j.ffa.2014.12.007}, abstractNote={Pairings on elliptic curves play an important role in cryptography. We provide an explicit formula for vectors of polynomials describing optimal ate pairings over cyclotomic family of elliptic curves. The explicit formula is simple in that it only involves partitioning a certain cyclotomic polynomial. The simplicity of the formula allows us to analyze the sparsity of the vector.}, journal={FINITE FIELDS AND THEIR APPLICATIONS}, publisher={Elsevier BV}, author={Hong, Hoon and Lee, Eunjeong and Lee, Hyang-Sook}, year={2015}, month={Jul}, pages={45–74} }
 @article{hong_tang_xia_2015, title={Special algorithm for stability analysis of multistable biological regulatory systems}, volume={70}, ISSN={["0747-7171"]}, url={http://dx.doi.org/10.1016/j.jsc.2014.09.039}, DOI={10.1016/j.jsc.2014.09.039}, abstractNote={We consider the problem of counting (stable) equilibriums of an important family of algebraic differential equations modeling multistable biological regulatory systems. The problem can be solved, in principle, using real quantifier elimination algorithms, in particular real root classification algorithms. However, it is well known that they can handle only very small cases due to the enormous computing time requirements. In this paper, we present a special algorithm which is much more efficient than the general methods. Its efficiency comes from the exploitation of certain interesting structures of the family of differential equations.}, journal={JOURNAL OF SYMBOLIC COMPUTATION}, publisher={Elsevier BV}, author={Hong, Hoon and Tang, Xiaoxian and Xia, Bican}, year={2015}, pages={112–135} }
 @inbook{yang_wang_hong_2014, title={ImUp: A Maple Package for Uniformity-Improved Reparameterization of Plane Curves}, volume={10}, ISBN={9783662437988 9783662437995}, url={http://dx.doi.org/10.1007/978-3-662-43799-5_29}, DOI={10.1007/978-3-662-43799-5_29}, booktitle={Computer Mathematics}, publisher={Springer Berlin Heidelberg}, author={Yang, Jing and Wang, Dongming and Hong, Hoon}, year={2014}, pages={437–451} }
 @book{hong_yap_2014, title={Mathematical Software – ICMS 2014: 4th International Congress, Seoul, South Korea, August 5-9, 2014. Proceedings}, volume={8592}, url={https://doi.org/10.1007/978-3-662-44199-2}, DOI={10.1007/978-3-662-44199-2}, abstractNote={This book constitutes the proceedings of the 4th International Conference on Mathematical Software, ICMS 2014, held in Seoul, South Korea, in August 2014. The 108 papers included in this volume were c}, journal={Lecture Notes in Computer Science}, publisher={Springer}, author={Hong, Hoon and Yap, Chee}, editor={Hong, Hoon and Yap, CheeEditors}, year={2014}, month={Jan} }
 @book{hong_yap_2014, title={Mathematical software - ICMS 2014: 4th International Congress, Seoul, South Korea, August 5-9, 2014 : Proceedings}, publisher={Heidelberg: Springer}, year={2014} }
 @inbook{chang_hong_lee_lee_2014, title={Pairing Inversion via Non-degenerate Auxiliary Pairings}, ISBN={9783319048727 9783319048734}, ISSN={0302-9743 1611-3349}, url={http://dx.doi.org/10.1007/978-3-319-04873-4_5}, DOI={10.1007/978-3-319-04873-4_5}, abstractNote={The security of pairing-based cryptosystems is closely related to the difficulty of the pairing inversion problem(PI). In this paper, we discuss the difficulty of pairing inversion on the generalized ate pairings of Vercauteren. First, we provide a simpler approach for PI by generalizing and simplifying Kanayama-Okamoto’s approach; our approach involves modifications of exponentiation inversion(EI) and Miller inversion(MI), via an “auxiliary” pairing. Then we provide a complexity of the modified MI, showing that the complexity depends on the sum-norm of the integer vector defining the auxiliary pairing. Next, we observe that degenerate auxiliary pairings expect to make modified EI harder. We provide a sufficient condition on the integer vector, in terms of its max norm, so that the corresponding auxiliary paring is non-degenerate. Finally, we define an infinite set of curve parameters, which includes those of typical pairing friendly curves, and we show that, within those parameters, PI of arbitrarily given generalized ate pairing can be reduced to modified EI in polynomial time.}, booktitle={Pairing-Based Cryptography – Pairing 2013}, publisher={Springer International Publishing}, author={Chang, Seunghwan and Hong, Hoon and Lee, Eunjeong and Lee, Hyang-Sook}, year={2014}, pages={77–96} }
 @inproceedings{eraşcu_hong_2014, place={New York, NY, USA}, title={Synthesis of Optimal Numerical Algorithms Using Real Quantifier Elimination (Case Study: Square Root Computation)}, url={https://doi.org/10.1145/2608628.2608654}, DOI={10.1145/2608628.2608654}, abstractNote={We report on on-going efforts to apply real quantifier elimination to the synthesis of optimal numerical algorithms. In particular, we describe a case study on the square root problem: given a real number x and an error bound ε, find a real interval such that it contains [EQUATION] and its width is less than or equal to ε.
 A typical numerical algorithm starts with an initial interval and repeatedly updates it by applying a "refinement map" on it until it becomes narrow enough. Thus the synthesis amounts to finding a refinement map that ensures the correctness and optimality of the resulting algorithm.
 This problem can be formulated as a real quantifier elimination. Hence, in principle, the synthesis can be carried out automatically. However, the computational requirement is huge, making the automatic synthesis practically impossible with the current general real quantifier elimination software.
 We overcame the difficulty by (1) carefully reducing a complicated quantified formula into several simpler ones and (2) automatically eliminating the quantifiers from the resulting ones using the state of the art quantifier elimination software.
