Papers by Mark van Hoeij

Recent additions are marked in yellow.

Algebraic Curves

An algorithm for computing an integral basis in an algebraic function field. J. Symb. Comput., 18, 353-363 (1994).

Computing parametrizations of rational algebraic curves. ISSAC '94 Proceedings, 187-190 (1994).

An algorithm for computing the Weierstrass normal form. ISSAC '95 Proceedings, 90-95 (1995).
      See also: same problem for the hyperelliptic case (4 pages, not submitted).

Rational Parametrizations of Algebraic Curves using a Canonical Divisor. J. Symb. Comput., 23, 209-227 (1997).

Computing Riemann matrices of algebraic curves. Joint work with Bernard Deconinck. PhysicaD, 152, 28-46 (2001).

Computing Riemann Theta Functions. Bernard Deconinck, Matthias Heil, Alexander Bobenko, Mark van Hoeij and Markus Schmies. Math. Comp. 73, 1417-1442 (2004).

A Reduction Algorithm for Algebraic Function Fields, joint with Andy Novocin. Not published. (2005).

Solving conics over function fields. Joint work with John Cremona. Journal de Theorie des Nombres de Bordeaux, 18, p. 595-606 (2006).

Low Degree Places on the Modular Curve X1(N) (5 pages, not submitted). Finding elliptic curves with torsion points of order N, defined over number fields of lower degree than the Q-gonality of X1(N). (2012-2014). For more on this topic see Sporadic Cubic Torsion below.

Computing Puiseux Expansions at Cusps of the Modular Curve X0(N), (4 pages, not submitted). Implementation (the files PuiseuxX0N and PuiseuxX0N_Examples) (2013).

Gonality of the modular curve X1(N), joint work with Maarten Derickx, J. Algebra, 417, p. 52-71 (2014).
      See also: accompanying files, and proof of Conjecture 1.

Computing Galois representations and equations for modular curves X_H(l). With Maarten Derickx and Jinxiang Zeng. (2013).

Computing an Integral Basis for an Algebraic Function Field (with Mike Stillman). Slides for a conference at Georgia Tech, April 11, 2015.

Computing an order complete basis for M^infty(N) and Applications, joint with Cristian-Silviu Radu (2019). implementation.

A Divisor Formula and a Bound on the Q-gonality of the Modular Curve X1(N), joint with Hanson Smith (2020). implementation.

Sporadic Cubic Torsion, Maarten Derickx, Anastassia Etropolski, Mark van Hoeij, Jackson S. Morrow, David Zureick-Brown (2020). accompanying files.

Appendix in: Towards a Classification of Isolated j-invariants. Paper by A. Bourdon et. al, appendix with M. Derickx. Math. Comp. 94 (2025), 447-473.

Difference Equations

Rational Solutions of Linear Difference Equations. ISSAC '98 Proceedings, 120-123 (1998).

Finite Singularities and Hypergeometric Solutions of Linear Recurrence Equations. J. Pure Appl. Algebra, 139, 109-131 (1999).

Desingularization of linear difference operators with polynomial coefficients. Joint work with S.A. Abramov. ISSAC '99 Proceedings, 269-275 (1999) dvi file.
      Related results can be found in the next paper:

Set of Poles of Solutions of Linear Difference Equations with Polynomial Coefficients. Joint work with S.A. Abramov. Computational Mathematics and Mathematical Physics, Vol. 43, No. 1, p. 57--62 (2003) dvi file.

A conjecture in the problem of rational definite summation. 4 pages (not submitted). See also the preliminary SumRat implementation. (2002).

Computing Hypergeometric Solutions of Linear Recurrence Equations. Joint work with Thomas Cluzeau. AAECC, Vol. 17, 83-115 (2006).

Apparent Singularities of Linear Difference Equations with Polynomial Coefficients. Joint work with S.A. Abramov and M. A. Barkatou. AAECC, Vol. 17, 117-133 (2006).

