@article{harris_hauenstein_szanto_2020, title={Smooth Points on Semi-algebraic Sets}, volume={54}, ISSN={["1932-2240"]}, DOI={10.1145/3457341.3457347}, abstractNote={
Many algorithms for determining properties of semi-algebraic sets rely upon the ability to compute smooth points [1]. We present a simple procedure based on computing the critical points of some well-chosen function that guarantees the computation of smooth points in each connected bounded component of a real atomic semi-algebraic set. Our technique is intuitive in principal, performs well on previously difficult examples, and is straightforward to implement using existing numerical algebraic geometry software. The practical efficiency of our approach is demonstrated by solving a conjecture on the number of equilibria of the Kuramoto model for the
n
= 4 case. We also apply our method to design an efficient algorithm to compute the real dimension of algebraic sets, the original motivation for this research.
}, number={3}, journal={ACM COMMUNICATIONS IN COMPUTER ALGEBRA}, author={Harris, Katherine and Hauenstein, Jonathan D. and Szanto, Agnes}, year={2020}, month={Sep}, pages={105–108} }
@article{brake_hauenstein_vinzant_2019, title={Computing complex and real tropical curves using monodromy}, volume={223}, ISSN={["1873-1376"]}, DOI={10.1016/j.jpaa.2019.03.019}, abstractNote={Tropical varieties capture combinatorial information about how coordinates of points in a classical variety approach zero or infinity. We present algorithms for computing the rays of a complex and real tropical curve defined by polynomials with constant coefficients. These algorithms rely on homotopy continuation, monodromy loops, and Cauchy integrals. Several examples are presented which are computed using an implementation that builds on the numerical algebraic geometry software Bertini.}, number={12}, journal={JOURNAL OF PURE AND APPLIED ALGEBRA}, author={Brake, Danielle A. and Hauenstein, Jonathan D. and Vinzant, Cynthia}, year={2019}, month={Dec}, pages={5232–5250} }
@inbook{hauenstein_pan_szanto_2014, title={A Note on Global Newton Iteration Over Archimedean and Non-Archimedean Fields}, volume={8660}, ISBN={9783319105147 9783319105154}, ISSN={0302-9743 1611-3349}, url={http://dx.doi.org/10.1007/978-3-319-10515-4_15}, DOI={10.1007/978-3-319-10515-4_15}, abstractNote={In this paper, we study iterative methods on the coefficients of the rational univariate representation (RUR) of a given algebraic set, called a global Newton iterations. We compare two natural approaches to define locally quadratically convergent iterations: the first one involves Newton iteration applied to the approximate roots individually and then interpolation to find the RUR of these approximate roots; the second one considers the coefficients in the exact RUR as zeroes of a high dimensional map defined by polynomial reduction and applies Newton iteration on this map. We prove that over fields with a p-adic valuation these two approaches give the same iteration function. However, over fields equipped with the usual Archimedean absolute value they are not equivalent. In the latter case, we give explicitly the iteration function for both approaches. Finally, we analyze the parallel complexity of the different versions of the global Newton iteration, compare them, and demonstrate that they can be efficiently computed. The motivation for this study comes from the certification of approximate roots of overdetermined and singular polynomial systems via the recovery of an exact RUR from approximate numerical data.}, booktitle={Computer Algebra in Scientific Computing}, publisher={Springer International Publishing}, author={Hauenstein, Jonathan D. and Pan, Victor Y. and Szanto, Agnes}, year={2014}, pages={202–217} }
@article{hao_hauenstein_hu_sommese_2014, title={A bootstrapping approach for computing multiple solutions of differential equations}, volume={258}, ISSN={["1879-1778"]}, DOI={10.1016/j.cam.2013.09.007}, abstractNote={Discretizing systems of nonlinear algebraic differential equations yields polynomial systems. When using a fine discretization, the resulting polynomial system is often too large to solve using a direct solving approach. Our approach for solving such systems is to utilize a homotopy continuation based method arising from domain decomposition. This method solves polynomial systems arising from subdomains and then uses homotopy continuation to build solutions of the original polynomial system. We illustrate this approach on both one- and two-dimensional problems.}, journal={JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS}, author={Hao, Wenrui and Hauenstein, Jonathan D. and Hu, Bei and Sommese, Andrew J.}, year={2014}, month={Mar}, pages={181–190} }