 As the result, we were able to synthesize semi-automatically, under mild assumptions, a class of optimal maps, which are significantly better than the well known hand-crafted Secant-Newton map. Interestingly, the optimal synthesized maps are not contracting as one would naturally expect.}, booktitle={Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation}, publisher={Association for Computing Machinery}, author={Eraşcu, Mădălina and Hong, Hoon}, editor={Nabeshima, Katsusuke and Nagasaka, Kosaku and Winkler, Franz and Szántó, ÁgnesEditors}, year={2014}, pages={162–169} }
 @article{hong_dongming_jing_2013, title={A framework for improving uniformity of parameterizations of curves}, volume={56}, ISSN={["1869-1919"]}, url={http://dx.doi.org/10.1007/s11432-013-4924-4}, DOI={10.1007/s11432-013-4924-4}, number={10}, journal={SCIENCE CHINA-INFORMATION SCIENCES}, publisher={Springer Science and Business Media LLC}, author={Hong, Hoon and DongMing, Wang and Jing, Yang}, year={2013}, month={Oct} }
 @inbook{yang_wang_hong_2013, title={Improving Angular Speed Uniformity by C 1 Piecewise Reparameterization}, ISBN={9783642406713 9783642406720}, ISSN={0302-9743 1611-3349}, url={http://dx.doi.org/10.1007/978-3-642-40672-0_3}, DOI={10.1007/978-3-642-40672-0_3}, abstractNote={We show how to compute a C 1 piecewise-rational reparameterization that closely approximates to the arc-angle parameterization of any plane curve by C 1 piecewise Möbius transformation. By making use of the information provided by the first derivative of the angular speed function, the unit interval is partitioned such that the obtained reparameterization has high uniformity and continuous angular speed. An iteration process is used to refine the interval partition. Experimental results are presented to show the performance of the proposed method and the geometric behavior of the computed reparameterizations.}, booktitle={Automated Deduction in Geometry}, publisher={Springer Berlin Heidelberg}, author={Yang, Jing and Wang, Dongming and Hong, Hoon}, year={2013}, pages={33–47} }
 @article{yang_wang_hong_2013, title={Improving angular speed uniformity by reparameterization}, volume={30}, url={http://dx.doi.org/10.1016/j.cagd.2013.04.001}, DOI={10.1016/j.cagd.2013.04.001}, abstractNote={We introduce the notion of angular speed uniformity as a quality measure for parameter-izations of plane curves and propose an algorithm to compute uniform reparameterizations for quadratic and cubic curves. We prove that only straight lines have uniform rational parameterizations. For any plane curve other than lines, we show how to find a rational reparameterization that has the maximum uniformity among all the rational parameterizations of the same degree. We also establish specific results for quadratic and certain cubic Bézier curves.}, number={7}, journal={Computer Aided Geometric Design}, publisher={Elsevier BV}, author={Yang, Jing and Wang, Dongming and Hong, Hoon}, year={2013}, month={Oct}, pages={636–652} }
 @article{burdis_kogan_hong_2013, title={Object-Image Correspondence for Algebraic Curves under Projections}, volume={9}, ISSN={["1815-0659"]}, url={http://dx.doi.org/10.3842/sigma.2013.023}, DOI={10.3842/sigma.2013.023}, abstractNote={We present a novel algorithm for deciding whether a given planar curve is an image of a given spatial curve, obtained by a central or a parallel projection with unknown parameters.The motivation comes from the problem of establishing a correspondence between an object and an image, taken by a camera with unknown position and parameters.A straightforward approach to this problem consists of setting up a system of conditions on the projection parameters and then checking whether or not this system has a solution.The computational advantage of the algorithm presented here, in comparison to algorithms based on the straightforward approach, lies in a significant reduction of a number of real parameters that need to be eliminated in order to establish existence or non-existence of a projection that maps a given spatial curve to a given planar curve.Our algorithm is based on projection criteria that reduce the projection problem to a certain modification of the equivalence problem of planar curves under affine and projective transformations.To solve the latter problem we make an algebraic adaptation of signature construction that has been used to solve the equivalence problems for smooth curves.We introduce a notion of a classifying set of rational differential invariants and produce explicit formulas for such invariants for the actions of the projective and the affine groups on the plane.}, journal={SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS}, publisher={SIGMA (Symmetry, Integrability and Geometry: Methods and Application)}, author={Burdis, Joseph M. and Kogan, Irina A. and Hong, Hoon}, year={2013} }
 @inproceedings{chang_hong_lee_lee_2013, title={Pairing Inversion via Non-degenerate Auxiliary Pairings}, volume={8365}, url={https://doi.org/10.1007/978-3-319-04873-4\_5}, DOI={10.1007/978-3-319-04873-4\_5}, booktitle={Pairing-Based Cryptography - Pairing 2013 - 6th International Conference,
               Beijing, China, November 22-24, 2013, Revised Selected Papers}, publisher={Springer}, author={Chang, Seunghwan and Hong, Hoon and Lee, Eunjeong and Lee, Hyang-Sook}, editor={Cao, Zhenfu and Zhang, FangguoEditors}, year={2013}, pages={77–96} }
 @article{chang_hong_lee_lee_2013, title={Reducing Pairing Inversion to Exponentiation Inversion using Non-degenerate
               Auxiliary Pairing}, volume={2013}, url={http://eprint.iacr.org/2013/313}, journal={IACR Cryptol. ePrint Arch.}, author={Chang, Seunghwan and Hong, Hoon and Lee, Eunjeong and Lee, Hyang-Sook}, year={2013}, pages={313} }
 @article{hong_lee_lee_park_2013, title={Simple and exact formula for minimum loop length in Ate (i) pairing based on Brezing-Weng curves}, volume={67}, ISSN={["0925-1022"]}, url={http://dx.doi.org/10.1007/s10623-011-9605-y}, DOI={10.1007/s10623-011-9605-y}, abstractNote={We provide a simple and exact formula for the minimum Miller loop length in Ate i pairing based on Brezing–Weng curves, in terms of the involved parameters, under a mild condition on the parameters. It will also be shown that almost all cryptographically useful/meaningful parameters satisfy the mild condition. Hence the simple and exact formula is valid for them. It will also turn out that the formula depends only on essentially two parameters, providing freedom to choose the other parameters to address the design issues other than minimizing the loop length.}, number={2}, journal={DESIGNS CODES AND CRYPTOGRAPHY}, publisher={Springer Science and Business Media LLC}, author={Hong, Hoon and Lee, Eunjeong and Lee, Hyang-Sook and Park, Cheol-Min}, year={2013}, month={May}, pages={271–292} }
 @article{erascu_hong_2013, title={The Secant-Newton Map is Optimal Among Contracting Quadratic Maps
               for Square Root Computation}, volume={18}, url={http://interval.louisiana.edu/reliable-computing-journal/volume-18/reliable-computing-18-pp-073-081.pdf}, journal={Reliab. Comput.}, author={Erascu, Madalina and Hong, Hoon}, year={2013}, pages={73–81} }
 @inbook{yang_wang_hong_2012, title={Improving Angular Speed Uniformity by Optimal C 0 Piecewise Reparameterization}, ISBN={9783642329722 9783642329739}, ISSN={0302-9743 1611-3349}, url={http://dx.doi.org/10.1007/978-3-642-32973-9_29}, DOI={10.1007/978-3-642-32973-9_29}, abstractNote={We adapt the C 0 piecewise Möbius transformation to compute a C 0 piecewise-rational reparameterization of any plane curve that approximates to the arc-angle parameterization of the curve. The method proposed on the basis of this transformation can achieve highly accurate approximation to the arc-angle parameterization. A mechanism is developed to optimize the transformation using locally optimal partitioning of the unit interval. Experimental results are provided to show the effectiveness and efficiency of the reparameterization method.}, booktitle={Computer Algebra in Scientific Computing}, publisher={Springer Berlin Heidelberg}, author={Yang, Jing and Wang, Dongming and Hong, Hoon}, year={2012}, pages={349–360} }
 @inproceedings{yang_wang_hong_2012, title={Improving Angular Speed Uniformity by Optimal C 0 Piecewise Reparameterization}, volume={7442}, url={https://doi.org/10.1007/978-3-642-32973-9\_29}, DOI={10.1007/978-3-642-32973-9\_29}, booktitle={Computer Algebra in Scientific Computing - 14th International Workshop,
               CASC 2012, Maribor, Slovenia, September 3-6, 2012. Proceedings}, publisher={Springer}, author={Yang, Jing and Wang, Dongming and Hong, Hoon}, editor={Gerdt, Vladimir P. and Koepf, Wolfram and Mayr, Ernst W. and Vorozhtsov, Evgenii V.Editors}, year={2012}, pages={349–360} }
 @article{hong_lee_lee_park_2012, title={Maximum gap in (inverse) cyclotomic polynomial}, volume={132}, ISSN={["1096-1658"]}, url={http://dx.doi.org/10.1016/j.jnt.2012.04.008}, DOI={10.1016/j.jnt.2012.04.008}, abstractNote={Let g(f) denote the maximum of the differences (gaps) between two consecutive exponents occurring in a polynomial f. Let Φn denote the n-th cyclotomic polynomial and let Ψn denote the n-th inverse cyclotomic polynomial. In this note, we study g(Φn) and g(Ψn) where n is a product of odd primes, say p1<p2<p3, etc. It is trivial to determine g(Φp1), g(Ψp1) and g(Ψp1p2). Hence the simplest non-trivial cases are g(Φp1p2) and g(Ψp1p2p3). We provide an exact expression for g(Φp1p2). We also provide an exact expression for g(Ψp1p2p3) under a mild condition. The condition is almost always satisfied (only finite exceptions for each p1). We also provide a lower bound and an upper bound for g(Ψp1p2p3).}, number={10}, journal={JOURNAL OF NUMBER THEORY}, publisher={Elsevier BV}, author={Hong, Hoon and Lee, Eunjeong and Lee, Hyang-Sook and Park, Cheol-Min}, year={2012}, month={Oct}, pages={2297–2315} }
