@article{chen_feng_li_singer_watt_2024, title={Telescopers for differential forms with one parameter}, volume={30}, ISSN={["1420-9020"]}, DOI={10.1007/s00029-024-00926-6}, number={3}, journal={SELECTA MATHEMATICA-NEW SERIES}, author={Chen, Shaoshi and Feng, Ruyong and Li, Ziming and Singer, Michael F. and Watt, Stephen M.}, year={2024}, month={Jul} }
 @book{raab_singer_2022, place={Cham, Switzerland}, series={Texts & Monographs in Symbolic Computation}, title={Integration in Finite Terms: Fundamental Sources}, ISBN={9783030987664 9783030987671}, ISSN={0943-853X 2197-8409}, url={http://dx.doi.org/10.1007/978-3-030-98767-1}, DOI={10.1007/978-3-030-98767-1}, publisher={Springer International Publishing}, year={2022}, collection={Texts & Monographs in Symbolic Computation} }
 @article{hardouin_singer_2021, title={On differentially algebraic generating series for walks in the quarter plane}, volume={27}, ISSN={["1420-9020"]}, DOI={10.1007/s00029-021-00703-9}, abstractNote={We refine necessary and sufficient conditions for the generating series of a weighted model of a quarter plane walk to be differentially algebraic. In addition, we give algorithms based on the theory of Mordell–Weil lattices, that, for each weighted model, yield polynomial conditions on the weights determining this property of the associated generating series.}, number={5}, journal={SELECTA MATHEMATICA-NEW SERIES}, author={Hardouin, Charlotte and Singer, Michael F.}, year={2021}, month={Nov} }
 @article{dreyfus_hardouin_roques_singer_2021, title={On the Kernel Curves Associated with Walks in the Quarter Plane}, volume={373}, ISBN={["978-3-030-84303-8"]}, ISSN={["2194-1009"]}, DOI={10.1007/978-3-030-84304-5_3}, abstractNote={The kernel method is an essential tool for the study of generating series of walks in the quarter plane. This method involves equating to zero a certain polynomial - the kernel polynomial - and using properties of the curve - the kernel curve - this defines. In the present paper, we investigate the basic properties of the kernel curve (irreducibility, singularities, genus, uniformization, etc.).}, journal={TRANSCENDENCE IN ALGEBRA, COMBINATORICS, GEOMETRY AND NUMBER THEORY, TRANS19}, author={Dreyfus, Thomas and Hardouin, Charlotte and Roques, Julien and Singer, Michael F.}, year={2021}, pages={61–89} }
 @article{hubert_singer_2021, title={Sparse Interpolation in Terms of Multivariate Chebyshev Polynomials}, ISSN={["1615-3383"]}, DOI={10.1007/s10208-021-09535-7}, abstractNote={Sparse interpolation refers to the exact recovery of a function as a short linear combination of basis functions from a limited number of evaluations. For multivariate functions, the case of the monomial basis is well studied, as is now the basis of exponential functions. Beyond the multivariate Chebyshev polynomial obtained as tensor products of univariate Chebyshev polynomials, the theory of root systems allows to define a variety of generalized multivariate Chebyshev polynomials that have connections to topics such as Fourier analysis and representations of Lie algebras. We present a deterministic algorithm to recover a function that is the linear combination of at most r such polynomials from the knowledge of r and an explicitly bounded number of evaluations of this function.}, journal={FOUNDATIONS OF COMPUTATIONAL MATHEMATICS}, author={Hubert, Evelyne and Singer, Michael F.}, year={2021}, month={Sep} }
 @article{jimenez-pastor_pillwein_singer_2020, title={Some structural results on D-n-finite functions}, volume={117}, ISSN={["1090-2074"]}, DOI={10.1016/j.aam.2020.102027}, abstractNote={D-finite (or holonomic) functions satisfy linear differential equations with polynomial coefficients. They form a large class of functions that appear in many applications in Mathematics or Physics. It is well-known that these functions are closed under certain operations and these closure properties can be executed algorithmically. Recently, the notion of D-finite functions has been generalized to differentially definable or Dn-finite functions. Also these functions are closed under operations such as forming (anti)derivative, addition or multiplication and, again, these can be implemented. In this paper we investigate how Dn-finite functions behave under composition and how they are related to algebraic and differentially algebraic functions.}, journal={ADVANCES IN APPLIED MATHEMATICS}, author={Jimenez-Pastor, Antonio and Pillwein, Veronika and Singer, Michael F.}, year={2020}, month={Jun} }