Slides on "Factorization and hypergeometric solutions of linear recurrence systems" at Manuel Bronstein's conference (July 2006).

Subanalytic Solutions of Linear Difference Equations and Multidimensional Hypergeometric Sequences, joint work with S. Abramov, M. Barkatou, and M. Petkovsek. Journal of Symbolic Computation 46, p. 1205-1228, (2011).

Liouvillian Solutions of Irreducible Linear Difference Equations, ISSAC'2009 Proceedings, p. 87-93, joint work with YongJae Cha. slides, Distinguished Student Author Award

Liouvillian Solutions of Irreducible Second Order Linear Difference Equations, joint work with Giles Levy, ISSAC'2010 Proceedings, p. 297-302.
      More can be be found in the Ph.D thesis of Giles Levy. Here are the slides of the talk.

Solving Recurrence Relations using Local Invariants, joint work with YongJae Cha and Giles Levy, ISSAC'2010 Proceedings, p. 303-310.
      More can be found in the Ph.D thesis of Yongjae Cha.

Abstract for: Software demonstration at ISSAC'2010, presented by Giles Levy pdf files.

Rational elements of the tensor product of solutions of difference operators, joint with YongJae Cha, In: Proceedings of the Tenth Asian Symposium on Computer Mathematics (2012).

Factoring Linear Recurrence Operators, presentation Brasov Romania, May 2019.

Desingularization and p-Curvature of Recurrence Operators, Yi Zhou and Mark van Hoeij, ISSAC'2022, slides.

A Family of Denominator Bounds for First Order Linear Recurrence Systems, joint with Moulay Barkatou and Johannes Middeke. Not published. Useful combined with the next paper:

Hypergeometric Solutions of Linear Difference Systems, Moulay Barkatou, Mark van Hoeij, Johannes Middeke, Yi Zhou. Accepted to Journal of Symbolic Computation. Implementation

Submodule approach to creative telescoping, Mark van Hoeij. Journal of Symbolic Computation, Vol. 126, 2025. Accompanying computer examples

Solving Order 3 Difference Equations Heba Bou KaedBey, Mark van Hoeij, Man Cheung Tsui, submitted to JSC 2024.

Solving Third Order Linear Difference Equations in Terms of Second Order Equations, Heba Bou KaedBey, Mark van Hoeij, Man Cheung Tsui, published in the ISSAC'2024 proceedings.

Differential Equations

Formal Solutions and Factorization of Differential Operators with Power Series Coefficients. J. Symb. Comput., 24, 1-30 (1997).

Factorization of Differential Operators with Rational Functions Coefficients. J. Symb. Comput., 24, 537-561 (1997).

An algorithm for computing invariants of differential Galois groups. Joint work with J.A. Weil. J. Pure Appl. Algebra, 117&118, 353-379 (1997).

Rational Solutions of the Mixed Differential Equation and its Application to Factorization of Differential Operators. ISSAC '96 Proceedings, 219-225 (1996).

Note: the 4 papers above are also included in my Ph.D thesis postscript file, pdf file (1996, written in English with a short summary in Dutch). Here is the list of errata. It is a custom in the Netherlands to include a number of statements, ``stellingen'', with the thesis (some of these are in English, some in Dutch). Here is my masters thesis (1992, an integral basis algorithm, written in Dutch).

A method for the Integration of Solutions of Ore Equations. Joint work with S.A. Abramov. ISSAC '97 Proceedings, 172-175 (1997).
      A newer version of this paper which has appeared as Integration of Solutions of Linear Functional Equations in: Integral Transforms and Special Functions, 1999, Vol.8, No 1-2, pp. 3-12.

Liouvillian solutions of linear differential equations of order three and higher. Joint work with Jean-Francois Ragot, Felix Ulmer and Jacques-Arthur Weil. J. Symb. Comput., 28, 589-609 (1999).