 @inproceedings{zhao_hong_wang_aubry_2012, title={Real solution formulas of cubic and quartic equations applied to generate dynamic diagrams with inequality constraints}, url={http://dx.doi.org/10.1145/2245276.2245297}, DOI={10.1145/2245276.2245297}, abstractNote={The approach of solving geometric constraints involving inequalities proposed by Hong and others uses triangular decomposition, solution formulas, and quantifier elimination. We show that for generating dynamic diagrams automatically the performance of this approach can be enhanced, in terms of stability of numeric computation and quality of generated diagrams, when the used solution formulas of cubic and quartic equations are replaced by newly introduced real solution formulas with inequality constraints. Several examples are presented to illustrate the enhanced approach and to demonstrate the advantages and effectiveness of the new solution formulas. An implementation of the enhanced approach in Java with interface to Epsilon and QEPCAD for automated generation of dynamic diagrams is outlined and some experimental data are provided.}, booktitle={Proceedings of the 27th Annual ACM Symposium on Applied Computing - SAC '12}, publisher={ACM Press}, author={Zhao, Ting and Hong, Hoon and Wang, Dongming and Aubry, Philippe}, editor={Ossowski, Sascha and Lecca, PaolaEditors}, year={2012}, pages={94–101} }
 @article{hong_din_2012, title={Variant quantifier elimination}, volume={47}, url={http://dx.doi.org/10.1016/j.jsc.2011.05.014}, DOI={10.1016/j.jsc.2011.05.014}, abstractNote={We describe an algorithm (VQE) for a variant of the real quantifier elimination problem (QE). The variant problem requires the input to satisfy a certain extra condition, and allows the output to be almost equivalent to the input. The motivation/rationale for studying such a variant QE problem is that many quantified formulas arising in applications do satisfy the extra conditions. Furthermore, in most applications, it is sufficient that the output formula is almost equivalent to the input formula. The main idea underlying the algorithm is to substitute the repeated projection step of CAD by a single projection without carrying out a parametric existential decision over the reals. We find that the algorithm can tackle important and challenging problems, such as numerical stability analysis of the widely-used MacCormack’s scheme. The problem has been practically out of reach for standard QE algorithms in spite of many attempts to tackle it. However, the current implementation of VQE can solve it in about 12 hours. This paper extends the results reported at the conference ISSAC 2009.}, number={7}, journal={Journal of Symbolic Computation}, publisher={Elsevier BV}, author={Hong, Hoon and Din, Mohab Safey El}, year={2012}, month={Jul}, pages={883–901} }
 @article{zhao_wang_hong_2011, title={Solution formulas for cubic equations without or with constraints}, volume={46}, ISSN={["0747-7171"]}, url={http://dx.doi.org/10.1016/j.jsc.2011.02.001}, DOI={10.1016/j.jsc.2011.02.001}, abstractNote={HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not.The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Solution Formulas for Cubic Equations Without or With Constraints}, number={8}, journal={JOURNAL OF SYMBOLIC COMPUTATION}, publisher={Elsevier BV}, author={Zhao, Ting and Wang, Dongming and Hong, Hoon}, year={2011}, month={Aug}, pages={904–918} }
 @inproceedings{li_hong_2011, title={WTHD optimization for single phase multilevel converters with step modulation}, url={http://dx.doi.org/10.1109/iecon.2011.6120043}, DOI={10.1109/iecon.2011.6120043}, abstractNote={This paper proposed a new approach for switching angle calculation to achieve the WTHD minimization of the output voltage for single phase multilevel converters with step modulation. The mathematical derivation for the switching angle calculation and the proof of the minimized WTHD are presented in detail. The switching angles of the converters can be selected from a pre-calculated look-up table, which is obtained by solving a set of nonlinear equations. The proposed method is compared with other switching angle calculation methods, and the results show that the proposed method always provides the minimized WTHD, which implies that it will minimize the harmonic content of the output current and the harmonic losses in the inductive loads of the system. Simulation results verify the proposed method.}, booktitle={IECON 2011 - 37th Annual Conference of the IEEE Industrial Electronics Society}, publisher={IEEE}, author={Li, Jun and Hong, Hoon}, year={2011}, month={Nov} }
 @inproceedings{hong_2010, title={Connectivity in Semi-algebraic Sets}, url={http://dx.doi.org/10.1109/synasc.2010.91}, DOI={10.1109/synasc.2010.91}, abstractNote={We consider the problem of deciding whether two given points in a semialgebraic set can be connected, that is, whether the two points lie in a same connected component. In particular, we consider a semialgebraic set consisting of points where a given polynomial is non-zero. We will describe a method based on gradient fields, eigenvectors and interval analysis}, booktitle={2010 12th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing}, publisher={IEEE}, author={Hong, Hoon}, editor={Ida, Tetsuo and Negru, Viorel and Jebelean, Tudor and Petcu, Dana and Watt, Stephen M. and Zaharie, DanielaEditors}, year={2010}, month={Sep}, pages={4–7} }
 @article{liu_hong_huang_2009, title={Real-Time Algorithm for Minimizing THD in Multilevel Inverters With Unequal or Varying Voltage Steps Under Staircase Modulation}, volume={56}, ISSN={["1557-9948"]}, url={http://dx.doi.org/10.1109/tie.2009.2015360}, DOI={10.1109/TIE.2009.2015360}, abstractNote={Different modulation methods have been proposed to control the multilevel inverters. This paper focuses on the staircase modulation that is very popular for multilevel inverters with low switching frequencies. Nonreal-time algorithms for the staircase modulation cannot be applied practically in multilevel inverters with varying voltage steps, since the sizes of lookup tables would be huge. We propose a real-time algorithm for multilevel inverters with unequal or varying voltage steps under the staircase modulation. The algorithm results in the minimal total harmonic distortion (THD) of the output voltage of the inverter, which is proved by rigorous mathematical derivations in this paper. A new expression of THD is presented to simplify the derivation significantly. Computational complexity of the algorithm is analyzed to estimate the time cost of the calculation. We implemented the algorithm on a digital signal processor and provide experimental results that verify the performance of the proposed algorithm.}, number={6}, journal={IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS}, publisher={Institute of Electrical and Electronics Engineers (IEEE)}, author={Liu, Yu and Hong, Hoon and Huang, Alex Q.}, year={2009}, month={Jun}, pages={2249–2258} }
 @article{liu_hong_huang_2009, title={Real-Time Calculation of Switching Angles Minimizing THD for Multilevel Inverters With Step Modulation}, volume={56}, ISSN={["0278-0046"]}, url={http://dx.doi.org/10.1109/tie.2008.918461}, DOI={10.1109/TIE.2008.918461}, abstractNote={Multilevel inverters have been widely applied in industries. A family of optimal pulsewidth modulation (PWM) methods for multilevel inverters, such as step modulation, can generate output voltage with less harmonic distortion than popular modulation strategies, such as the carrier-based sinusoidal PWM or the space vector PWM. However, some drawbacks limit the application of optimal PWM. One of such crucial drawback is that the optimal switching angles could not be calculated in real-time and one has to rely on lookup tables with precalculated angles. We propose a novel real-time algorithm for calculating switching angles that minimizes total harmonic distortion (THD) for step modulation. We give a mathematical proof that the output voltage has the minimum THD. We implemented the algorithm on a digital signal processor and provide experimental results that verify the performance of the proposed algorithm.}, number={2}, journal={IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS}, publisher={Institute of Electrical and Electronics Engineers (IEEE)}, author={Liu, Yu and Hong, Hoon and Huang, Alex Q.}, year={2009}, month={Feb}, pages={285–293} }
 @article{d’andrea_hong_krick_szanto_2009, title={Sylvester’s double sums: The general case}, volume={44}, ISSN={0747-7171}, url={http://dx.doi.org/10.1016/j.jsc.2008.02.011}, DOI={10.1016/j.jsc.2008.02.011}, abstractNote={In 1853 Sylvester introduced a family of double-sum expressions for two finite sets of indeterminates and showed that some members of the family are essentially the polynomial subresultants of the monic polynomials associated with these sets. A question naturally arises: What are the other members of the family? This paper provides a complete answer to this question. The technique that we developed to answer the question turns out to be general enough to characterize all members of the family, providing a uniform method.}, number={9}, journal={Journal of Symbolic Computation}, publisher={Elsevier BV}, author={D’Andrea, Carlos and Hong, Hoon and Krick, Teresa and Szanto, Agnes}, year={2009}, month={Sep}, pages={1164–1175} }
 @inproceedings{hong_din_2009, title={Variant real quantifier elimination}, url={http://dx.doi.org/10.1145/1576702.1576729}, DOI={10.1145/1576702.1576729}, abstractNote={We study a variant of the real quantifier elimination problem (QE). The variant problem requires the input to satisfy a certain extra condition, and allows the ouput to be almost equivalent to the input. In a sense, we are strengthening the pre-condition and weakening the post-condition of the standard QE problem.