 @article{dreyfus_hardouin_roques_singer_2020, title={Walks in the quarter plane: Genus zero case}, volume={174}, ISSN={["1096-0899"]}, DOI={10.1016/j.jcta.2020.105251}, abstractNote={We use Galois theory of difference equations to study the nature of the generating series of (weighted) walks in the quarter plane with genus zero kernel curve. Using this approach, we prove that the generating series do not satisfy any nontrivial (possibly nonlinear) algebraic differential equation with rational coefficients.}, journal={JOURNAL OF COMBINATORIAL THEORY SERIES A}, author={Dreyfus, Thomas and Hardouin, Charlotte and Roques, Julien and Singer, Michael F.}, year={2020}, month={Aug} }
 @article{schafke_singer_2019, title={Consistent systems of linear differential and difference equations}, volume={21}, ISSN={["1435-9855"]}, DOI={10.4171/JEMS/891}, abstractNote={We consider systems of linear differential and difference equations \begin{eqnarray*} \partial Y(x) =A(x)Y(x), \sigma Y(x) =B(x)Y(x) \end{eqnarray*} with $\partial = \frac{d}{dx}$, $\sigma$ a shift operator $\sigma(x) = x+a$, $q$-dilation operator $\sigma(x) = qx$ or Mahler operator $\sigma(x) = x^p$ and systems of two linear difference equations \begin{eqnarray*} \sigma_1 Y(x) =A(x)Y(x), \sigma_2 Y(x) =B(x)Y(x) \end{eqnarray*} with $(\sigma_1,\sigma_2)$ a sufficiently independent pair of shift operators, pair of $q$-dilation operators or pair of Mahler operators. Here $A(x)$ and $B(x)$ are $n\times n$ matrices with rational function entries. Assuming a consistency hypothesis, we show that such system can be reduced to a system of a very simple form. Using this we characterize functions satisfying two linear scalar differential or difference equations with respect to these operators. We also indicate how these results have consequences both in the theory of automatic sets, leading to a new proof of Cobham's Theorem, and in the Galois theories of linear difference and differential equations, leading to hypertranscendence results.}, number={9}, journal={JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY}, author={Schafke, Reinhard and Singer, Michael F.}, year={2019}, pages={2751–2792} }
 @article{dreyfus_hardouin_roques_singer_2018, title={On the nature of the generating series of walks in the quarter plane}, volume={213}, ISSN={["1432-1297"]}, DOI={10.1007/s00222-018-0787-z}, abstractNote={In the present paper, we introduce a new approach, relying on the Galois theory of difference equations, to study the nature of the generating series of walks in the quarter plane. Using this approach, we are not only able to recover many of the recent results about these series, but also to go beyond them. For instance, we give for the first time hypertranscendency results, i.e., we prove that certain of these generating series do not satisfy any nontrivial nonlinear algebraic differential equation with rational function coefficients.}, number={1}, journal={INVENTIONES MATHEMATICAE}, author={Dreyfus, Thomas and Hardouin, Charlotte and Roques, Julien and Singer, Michael F.}, year={2018}, month={Jul}, pages={139–203} }
 @article{arreche_singer_2017, title={Galois groups for integrable and projectively integrable linear difference equations}, volume={480}, ISSN={["1090-266X"]}, DOI={10.1016/j.jalgebra.2017.02.032}, abstractNote={We consider first-order linear difference systems over C(x), with respect to a difference operator σ that is either a shift σ:x↦x+1, q-dilation σ:x↦qx with q∈C× not a root of unity, or Mahler operator σ:x↦xq with q∈Z≥2. Such a system is integrable if its solutions also satisfy a linear differential system; it is projectively integrable if it becomes integrable “after moding out by scalars.” We apply recent results of Schäfke and Singer to characterize which groups can occur as Galois groups of integrable or projectively integrable linear difference systems. In particular, such groups must be solvable. Finally, we give hypertranscendence criteria.}, journal={JOURNAL OF ALGEBRA}, author={Arreche, Carlos K. and Singer, Michael F.}, year={2017}, month={Jun}, pages={423–449} }
 @article{chen_kauers_singer_2016, title={Desingularization of Ore operators}, volume={74}, ISSN={["0747-7171"]}, DOI={10.1016/j.jsc.2015.11.001}, abstractNote={We show that Ore operators can be desingularized by calculating a least common left multiple with a random operator of appropriate order, thereby turning a heuristic used for many years in several computer algebra systems into an algorithm. Our result can be viewed as a generalization of a classical result about apparent singularities of linear differential equations.}, journal={JOURNAL OF SYMBOLIC COMPUTATION}, author={Chen, Shaoshi and Kauers, Manuel and Singer, Michael F.}, year={2016}, pages={617–626} }