@article{mehta_hauenstein_wales_2014, title={Certification and the potential energy landscape}, volume={140}, ISSN={["1089-7690"]}, DOI={10.1063/1.4881638}, abstractNote={Typically, there is no guarantee that a numerical approximation obtained using standard nonlinear equation solvers is indeed an actual solution, meaning that it lies in the quadratic convergence basin. Instead, it may lie only in the linear convergence basin, or even in a chaotic region, and hence not converge to the corresponding stationary point when further optimization is attempted. In some cases, these non-solutions could be misleading. Proving that a numerical approximation will quadratically converge to a stationary point is termed certification. In this report, we provide details of how Smale's α-theory can be used to certify numerically obtained stationary points of a potential energy landscape, providing a mathematical proof that the numerical approximation does indeed correspond to an actual stationary point, independent of the precision employed.}, number={22}, journal={JOURNAL OF CHEMICAL PHYSICS}, author={Mehta, Dhagash and Hauenstein, Jonathan D. and Wales, David J.}, year={2014}, month={Jun} }
@article{bates_decker_hauenstein_peterson_pfister_schreyer_sommese_wampler_2014, title={Comparison of probabilistic algorithms for analyzing the components of an affine algebraic variety}, volume={231}, ISSN={["1873-5649"]}, DOI={10.1016/j.amc.2013.12.165}, abstractNote={Systems of polynomial equations arise throughout mathematics, engineering, and the sciences. It is therefore a fundamental problem both in mathematics and in application areas to find the solution sets of polynomial systems. The focus of this paper is to compare two fundamentally different approaches to computing and representing the solutions of polynomial systems: numerical homotopy continuation and symbolic computation. Several illustrative examples are considered, using the software packages Bertini and Singular.}, journal={APPLIED MATHEMATICS AND COMPUTATION}, author={Bates, Daniel J. and Decker, Wolfram and Hauenstein, Jonathan D. and Peterson, Chris and Pfister, Gerhard and Schreyer, Frank-Olaf and Sommese, Andrew J. and Wampler, Charles W.}, year={2014}, month={Mar}, pages={619–633} }
@article{mehta_hauenstein_wales_2013, title={Communication: Certifying the potential energy landscape}, volume={138}, ISSN={["0021-9606"]}, DOI={10.1063/1.4803162}, abstractNote={It is highly desirable for numerical approximations to stationary points for a potential energy landscape to lie in the corresponding quadratic convergence basin. However, it is possible that an approximation may lie only in the linear convergence basin, or even in a chaotic region, and hence not converge to the actual stationary point when further optimization is attempted. Proving that a numerical approximation will quadratically converge to the associated stationary point is termed certification. Here, we apply Smale's α-theory to stationary points, providing a certification serving as a mathematical proof that the numerical approximation does indeed correspond to an actual stationary point, independent of the precision employed. As a practical example, employing recently developed certification algorithms, we show how the α-theory can be used to certify all the known minima and transition states of Lennard-Jones LJN atomic clusters for N = 7, …, 14.}, number={17}, journal={JOURNAL OF CHEMICAL PHYSICS}, author={Mehta, Dhagash and Hauenstein, Jonathan D. and Wales, David J.}, year={2013}, month={May} }
@article{hauenstein_ikenmeyer_landsberg_2013, title={Equations for Lower Bounds on Border Rank}, volume={22}, ISSN={["1944-950X"]}, DOI={10.1080/10586458.2013.825892}, abstractNote={We present new methods for determining polynomials in the ideal of the variety of bilinear maps of border rank at most r. We apply these methods to several cases including the case r=6 in the space of bilinear maps . This space of bilinear maps includes the matrix multiplication operator M 2 for 2×2 matrices. We show that these newly obtained polynomials do not vanish on the matrix multiplication operator M 2, which gives a new proof that the border rank of the multiplication of 2×2 matrices is seven. Other examples are considered along with an explanation of how to implement the methods.}, number={4}, journal={EXPERIMENTAL MATHEMATICS}, author={Hauenstein, Jonathan D. and Ikenmeyer, Christian and Landsberg, J. M.}, year={2013}, month={Oct}, pages={372–383} }
@article{hauenstein_wampler_2013, title={Isosingular Sets and Deflation}, volume={13}, ISSN={["1615-3383"]}, DOI={10.1007/s10208-013-9147-y}, abstractNote={This article introduces the concept of isosingular sets, which are irreducible algebraic subsets of the set of solutions to a system of polynomial equations constructed by taking the closure of points with a common singularity structure. The definition of these sets depends on deflation, a procedure that uses differentiation to regularize solutions. A weak form of deflation has proven useful in regularizing algebraic sets, making them amenable to treatment by the algorithms of numerical algebraic geometry. We introduce a strong form of deflation and define deflation sequences, which, in a different context, are the sequences arising in Thom–Boardman singularity theory. We then define isosingular sets in terms of deflation sequences. We also define the isosingular local dimension and examine the properties of isosingular sets. While isosingular sets are of theoretical interest as constructs for describing singularity structures of algebraic sets, they also expand the kinds of algebraic set that can be investigated with methods from numerical algebraic geometry.}, number={3}, journal={FOUNDATIONS OF COMPUTATIONAL MATHEMATICS}, author={Hauenstein, Jonathan D. and Wampler, Charles W.}, year={2013}, month={Jun}, pages={371–403} }