 The motivation/rationale for studying such a variant QE problem is that many quantified formulas arising in applications do satisfy the extra conditions. Furthermore, in most applications, it is sufficient that the ouput formula is almost equivalent to the input formula. Thus, we propose to solve a variant of the initial quantifier elimination problem.
 We present an algorithm (VQE), that exploits the strengthened pre-condition and the weakened post-condition. The main idea underlying the algorithm is to substitute the repeated projection step of CAD by a single projection without carrying out a parametric existential decision over the reals.
 We find that the algorithm VQE can tackle important and challenging problems, such as numerical stability analysis of the widely-used MacCormack's scheme. The problem has been practically out of reach for standard QE algorithms in spite of many attempts to tackle it. However the current implementation of VQE can solve it in about 1 day.}, booktitle={Proceedings of the 2009 international symposium on Symbolic and algebraic computation - ISSAC '09}, publisher={ACM Press}, author={Hong, Hoon and Din, Mohab Safey El}, editor={Johnson, Jeremy R. and Park, Hyungju and Kaltofen, ErichEditors}, year={2009}, pages={183–190} }
 @article{hong_perry_2008, title={Are Buchberger's criteria necessary for the chain condition? (vol 42, pg 717, 2007)}, volume={43}, number={3}, journal={Journal of Symbolic Computation}, author={Hong, H. and Perry, J.}, year={2008}, pages={233–233} }
 @article{quinn_hong_2008, title={Connectivity in semialgebraic sets (abstract only)}, volume={42}, url={https://doi.org/10.1145/1394042.1394094}, DOI={10.1145/1394042.1394094}, abstractNote={A semialgebraic set is a subset of real space defined by polynomial equations and inequalities. A semialgebraic set is a union of finitely many maximally connected components. In this talk, we consider the problem of deciding whether two given points in a semialgebraic set are connected, that is, whether the two points lie in the same connected component. In particular, we consider the semialgebraic set defined by f not equal 0 where f is a given bivariate polynomial. The motivation comes from the observation that many important/non-trivial problems in science and engineering can be often reduced to that of connectivity. Due to its importance, there has been intense research effort on the problem. We will describe a method based on gradient fields and provide a sketch of the proof of correctness based on the Morse complex. The method seems to be more efficient than the previous methods in practice.}, number={1-2}, journal={ACM Commun. Comput. Algebra}, author={Quinn, Robert and Hong, Hoon}, year={2008}, pages={86} }
 @article{hong_perry_2008, title={Corrigendum to “Are Buchberger’s criteria necessary for the chain condition?” [J. Symbolic Comput. 42 (2007) 717–732]}, volume={43}, ISSN={0747-7171}, url={http://dx.doi.org/10.1016/j.jsc.2007.10.002}, DOI={10.1016/j.jsc.2007.10.002}, number={3}, journal={Journal of Symbolic Computation}, publisher={Elsevier BV}, author={Hong, Hoon and Perry, John}, year={2008}, month={Mar}, pages={233} }
 @article{d'andrea_hong_krick_szanto_2007, title={An elementary proof of Sylvester's double sums for subresultants}, volume={42}, ISSN={["0747-7171"]}, url={http://dx.doi.org/10.1016/j.jsc.2006.09.003}, DOI={10.1016/j.jsc.2006.09.003}, abstractNote={In 1853 Sylvester stated and proved an elegant formula that expresses the polynomial subresultants in terms of the roots of the input polynomials.Sylvester's formula was also recently proved by Lascoux and Pragacz by using multi-Schur functions and divided differences.In this paper, we provide an elementary proof that uses only basic properties of matrix multiplication and Vandermonde determinants.}, number={3}, journal={JOURNAL OF SYMBOLIC COMPUTATION}, publisher={Elsevier BV}, author={D'Andrea, Carlos and Hong, Hoon and Krick, Teresa and Szanto, Agnes}, year={2007}, month={Mar}, pages={290–297} }
 @article{hong_perry_2007, title={Are Buchberger’s criteria necessary for the chain condition?}, volume={42}, ISSN={0747-7171}, url={http://dx.doi.org/10.1016/j.jsc.2007.02.002}, DOI={10.1016/j.jsc.2007.02.002}, abstractNote={Buchberger’s Gröbner basis theory plays a fundamental role in symbolic computation. The resulting algorithms essentially carry out several S-polynomial reductions. In his Ph.D. thesis and later publication Buchberger showed that sometimes one can skip S-polynomial reductions if the leading terms of polynomials satisfy certain criteria. A question naturally arises: Are Buchberger’s criteria also necessary for skipping S-polynomial reductions? In this paper, after making the question more precise (in terms of a chain condition), we show the answer to be “almost, but not quite”: necessary when there are four or more polynomials, but not necessary when there are exactly three polynomials. For that case, we found an extension to Buchberger’s criteria that is necessary as well as sufficient.}, number={7}, journal={Journal of Symbolic Computation}, publisher={Elsevier BV}, author={Hong, Hoon and Perry, John}, year={2007}, month={Jul}, pages={717–732} }
 @article{yap_hong_2007, title={Foreword}, volume={1}, url={http://dx.doi.org/10.1007/s11786-007-0009-3}, DOI={10.1007/s11786-007-0009-3}, number={1}, journal={Mathematics in Computer Science}, publisher={Springer Science and Business Media LLC}, author={Yap, Chee K. and Hong, Hoon}, year={2007}, month={Dec}, pages={3–7} }
 @book{hong_wang_2006, title={Automated Deduction in Geometry, 5th International Workshop, ADG 2004, Gainesville, FL, USA, September 16-18, 2004, Revised Papers}, volume={3763}, url={http://dx.doi.org/10.1007/11615798}, DOI={10.1007/11615798}, journal={Springer Berlin Heidelberg}, publisher={Springer Berlin Heidelberg}, author={Hong, Hoon and Wang, Dongming}, editor={Hong, Hoon and Wang, DongmingEditors}, year={2006}, month={Jan} }
 @article{hong_kapur_paule_winkler_2006, title={Bruno Buchberger - A life devoted to symbolic computation}, volume={41}, ISSN={["0747-7171"]}, url={https://doi.org/10.1016/j.jsc.2005.09.005}, DOI={10.1016/j.jsc.2005.09.005}, number={3-4}, journal={JOURNAL OF SYMBOLIC COMPUTATION}, author={Hong, H and Kapur, D and Paule, P and Winkler, F}, year={2006}, pages={255–258} }
 @inbook{hong_li_liang_wang_2006, title={Solving Dynamic Geometric Constraints Involving Inequalities}, ISBN={9783540397281 9783540397304}, ISSN={0302-9743 1611-3349}, url={http://dx.doi.org/10.1007/11856290_17}, DOI={10.1007/11856290_17}, abstractNote={This paper presents a specialized method for solving dynamic geometric constraints involving equalities and inequalities. The method works by decomposing the system of constraints into finitely many explicit solution representations in terms of parameters with radicals using triangular decomposition and real quantifier elimination. For any given values of the parameters, if they verify some set of computed relations, the values of the dependent variables may be easily computed by direct evaluation of the corresponding explicit expressions. The effectiveness of our method and its experimental implementation is illustrated by some examples of diagram generation.}, booktitle={Artificial Intelligence and Symbolic Computation}, publisher={Springer Berlin Heidelberg}, author={Hong, Hoon and Li, Liyun and Liang, Tielin and Wang, Dongming}, year={2006}, pages={181–195} }