 @article{minchenko_ovchinnikov_singer_2015, title={Reductive Linear Differential Algebraic Groups and the Galois Groups of Parameterized Linear Differential Equations}, volume={2015}, ISSN={["1687-0247"]}, DOI={10.1093/imrn/rnt344}, abstractNote={We develop the representation theory for reductive linear differential algebraic groups (LDAGs). In particular, we exhibit an explicit sharp upper bound for orders of derivatives in differential representations of reductive LDAGs, extending existing results, which were obtained for SL(2) in the case of just one derivation. As an application of the above bound, we develop an algorithm that tests whether the parameterized differential Galois group of a system of linear differential equations is reductive and, if it is, calculates it.}, number={7}, journal={INTERNATIONAL MATHEMATICS RESEARCH NOTICES}, author={Minchenko, Andrey and Ovchinnikov, Alexey and Singer, Michael F.}, year={2015}, pages={1733–1793} }
 @article{chen_singer_2014, title={On the summability of bivariate rational functions}, volume={409}, ISSN={["1090-266X"]}, DOI={10.1016/j.jalgebra.2014.03.023}, abstractNote={We present criteria for deciding whether a bivariate rational function in two variables can be written as a sum of two (q-)differences of bivariate rational functions. Using these criteria, we show how certain double sums can be evaluated, first, in terms of single sums and, finally, in terms of values of special functions.}, journal={JOURNAL OF ALGEBRA}, author={Chen, Shaoshi and Singer, Michael F.}, year={2014}, month={Jul}, pages={320–343} }
 @article{minchenko_ovchinnikov_singer_2014, title={UNIPOTENT DIFFERENTIAL ALGEBRAIC GROUPS AS PARAMETERIZED DIFFERENTIAL GALOIS GROUPS}, volume={13}, ISSN={["1475-3030"]}, DOI={10.1017/s1474748013000200}, abstractNote={Abstract}, number={4}, journal={JOURNAL OF THE INSTITUTE OF MATHEMATICS OF JUSSIEU}, author={Minchenko, Andrey and Ovchinnikov, Alexey and Singer, Michael F.}, year={2014}, month={Oct}, pages={671–700} }
 @article{singer_2013, title={Linear algebraic groups as parameterized Picard-Vessiot Galois groups}, volume={373}, ISSN={["1090-266X"]}, DOI={10.1016/j.jalgebra.2012.09.037}, abstractNote={We show that a linear algebraic group is the Galois group of a parameterized Picard–Vessiot extension of k(x), x′=1, for certain differential fields k, if and only if its identity component has no one-dimensional quotient as a linear algebraic group.}, journal={JOURNAL OF ALGEBRA}, author={Singer, Michael F.}, year={2013}, month={Jan}, pages={153–161} }
 @article{mitschi_singer_2013, title={PROJECTIVE ISOMONODROMY AND GALOIS GROUPS}, volume={141}, ISSN={["1088-6826"]}, DOI={10.1090/s0002-9939-2012-11499-6}, abstractNote={In this article we introduce the notion of projective isomonodromy, which is a special type of monodromy-evolving deformation of linear differential equations, based on the example of the Darboux-Halphen equation. We give an algebraic condition for a parameterized linear differential equation to be projectively isomonodromic, in terms of the derived group of its parameterized Picard-Vessiot group.}, number={2}, journal={PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY}, author={Mitschi, Claude and Singer, Michael F.}, year={2013}, month={Feb}, pages={605–617} }
 @article{mitschi_singer_2012, title={Monodromy groups of parameterized linear differential equations with regular singularities}, volume={44}, ISSN={["1469-2120"]}, DOI={10.1112/blms/bds021}, abstractNote={We study the notion of regular singularities for parameterized complex ordinary linear differential systems, prove an analogue of the Schlesinger theorem for systems with regular singularities and solve both a parameterized version of the weak Riemann–Hilbert Problem and a special case of the inverse problem in parameterized Picard–Vessiot theory.}, journal={BULLETIN OF THE LONDON MATHEMATICAL SOCIETY}, author={Mitschi, Claude and Singer, Michael F.}, year={2012}, month={Oct}, pages={913–930} }
 @article{chen_singer_2012, title={Residues and telescopers for bivariate rational functions}, volume={49}, ISSN={["1090-2074"]}, DOI={10.1016/j.aam.2012.04.003}, abstractNote={We give necessary and sufficient conditions for the existence of telescopers for rational functions of two variables in the continuous, discrete and q-discrete settings and characterize which operators can occur as telescopers. Using this latter characterization, we reprove results of Furstenberg and Zeilberger concerning diagonals of power series representing rational functions. The key concept behind these considerations is a generalization of the notion of residue in the continuous case to an analogous concept in the discrete and q-discrete cases.}, number={2}, journal={ADVANCES IN APPLIED MATHEMATICS}, author={Chen, Shaoshi and Singer, Michael F.}, year={2012}, month={Aug}, pages={111–133} }