@article{hauenstein_sommese_2013, title={Membership tests for images of algebraic sets by linear projections}, volume={219}, ISSN={["1873-5649"]}, DOI={10.1016/j.amc.2012.12.060}, abstractNote={Given a witness set for an irreducible variety V and a linear map π, we describe membership tests for both the constructible algebraic set π(V) and the algebraic set π(V)¯. We also provide applications and examples of these new tests including computing the codimension one components of π(V)¯⧹π(V)¯. Additionally, we also describe computing the geometric genus of a curve section of an irreducible component of the solution set of a polynomial system and a test for deciding whether a plane quartic curve is a Lüroth quartic.}, number={12}, journal={APPLIED MATHEMATICS AND COMPUTATION}, author={Hauenstein, Jonathan D. and Sommese, Andrew J.}, year={2013}, month={Feb}, pages={6809–6818} }
@article{hauenstein_he_mehta_2013, title={Numerical elimination and moduli space of vacua}, ISSN={["1029-8479"]}, DOI={10.1007/jhep09(2013)083}, abstractNote={We propose a new computational method to understand the vacuum moduli space of (supersymmetric) field theories. By combining numerical algebraic geometry (NAG) and elimination theory, we develop a powerful, efficient, and parallelizable algorithm to extract important information such as the dimension, branch structure, Hilbert series and subsequent operator counting, as well as variation according to coupling constants and mass parameters. We illustrate this method on a host of examples from gauge theory, string theory, and algebraic geometry.}, number={9}, journal={JOURNAL OF HIGH ENERGY PHYSICS}, author={Hauenstein, Jonathan and He, Yang-Hui and Mehta, Dhagash}, year={2013}, month={Sep} }
@article{bates_hauenstein_mccoy_peterson_sommese_2013, title={Recovering Exact Results from Inexact Numerical Data in Algebraic Geometry}, volume={22}, ISSN={["1944-950X"]}, DOI={10.1080/10586458.2013.737640}, abstractNote={Let be a set of homogeneous polynomials. Let Z denote the complex projective algebraic set determined by the zero locus of . Numerical-continuation-based methods can be used to produce arbitrary-precision numerical approximations of generic points on each irreducible component of Z. Consider the ideal and the prime decomposition over . This article illustrates how lattice-reduction algorithms may take as input numerically approximated generic points on Z and effectively extract exact elements for each Pi . A collection of examples serves to illustrate the approach and indicate some of the application areas for which this technique is valuable.}, number={1}, journal={EXPERIMENTAL MATHEMATICS}, author={Bates, Daniel J. and Hauenstein, Jonathan D. and McCoy, Timothy M. and Peterson, Chris and Sommese, Andrew J.}, year={2013}, month={Jan}, pages={38–50} }
@article{hao_hauenstein_shu_sommese_xu_zhang_2013, title={A homotopy method based on WENO schemes for solving steady state problems of hyperbolic conservation laws}, volume={250}, DOI={10.21236/ada568170}, journal={Journal of Computational Physics}, author={Hao, W. R. and Hauenstein, J. D. and Shu, C. W. and Sommese, A. J. and Xu, Z. L. and Zhang, Y. T.}, year={2013}, pages={332–346} }
@article{besana_di rocco_hauenstein_sommese_wampler_2013, title={Cell decomposition of almost smooth real algebraic surfaces}, volume={63}, ISSN={["1572-9265"]}, DOI={10.1007/s11075-012-9646-y}, number={4}, journal={NUMERICAL ALGORITHMS}, author={Besana, Gian Mario and Di Rocco, Sandra and Hauenstein, Jonathan D. and Sommese, Andrew J. and Wampler, Charles W.}, year={2013}, month={Aug}, pages={645–678} }
@article{hauenstein_2013, title={Numerically Computing Real Points on Algebraic Sets}, volume={125}, ISSN={["1572-9036"]}, DOI={10.1007/s10440-012-9782-3}, abstractNote={Given a polynomial system f, a fundamental question is to determine if f has real roots. Many algorithms involving the use of infinitesimal deformations have been proposed to answer this question. In this article, we transform an approach of Rouillier, Roy, and Safey El Din, which is based on a classical optimization approach of Seidenberg, to develop a homotopy based approach for computing at least one point on each connected component of a real algebraic set. Examples are presented demonstrating the effectiveness of this parallelizable homotopy based approach.}, number={1}, journal={ACTA APPLICANDAE MATHEMATICAE}, author={Hauenstein, Jonathan D.}, year={2013}, month={Jun}, pages={105–119} }