 @inproceedings{hong_li_liang_wang_2006, place={Berlin, Heidelberg}, title={Solving Dynamic Geometric Constraints Involving Inequalities}, booktitle={Artificial Intelligence and Symbolic Computation}, publisher={Springer Berlin Heidelberg}, author={Hong, Hoon and Li, Liyun and Liang, Tielin and Wang, Dongming}, editor={Calmet, Jacques and Ida, Tetsuo and Wang, DongmingEditors}, year={2006}, pages={181–195} }
 @article{hong_2004, title={Note on Jacobi's method for approximating dominant roots}, volume={37}, ISSN={["0747-7171"]}, url={http://dx.doi.org/10.1016/j.jsc.2003.07.003}, DOI={10.1016/j.jsc.2003.07.003}, abstractNote={In 1834 Jacobi gave a method for approximating dominant roots of a polynomial. In 2002 Mignotte and Stefanescu showed that Jacobi’s method works only when the dominant roots are simple. In this note, we show that Jacobi’s method can still be useful even when the dominant roots are not simple, if we use it for approximating the “distinct” dominant roots.}, number={4}, journal={JOURNAL OF SYMBOLIC COMPUTATION}, publisher={Elsevier BV}, author={Hong, H}, year={2004}, month={Apr}, pages={449–453} }
 @article{hong_minimair_2002, title={Sparse resultant of composed polynomials I* mixed-unmixed case}, volume={33}, ISSN={["0747-7171"]}, url={http://dx.doi.org/10.1006/jsco.2001.0516}, DOI={10.1006/jsco.2001.0516}, abstractNote={The main question of this paper is: What happens to sparse resultants under composition? More precisely, let f1,…, fnbe homogeneous sparse polynomials in the variables y1,…, yn and g1,…, gn be homogeneous sparse polynomials in the variables x1,…,xn. Let fiο(g1,…,gn) be the sparse homogeneous polynomial obtained from fi by replacing yj by gj. Naturally a question arises: Is the sparse resultant of f1ο(g1,…,gn),…,f n(g1,…,gn) in any way related to the (sparse) resultants of f1,…,fn and g1,…,gn? The main contribution of this paper is to provide an answer for the case when g1,…,gn are unmixed, namely, 
Resc1,…,cn (f1 ο (g1,…,gn),…,fn ο (g1,…,gn)) = Resd1,…,dn (f1,…,fn)Vol(Q)ResB(g1) 
where Resd1,…,dn stands for the dense (Macaulay) resultant with respect to the total degrees di of the fi's, B stands for the unmixed sparse resultant with respect to the support B of the gj's,ResC1,…,Cn stands for the mixed sparse resultant with respect to the naturally induced supports Ci of the fi ο(g1,…,gn)'s, and Vol(Q) for the normalized volume of the Newton polytope of the gj. The above expression can be applied to compute sparse resultants of composed polynomials with improved efficience.}, number={4}, journal={JOURNAL OF SYMBOLIC COMPUTATION}, publisher={Elsevier BV}, author={Hong, H and Minimair, M}, year={2002}, month={Apr}, pages={447–465} }
 @article{hong_2001, title={Ore principal subresultant coefficients in solutions}, volume={11}, ISSN={["1432-0622"]}, url={http://dx.doi.org/10.1007/s002000000041}, DOI={10.1007/s002000000041}, number={3}, journal={APPLICABLE ALGEBRA IN ENGINEERING COMMUNICATION AND COMPUTING}, publisher={Springer Science and Business Media LLC}, author={Hong, H}, year={2001}, month={Feb}, pages={227–237} }
 @article{hong_2001, title={Ore subresultant coefficients in solutions}, volume={12}, ISSN={["1432-0622"]}, url={http://dx.doi.org/10.1007/s002000100082}, DOI={10.1007/s002000100082}, number={5}, journal={APPLICABLE ALGEBRA IN ENGINEERING COMMUNICATION AND COMPUTING}, publisher={Springer Science and Business Media LLC}, author={Hong, H}, year={2001}, month={Oct}, pages={421–428} }
 @article{hong_2000, title={Editorial}, volume={29}, url={https://doi.org/10.1006/jsco.2000.0345}, DOI={10.1006/jsco.2000.0345}, number={1}, journal={J. Symb. Comput.}, author={Hong, Hoon}, year={2000}, pages={3–4} }
 @article{hong_schicho_1998, title={Algorithms for trigonometric curves (simplification, implicitization, parameterization)}, volume={26}, ISSN={["0747-7171"]}, url={http://dx.doi.org/10.1006/jsco.1998.0212}, DOI={10.1006/jsco.1998.0212}, abstractNote={A trigonometric curve is a real plane curve where each coordinate is given parametrically by a truncated Fourier series. The trigonometric curves frequently arise in various areas of mathematics, physics, and engineering. Some trigonometric curves can also be represented implicitly by bivariate polynomial equations. In this paper, we give algorithms for (a) simplifying a given parametric representation, (b) computing an implicit representation from a given parametric representation, and (c) computing a parametric representation from a given implicit representation.}, number={3}, journal={JOURNAL OF SYMBOLIC COMPUTATION}, publisher={Elsevier BV}, author={Hong, H and Schicho, J}, year={1998}, month={Sep}, pages={279–300} }
 @inbook{hong_1998, title={An Improvement of the Projection Operator in Cylindrical Algebraic Decomposition}, url={http://dx.doi.org/10.1007/978-3-7091-9459-1_8}, DOI={10.1007/978-3-7091-9459-1_8}, abstractNote={The cylindrical algebraic decomposition (CAD) of Collins (1975) provides a potentially powerful method for solving many important mathematical problems, provided that the required amount of computation can be sufficiently reduced. An important component of the CAD method is the projection operation. Given a set A of r-variate polynomials, the projection operation produces a certain set P of (r − l)-variate polynomials such that a CAD of r-dimensional space for A can be constructed from a CAD of (r − 1)-dimensional space for P. The CAD algorithm begins by applying the projection operation repeatedly, beginning with the input polynomials, until univariate polynomials are obtained. This process is called the projection phase.}, booktitle={Texts and Monographs in Symbolic Computation}, publisher={Springer Vienna}, author={Hong, Hoon}, year={1998}, pages={166–173} }
 @article{hong_1998, title={Bounds for Absolute Positiveness of Multivariate Polynomials}, volume={25}, url={https://doi.org/10.1006/jsco.1997.0189}, DOI={10.1006/jsco.1997.0189}, abstractNote={A multivariate polynomialP(x1, �,xn) with real coefficients is said to beabsolutely positivefrom a real numberBiff it and all of its non-zero partial derivatives of every order are positive forx1,�,xn�B. We call suchBaboundfor the absolute positiveness ofP. This paper provides a simple formula for computing such bounds. We also prove that the resulting bounds are guaranteed to be close to the optimal ones.}, number={5}, journal={J. Symb. Comput.}, author={Hong, Hoon}, year={1998}, pages={571–585} }
 @inproceedings{hong_sendra_1998, place={Vienna}, title={Computation of Variant Resultants}, booktitle={Quantifier Elimination and Cylindrical Algebraic Decomposition}, publisher={Springer Vienna}, author={Hong, Hoon and Sendra, J. Rafael}, editor={Caviness, Bob F. and Johnson, Jeremy R.Editors}, year={1998}, pages={327–340} }
 @article{editorial_1998, url={http://dx.doi.org/10.1016/s0378-4754(97)00119-5}, DOI={10.1016/s0378-4754(97)00119-5}, journal={Mathematics and Computers in Simulation}, year={1998}, month={Mar} }