 @article{cassidy_singer_2011, title={A Jordan-Holder Theorem for differential algebraic groups}, volume={328}, ISSN={["1090-266X"]}, DOI={10.1016/j.jalgebra.2010.08.019}, abstractNote={We show that a differential algebraic group can be filtered by a finite subnormal series of differential algebraic groups such that successive quotients are almost simple, that is have no normal subgroups of the same type. We give a uniqueness result, prove several properties of almost simple groups and, in the ordinary differential case, classify almost simple linear differential algebraic groups.}, number={1}, journal={JOURNAL OF ALGEBRA}, author={Cassidy, Phyllis J. and Singer, Michael F.}, year={2011}, month={Feb}, pages={190–217} }
 @article{feng_singer_wu_2010, title={An algorithm to compute Liouvillian solutions of prime order linear difference-differential equations}, volume={45}, DOI={10.1016/j.jsc.2009.09.002}, abstractNote={A normal form is given for integrable linear difference–differential equations { σ ( Y ) = A Y , δ ( Y ) = B Y } , which is irreducible over C ( x , t ) and solvable in terms of Liouvillian solutions. We refine this normal form and devise an algorithm to compute all Liouvillian solutions of such kinds of systems of prime order.}, number={3}, journal={Journal of Symbolic Computation}, author={Feng, R. Y. and Singer, M. F. and Wu, M.}, year={2010}, pages={306–323} }
 @article{feng_singer_wu_2010, title={Liouvillian solutions of linear difference-differential equations}, volume={45}, ISSN={["0747-7171"]}, DOI={10.1016/j.jsc.2009.09.001}, abstractNote={For a field k with an automorphism σ and a derivation δ, we introduce the notion of Liouvillian solutions of linear difference–differential systems {σ(Y)=AY,δ(Y)=BY} over k and characterize the existence of Liouvillian solutions in terms of the Galois group of the systems. In the forthcoming paper, we will propose an algorithm for deciding if linear difference–differential systems of prime order have Liouvillian solutions.}, number={3}, journal={JOURNAL OF SYMBOLIC COMPUTATION}, author={Feng, Ruyong and Singer, Michael F. and Wu, Min}, year={2010}, month={Mar}, pages={287–305} }
 @article{hardouin_singer_2008, title={Differential Galois theory of linear difference equations}, volume={342}, ISSN={["1432-1807"]}, DOI={10.1007/s00208-008-0238-z}, abstractNote={We present a Galois theory of difference equations designed to measure the differential dependencies among solutions of linear difference equations. With this we are able to reprove Hölder’s theorem that the Gamma function satisfies no polynomial differential equation and are able to give general results that imply, for example, that no differential relationship holds among solutions of certain classes of q-hypergeometric equations.}, number={2}, journal={MATHEMATISCHE ANNALEN}, author={Hardouin, Charlotte and Singer, Michael F.}, year={2008}, month={Oct}, pages={333–377} }
 @article{singer_2007, title={Model theory of partial differential fields: From commuting to noncommuting derivations}, volume={135}, ISSN={["1088-6826"]}, DOI={10.1090/S0002-9939-07-08653-4}, abstractNote={McGrail (2000) has shown the existence of a model completion for the universal theory of fields on which a finite number of commuting derivations act and, independently, Yaffe (2001) has shown the existence of a model completion for the univeral theory of fields on which a fixed Lie algebra acts as derivations. We show how to derive the second result from the first.}, number={6}, journal={PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY}, author={Singer, Michael F.}, year={2007}, pages={1929–1934} }
 @article{cook_mitschi_singer_2005, title={On the constructive inverse problem in differential Galois theory}, volume={33}, ISSN={["1532-4125"]}, DOI={10.1080/00927870500243304}, abstractNote={We give sufficient conditions for a differential equation to have a given semisimple group as its Galois group. For any group G with G 0 = G 1 · ··· · G r , where each G i is a simple group of type Aℓ, Cℓ, Dℓ, E6, or E7, we construct a differential equation over C(x) having Galois group G.}, number={10}, journal={COMMUNICATIONS IN ALGEBRA}, author={Cook, WJ and Mitschi, C and Singer, MF}, year={2005}, pages={3639–3665} }