 @article{hong_1998, title={Groebner Basis Under Composition I}, volume={25}, url={http://dx.doi.org/10.1006/jsco.1997.0192}, DOI={10.1006/jsco.1997.0192}, abstractNote={Composition is the operation of replacing variables in a polynomial with other polynomials. The main question of this paper is:When does composition commute with Groebner basis computation?We prove that this happens iff the composition is `compatible? with the term ordering and the nondivisibility. This has a natural application in the computation of Groebner bases of composed polynomials which often arises in real-life problems.}, number={5}, journal={Journal of Symbolic Computation}, publisher={Elsevier BV}, author={Hong, H.}, year={1998}, month={May}, pages={643–663} }
 @article{hong_jakuš_1998, title={Testing Positiveness of Polynomials}, volume={21}, url={https://doi.org/10.1023/A:1005983105493}, DOI={10.1023/A:1005983105493}, number={1}, journal={Journal of Automated Reasoning}, publisher={Springer Science and Business Media LLC}, author={Hong, Hoon and Jakuš, Dalibor}, year={1998}, month={Aug}, pages={23–38} }
 @article{hong_1997, title={Comparison of Several Decision Algorithms for the Existential Theory of the Reals}, volume={5}, author={Hong, Hoon}, year={1997}, month={May} }
 @article{hong_1997, title={Heuristic Search and Pruning in Polynomial Constraints Satisfaction.}, volume={19}, url={https://doi.org/10.1023/A:1018911907086}, DOI={10.1023/A:1018911907086}, number={3-4}, journal={Ann. Math. Artif. Intell.}, author={Hong, Hoon}, year={1997}, month={Apr}, pages={319–334} }
 @article{hong_1997, title={Implicitization of Nested Circular Curves}, volume={23}, url={http://dx.doi.org/10.1006/jsco.1996.0082}, DOI={10.1006/jsco.1996.0082}, abstractNote={Abstract A nested circular curve is a real plane curve traced by a point on a circle that rotates around another circle that again rotates around still another circle, and so on. In this paper, we give an efficient method for obtaining an implicit equation of a nested circular curve.}, number={2-3}, journal={Journal of Symbolic Computation}, publisher={Elsevier BV}, author={HONG, HOON}, year={1997}, month={Feb}, pages={177–189} }
 @article{hong_1997, title={Multivariate Resultants Under Composition}, volume={4}, author={Hong, Hoon}, year={1997}, month={Apr} }
 @inproceedings{hong_kaltofen_hitz_1997, title={Proceedings of the 2nd International Workshop on Parallel Symbolic
               Computation, PASCO 1997, July 20-22, 1997, Kihei, Hawaii, USA}, publisher={ACM}, year={1997} }
 @article{hong_liska_1997, title={Special Issue on Application of Quantifier Elimination. Foreword of the Guest Editors}, volume={24}, url={http://dx.doi.org/10.1006/jsco.1997.0117}, DOI={10.1006/jsco.1997.0117}, abstractNote={No abstract}, number={2}, journal={Journal of Symbolic Computation}, publisher={Elsevier BV}, author={Hong, Hoon and Liska, Richard}, year={1997}, month={Aug}, pages={123} }
 @article{hong_1997, title={Subresultants Under Composition}, volume={23}, url={http://dx.doi.org/10.1006/jsco.1996.0093}, DOI={10.1006/jsco.1996.0093}, abstractNote={Composition is an operation of replacing a variable in a polynomial with another polynomial. The main question of this paper is:What happens to subresultants under composition? The main contribution of the paper is to show that the subresultants ``almost'' commute with composition. This generalizes the well-known fact that the resultant is invariant under translation.}, number={4}, journal={Journal of Symbolic Computation}, publisher={Elsevier BV}, author={Hong, Hoon}, year={1997}, month={Apr}, pages={355–365} }
 @article{hong_liska_steinberg_1997, title={Testing Stability by Quantifier Elimination}, volume={24}, url={http://dx.doi.org/10.1006/jsco.1997.0121}, DOI={10.1006/jsco.1997.0121}, abstractNote={For initial and initial-boundary value problems described by differential equations, stability requires the solutions to behave well for large times. For linear constant-coefficient problems, Fourier and Laplace transforms are used to convert stability problems to questions about roots of polynomials. Many of these questions can be viewed, in a natural way, as quantifier-elimination problems. The Tarski?Seidenberg theorem shows that quantifier-elimination problems are solvable in a finite number of steps. However, the complexity of this algorithm makes it impractical for even the simplest problems. The newer Quantifier Elimination by Partial Algebraic Decomposition (QEPCAD) algorithm is far more practical, allowing the solution of some non-trivial problems. In this paper, we show how to write all common stability problems as quantifier-elimination problems, and develop a set of computer-algebra tools that allows us to find analytic solutions to simple stability problems in a few seconds, and to solve some interesting problems in from a few minutes to a few hours.}, number={2}, journal={Journal of Symbolic Computation}, publisher={Elsevier BV}, author={HONG, HOON and LISKA, RICHARD and STEINBERG, STANLY}, year={1997}, month={Aug}, pages={161–187} }
 @article{hong_1996, title={An efficient method for analyzing the topology of plane real algebraic curves}, volume={42}, url={http://dx.doi.org/10.1016/s0378-4754(96)00034-1}, DOI={10.1016/s0378-4754(96)00034-1}, abstractNote={A practically efficient algorithm for analyzing the topology of plane real algebraic curves is given. Given a bivariate polynomial, the algorithm produces a planar graph which is topologically equivalent to the real variety of the polynomial on the Euclidean plane. The method does not require the expensive computations of g.c.d., divisions, and root bounds of polynomials with real algebraic number coefficients. Further, it utilizes floating point arithmetic and interval arithmetic whenever possible. Experiments show that most benchmark curves found in the literature can be analyzed within a few seconds on a workstation. Timings on randomly generated polynomials also indicate that the algorithm is efficient to be useful in practice.}, number={4-6}, journal={Mathematics and Computers in Simulation}, publisher={Elsevier BV}, author={Hong, Hoon}, year={1996}, month={Nov}, pages={571–582} }
 @inproceedings{hong_1996, title={Groebner basis under composition II}, url={http://dx.doi.org/10.1145/236869.236906}, DOI={10.1145/236869.236906}, abstractNote={Composition is the operation of replacing variables in a polynomial with other polynomials. The main question of this paper is: When does composition commute with Groebner basis computation? We prove that this happens i the composition is ‘compatible’ with the term ordering and the nondivisibility. This has a natural application in the computation of Groebner bases of composed polynomials which often arises in real-life problems. c 1998 Academic Press Limited}, booktitle={Proceedings of the 1996 international symposium on Symbolic and algebraic computation  - ISSAC '96}, publisher={ACM Press}, author={Hong, Hoon}, editor={Engeler, Erwin and Caviness, B. F. and Lakshman, Yagati N.Editors}, year={1996}, pages={79–85} }
 @article{hong_1996, title={The exact region of stability for MacCormack scheme}, volume={56}, url={http://dx.doi.org/10.1007/bf02253461}, DOI={10.1007/bf02253461}, number={4}, journal={Computing}, publisher={Springer Science and Business Media LLC}, author={Hong, H.}, year={1996}, month={Dec}, pages={371–383} }
 @article{hong_1996, title={The exact stability region of the MacCormack scheme for the scalar advection equation}, volume={9}, url={http://dx.doi.org/10.1016/0893-9659(96)00059-6}, DOI={10.1016/0893-9659(96)00059-6}, abstractNote={This note announces that the stability region of the MacCormack scheme for the scalar advection equation ut = aux + buy is exactly given by aΔtΔx23 + bΔtΔy23 ≥ 1 where Δt, Δx and Δy are the grid distances. This improves upon the result obtained by Wendroff [1] and corrects a mistake in it.}, number={4}, journal={Applied Mathematics Letters}, publisher={Elsevier BV}, author={Hong, H.}, year={1996}, month={Jul}, pages={99–101} }
 @article{hong_stahl_1995, title={Bernstein form is inclusion monotone}, volume={55}, url={http://dx.doi.org/10.1007/bf02238236}, DOI={10.1007/bf02238236}, number={1}, journal={Computing}, publisher={Springer Science and Business Media LLC}, author={Hong, H. and Stahl, V.}, year={1995}, month={Mar}, pages={43–53} }
 @inproceedings{hong_1995, title={Implicitization of Curves Parameterized by Generalized Trigonometric
               Polynomials}, volume={948}, url={https://doi.org/10.1007/3-540-60114-7\_21}, DOI={10.1007/3-540-60114-7\_21}, booktitle={Applied Algebra, Algebraic Algorithms and Error-Correcting Codes,
               11th International Symposium, AAECC-11, Paris, France, July 17-22,
               1995, Proceedings}, publisher={Springer}, author={Hong, Hoon}, editor={Cohen, Gérard D. and Giusti, Marc and Mora, TeoEditors}, year={1995}, pages={285–296} }
 @article{hong_wang_winkler_1995, title={Preface}, volume={13}, url={http://dx.doi.org/10.1007/bf01531320}, DOI={10.1007/bf01531320}, number={1-2}, journal={Annals of Mathematics and Artificial Intelligence}, publisher={Springer Science and Business Media LLC}, author={Hong, Hoon and Wang, Dongming and Winkler, Franz}, year={1995}, month={Mar}, pages={I-II} }
 @article{hong_neubacher_schreiner_1995, title={The Design of the SACLIB/PACLIB Kernels}, volume={19}, url={http://dx.doi.org/10.1006/jsco.1995.1007}, DOI={10.1006/jsco.1995.1007}, abstractNote={This paper describes the design of the kernels of two variants of the SAC-2 computer algebra library: Saclib and Paclib. Saclib is a C version of SAC-2, supporting automatic garbage collection and embeddability. Paclib is a parallel version of Saclib, supporting lightweight concurrency, non-determinism, and parallel garbage collection.}, number={1-3}, journal={Journal of Symbolic Computation}, publisher={Elsevier BV}, author={Hong, Hoon and Neubacher, Andreas and Schreiner, Wolfgang}, year={1995}, month={Jan}, pages={111–132} }
 @inproceedings{hong_loidl_1994, title={Parallel Computation of Modular Multivariate Polynominal Resultants
               on a Shared Memory Machine}, volume={854}, url={https://doi.org/10.1007/3-540-58430-7\_29}, DOI={10.1007/3-540-58430-7\_29}, booktitle={Parallel Processing: CONPAR 94 - VAPP VI, Third Joint International
               Conference on Vector and Parallel Processing, Linz, Austria, September
               6-8, 1994, Proceedings}, publisher={Springer}, author={Hong, Hoon and Loidl, Hans-Wolfgang}, editor={Buchberger, Bruno and Volkert, JensEditors}, year={1994}, pages={325–336} }
 @inproceedings{hong_1994, title={RISC-CLP(CF) Constraint Logic Programming over Complex Functions}, volume={822}, url={https://doi.org/10.1007/3-540-58216-9\_32}, DOI={10.1007/3-540-58216-9\_32}, booktitle={Logic Programming and Automated Reasoning, 5th International Conference,
               LPAR'94, Kiev, Ukraine, July 16-22, 1994, Proceedings}, publisher={Springer}, author={Hong, Hoon}, editor={Pfenning, FrankEditor}, year={1994}, pages={99–113} }
 @article{hong_stahl_1994, title={Safe starting regions by fixed points and tightening}, volume={53}, url={http://dx.doi.org/10.1007/bf02307383}, DOI={10.1007/bf02307383}, number={3-4}, journal={Computing}, publisher={Springer Science and Business Media LLC}, author={Hong, H. and Stahl, V.}, year={1994}, month={Sep}, pages={323–335} }
 @inproceedings{schreiner_hong_1993, title={A New Library for Parallel Algebraic Computation}, booktitle={Proceedings of the Sixth SIAM Conference on Parallel Processing
               for Scientific Computing, PPSC 1993, Norfolk, Virginia, USA, March
               22-24, 1993}, publisher={SIAM}, author={Schreiner, Wolfgang and Hong, Hoon}, editor={Sincovec, Richard F. and Keyes, David E. and Leuze, Michael R. and Petzold, Linda R. and Reed, Daniel A.Editors}, year={1993}, pages={776–783} }
 @inproceedings{hong_1993, title={Parallelization of Quantifier Elimination on a Workstation Network}, volume={673}, url={https://doi.org/10.1007/3-540-56686-4\_42}, DOI={10.1007/3-540-56686-4\_42}, booktitle={Applied Algebra, Algebraic Algorithms and Error-Correcting Codes,
               10th International Symposium, AAECC-10, San Juan de Puerto Rico, Puerto
               Rico, May 10-14, 1993, Proceedings}, publisher={Springer}, author={Hong, Hoon}, editor={Cohen, Gérard D. and Mora, Teo and Moreno, OscarEditors}, year={1993}, pages={170–179} }
 @article{hong_1993, title={Quantifier Elimination for Formulas Constrained by Quadratic Equations via Slope Resultants}, volume={36}, url={http://dx.doi.org/10.1093/comjnl/36.5.439}, DOI={10.1093/comjnl/36.5.439}, abstractNote={An algorithm is given for eliminating the quantifier from a formula: (∃x∈R)[α 2 x 2 +α 1 z+α 0 =0ΛF], where F is a quantifier free formula in x 1 ,..., x r , x, and α 2 , α 1 , α 0 are polynomials in x 1 ,..., x r with real coefficients such that the system {α 2 =0, α 1 =0, α 0 =0} has no solution in R r . The output formulas are made of resultants and their variants, which we call slope resultants. The slope resultants can be, like the resultants, expressed as determinants of certain matrices}, number={5}, journal={The Computer Journal}, publisher={Oxford University Press (OUP)}, author={Hong, H.}, year={1993}, month={May}, pages={439–449} }
 @inproceedings{hong_1993, title={Quantifier elimination for formulas constrained by quadratic equations}, url={http://dx.doi.org/10.1145/164081.164140}, DOI={10.1145/164081.164140}, abstractNote={An algorithm is given for constructing a quantifier free formula (a boolean expression of polynomial equations and inequalities) equivalent to a given formula of the form: (% c R)[azzz + alz + a. = O A F], where F is a quantifier free formula in Z1, . . . . z~, z, and az, al, ao are polynomials in z 1, . . . . Xr with real coefficients such that the system {az = O, al = O, a. = O} has no solution in Rr. Formulas of this form frequently occur in the context of constraint logic programming over the real numbers. The output formulas are made of resultants and two variants, which we call trace and slope resultants. Both of these variant resultants can be expressed as determinants of certain matrices.}, booktitle={Proceedings of the 1993 international symposium on Symbolic and algebraic computation  - ISSAC '93}, publisher={ACM Press}, author={Hong, Hoon}, editor={Bronstein, ManuelEditor}, year={1993}, pages={264–274} }
 @inbook{hong_1993, place={Cambridge, MA, USA}, title={RISC-CLP (Real): Logic Programming with Non-Linear Constraints over the Reals}, booktitle={Constraint Logic Programming: Selected Research}, publisher={MIT Press}, author={Hong, Hoon}, year={1993}, pages={133–159} }