 @article{cormier_singer_trager_ulmer_2002, title={Linear differential operators for polynomial equations}, volume={34}, ISSN={["0747-7171"]}, DOI={10.1006/jsco.2002.0564}, abstractNote={Given a squarefree polynomial P?k0x,y ], k0a number field, we construct a linear differential operator that allows one to calculate the genus of the complex curve defined by P= 0 (when P is absolutely irreducible), the absolute factorization of P over the algebraic closure of k0, and calculate information concerning the Galois group of P over ___ k0(x) as well as overk0 (x).}, number={5}, journal={JOURNAL OF SYMBOLIC COMPUTATION}, author={Cormier, O and Singer, MF and Trager, BM and Ulmer, F}, year={2002}, month={Nov}, pages={355–398} }
 @article{singer_2001, title={Effective methods in algebraic geometry}, volume={164}, ISSN={["0022-4049"]}, DOI={10.1016/S0022-4049(00)00141-9}, number={1-2}, journal={JOURNAL OF PURE AND APPLIED ALGEBRA}, author={Singer, MF}, year={2001}, month={Oct}, pages={1–2} }
 @article{berman_singer_1999, title={Calculating the Galois group of L-1(L-2(y))=0, L-1,L-2 completely reducible operators}, volume={139}, ISSN={["0022-4049"]}, DOI={10.1016/S0022-4049(99)00003-1}, abstractNote={In Calculating Galois groups of completely reducible linear operators, Compoint and Singer describe a decision procedure that computes the Galois group of a completely reducible linear differential operator with rational or algebraic function coefficients (i.e., a linear differential operator that is the least common left multiple of irreducible operators or, equivalently, one whose Galois group is a reductive group). At present, it is unknown how to calculate the Galois group of a general operator. In this paper, we push beyond the completely reducible case by showing how to compute the Galois group of an operator of the form L1∘L2 where L1 and L2 are completely reducible and have rational function coefficients. We begin by showing how to compute the Galois group of an equation of the form L(y)=b with L completely reducible. This corresponds to the case of L1∘L2 where L1=D−b′/b. We then show how one can reduce the general case to the above case and give several examples.}, number={1-3}, journal={JOURNAL OF PURE AND APPLIED ALGEBRA}, author={Berman, PH and Singer, MF}, year={1999}, month={Jun}, pages={3–23} }
 @article{compoint_singer_1999, title={Computing Galois groups of completely reducible differential equations}, volume={28}, ISSN={["0747-7171"]}, DOI={10.1006/jsco.1999.0311}, abstractNote={We give an algorithm to calculate a presentation of the Picard?Vessiot extension associated to a completely reducible linear differential equation (i.e. an equation whose Galois group is reductive). Using this, we show how to compute the Galois group of such an equation as well as properties of the Galois groups of general equations.}, number={4-5}, journal={JOURNAL OF SYMBOLIC COMPUTATION}, author={Compoint, E and Singer, MF}, year={1999}, pages={473–494} }
 @article{hendriks_singer_1999, title={Solving difference equations in finite terms}, volume={27}, ISSN={["0747-7171"]}, DOI={10.1006/jsco.1998.0251}, abstractNote={We define the notion of a Liouvillian sequence and show that the solution space of a difference equation with rational function coefficients has a basis of Liouvillian sequences iff the Galois group of the equation is solvable. Using this we give a procedure to determine the Liouvillian solutions of such a difference equation.}, number={3}, journal={JOURNAL OF SYMBOLIC COMPUTATION}, author={Hendriks, PA and Singer, MF}, year={1999}, month={Mar}, pages={239–259} }
 @article{singer_ulmer_1997, title={Linear differential equations and products of linear forms}, volume={117}, ISSN={["1873-1376"]}, DOI={10.1016/S0022-4049(97)00027-3}, abstractNote={We show that liouvillian solutions of an n th-order linear differential equation L ( y ) = 0 are related to semi-invariant forms of the differential Galois group of L ( y ) = 0 which factor into linear forms. The logarithmic derivative of such a form F , evaluated in the solutions of L ( y ) = 0, is the first coefficient of a polynomial P ( u ) whose zeros are logarithmic derivatives of solutions of L ( y ) = 0. Together with the Brill equations, this characterization allows one to efficiently test if a semi-invariant corresponds to such a coefficient and to compute the other coefficients of P ( u ) via a factorization of the form F .}, journal={JOURNAL OF PURE AND APPLIED ALGEBRA}, author={Singer, MF and Ulmer, F}, year={1997}, month={May}, pages={549–563} }