 @article{hong_1993, title={Special Issue Editorial: Computational Quantifier Elimination}, volume={36}, url={https://doi.org/10.1093/comjnl/36.5.399}, DOI={10.1093/comjnl/36.5.399}, number={5}, journal={Comput. J.}, publisher={Oxford University Press (OUP)}, author={Hong, Hoon}, year={1993}, pages={399} }
 @inproceedings{schreiner_hong_1993, title={The Design of the PACLIB Kernel for Parallel Algebraic Computation}, volume={734}, url={https://doi.org/10.1007/3-540-57314-3\_17}, DOI={10.1007/3-540-57314-3\_17}, booktitle={Parallel Computation, Second International ACPC Conference, Gmunden,
               Austria, October 4-6, 1993, Proceedings}, publisher={Springer}, author={Schreiner, Wolfgang and Hong, Hoon}, editor={Volkert, JensEditor}, year={1993}, pages={204–218} }
 @inproceedings{hong_neubacher_schreiner_1993, title={The Design of the SACLIB/PACLIB Kernels}, volume={722}, url={https://doi.org/10.1007/BFb0013184}, DOI={10.1007/BFb0013184}, booktitle={Design and Implementation of Symbolic Computation Systems, International
               Symposium, DISCO '93, Gmunden, Austria, September 15-17, 1993, Proceedings}, publisher={Springer}, author={Hong, Hoon and Neubacher, Andreas and Schreiner, Wolfgang}, editor={Miola, AlfonsoEditor}, year={1993}, pages={288–302} }
 @inproceedings{hong_1992, title={Heuristic Search Strategies for Cylindrical Algebraic Decomposion}, volume={737}, url={https://doi.org/10.1007/3-540-57322-4\_10}, DOI={10.1007/3-540-57322-4\_10}, booktitle={Artificial Intelligence and Symbolic Mathematical Computation, International
               Conference, AISMC-1, Karlsruhe, Germany, August 3-6, 1992, Proceedings}, publisher={Springer}, author={Hong, Hoon}, editor={Calmet, Jacques and Campbell, John A.Editors}, year={1992}, pages={152–165} }
 @inproceedings{hong_1992, title={Non-linear Real Constraints in Constraint Logic Programming}, volume={632}, url={https://doi.org/10.1007/BFb0013827}, DOI={10.1007/BFb0013827}, booktitle={Algebraic and Logic Programming, Third International Conference, Volterra,
               Italy, September 2-4, 1992, Proceedings}, publisher={Springer}, author={Hong, Hoon}, editor={Kirchner, Hélène and Levi, GiorgioEditors}, year={1992}, pages={201–212} }
 @article{hong_schreiner_1992, title={Programming in PACLIB}, volume={26}, url={https://doi.org/10.1145/147105.147108}, DOI={10.1145/147105.147108}, abstractNote={
            This paper gives a short overview on PACLIB, a new system for
            p
            arallel
            a
            lgebraic
            c
            omputation on shared memory computers. PACLIB has been developed as a professional tool for the simple design and efficient implementation of parallel algorithms in computer algebra and related areas. It provides concurrency, shared memory communication, non-determinism, speculative parallelism, streams and pipelining and a parallelized garbage collection. PACLIB has been implemented on a Sequent Symmetry multiprocessor and is portable to other shared memory machines and workstations.
          }, number={4}, journal={SIGSAM Bull.}, author={Hong, Hoon and Schreiner, Wolfgang}, year={1992}, pages={1–6} }
 @inproceedings{hong_1992, title={Simple Solution Formula Construction in Cylindrical Algebraic Decomposition
               Based Quantifier Elimination}, url={https://doi.org/10.1145/143242.143306}, DOI={10.1145/143242.143306}, abstractNote={Since Tarski (1951) gave the first quantifier elimination algorithm for real closed fields, various improvements and new methods have been devised and analyzed (Arnon 1981, 1988b; Ben-Or et al. 1986; Boge 1980; Buchberger and Hong 1991; Canny 1988; Cohen 1969; Collins 1975; Collins and Hong 1991; Fitchas et al. 1990a; Grigor’ev 1988; Grigor’ev and Vorobjov 1988; Heintz et al. 1989a; Holthusen 1974; Hong 1989, 1990a, 1990b, 1991a, 1991b, 1991c; Johnson 1991; Langemyr 1990; Lazard 1990; McCallum 1984; Renegar 1992a, 1992b, 1992c; Seidenberg 1954).}, booktitle={Proceedings of the 1992 International Symposium on Symbolic and Algebraic
               Computation, ISSAC '92, Berkeley, CA, USA, July 27-29, 1992}, publisher={ACM}, author={Hong, Hoon}, editor={Wang, Paul S.Editor}, year={1992}, pages={177–188} }
 @article{collins_hong_1991, title={Partial Cylindrical Algebraic Decomposition for quantifier elimination}, volume={12}, url={http://dx.doi.org/10.1016/s0747-7171(08)80152-6}, DOI={10.1016/s0747-7171(08)80152-6}, abstractNote={The Cylindrical Algebraic Decomposition method (CAD) decomposes Rr into regions over which given polynomials have constant signs. An important application of CAD is quantifier elimination in elementary algebra and geometry. In this paper we present a method which intermingles CAD construction with truth evaluation so that parts of the CAD are constructed only as needed to further truth evaluation and aborts CAD construction as soon as no more truth evaluation is needed. The truth evaluation utilizes in an essential way any quantifiers which are present and additionally takes account of atomic formulas from which some variables are absent. Preliminary observations show that the new method is always more efficient than the original, and often significantly more efficient.}, number={3}, journal={Journal of Symbolic Computation}, publisher={Elsevier BV}, author={Collins, George E. and Hong, Hoon}, year={1991}, month={Sep}, pages={299–328} }
 @inproceedings{hong_1990, title={An improvement of the projection operator in cylindrical algebraic decomposition}, url={http://dx.doi.org/10.1145/96877.96943}, DOI={10.1145/96877.96943}, abstractNote={The Cylindrical Algebraic Decomposition (CAD) method of Collins [5] decomposes <italic>r</italic>-dimensional Euclidean space into regions over which a given set of polynomials have constant signs. An important component of the CAD method is the projection operation: given a set <italic>A</italic> of <italic>r</italic>-variate polynomials, the projection operation produces a set <italic>P</italic> of (<italic>r</italic> - 1)-variate polynomials such that a CAD of <italic>r</italic>-dimensional space for <italic>A</italic> can be constructed from a CAD of (<italic>r</italic>-1)-dimensional space for <italic>P</italic>. In this paper, we present an improvement to the projection operation. By generalizing a lemma on which the proof of the original projection operation is based, we are able to find another projection operation which produces a smaller number of polynomials. Let <italic>m</italic> be the number of polynomials contained in <italic>A</italic>, and let <italic>n</italic> be a bound for the degree of each polynomial in <italic>A</italic> in the projection variable. The number of polynomials produced by the original projection operation is dominated by <italic>m</italic><supscrpt>2</supscrpt><italic>n</italic><supscrpt>3</supscrpt> whereas the number of polynomials produced by our projection operation is dominated by <italic>m</italic><supscrpt>2</supscrpt><italic>n</italic><supscrpt>2</supscrpt>.}, booktitle={Proceedings of the international symposium on Symbolic and algebraic computation  - ISSAC '90}, publisher={ACM Press}, author={Hong, H.}, year={1990} }