The Minimum Polynomial of an Algebraic Solution of Abel's problem. Preprint FSU00-02.

Decomposing a 4'th order linear differential equation as a symmetric product. Banach Center Publications, 58, 89-96, (2002). abstract.

Closed Form Solutions of Linear Odes having Elliptic Function Coefficients. Joint work with Reinhold Burger and George Labahn. ISSAC'04 Proceedings, 58-64, (2004).

Descent for differential modules and skew fields. Joint work with Marius van der Put. Journal of Algebra, 296, pp. 18-55 (2006).

A Modular Algorithm to Compute the Exponential Solutions of a Linear Differential Operator. Joint work with Thomas Cluzeau. J. Symb. Comput., 38, 1043-1076 (2004). dvi file.

Solving Second Order Linear Differential Equations with Klein's Theorem. Joint work with Jacques-Arthur Weil. ISSAC'05 Proceedings, 340-347, (2005). draft implementation, slides of the talk.

The Fourth-Order Type Linear Ordinary Differential Equations. Joint work with W. N. Everitt and D. J. Smith. Preprint (not submitted for publication) (2006).

Solving Third Order Linear Differential Equations in Terms of Second Order Equations, ISSAC'07 Proceedings, 355-360, (2007). slides of the ISSAC'2007 talk.
      A longer version of this talk was presented at the DART workshop (DART slides).

Solving Differential Equations in Terms of Bessel Functions, Joint work with R. Debeerst and W. Koepf. ISSAC'08 Proceedings, 39-46, (2008).

Finding all Bessel type solutions for Linear Differential Equations with Rational Function Coefficients, joint with Quan Yuan, ISSAC'2010 Proceedings, p 37-44, (2010).

Explicit formula for the generating series of diagonal 3D rook paths, Alin Bostan, Frederic Chyzak, Mark van Hoeij, Lucien Pech. Seminaire Lotharingien de Combinatoire, B66a (2011).

Solving Linear Differential Equations in Terms of Special Functions, Project Description for NSF grant 1017880 (Sept. 2010 - 2013).

Linear Differential Equations with a Convergent Integer Series Solution, Project Description for NSF grant 1319547 (Sept. 2013 - 2016).

A-Hypergeometric Solutions of Linear Differential Equations, Project Description for NSF grant 1618657 (Sept. 2016 - 2019).

2-descent for Second Order Linear Differential Equations, joint work with Tingting Fang, ISSAC'2011 Proceedings, p. 107-114 (2011).

Second Order Differential Equations with Hypergeometric Solutions of Degree Three, joint with Vijay Jung Kunwar, ISSAC'2013.

Differential Equations with a Convergent Integer power Series Solution slides at Algebra, Geometry and Computation conference, Eindhoven, 2014.

Belyi functions for hyperbolic hypergeometric-to-Heun transformations, J. Algebra 2015 (DOI: 10.1016/j.jalgebra.2015.06.008), joint with Raimundas Vidunas.

Computing Hypergeometric Solutions of Second Order Linear Differential Equations using Quotients of Formal Solutions, joint with Erdal Imamoglu, implementation, ISSAC'2015 proceedings, 2015.

Hypergeometric Expressions for Generating Functions of Walks with Small Steps in the Quarter Plane, joint with A. Bostan, F. Chyzak, M. Kauers, L. Pech. European Journal of Combinatorics, Vol 61, p. 242-275 (2017).

Classifying (near)-Belyi maps with Five Exceptional Points. Website and paper, joint with Vijay Kunwar, Indag. Math. 30 p.136-156 (2019).

Computing Hypergeometric Solutions of Second Order Linear Differential Equations using Quotients of Formal Solutions and Integral Bases, joint with Erdal Imamoglu, implementation, Journal of Symbolic Computation, 83, p. 254-271 (2017).

Reduction-Based Creative Telescoping for Fuchsian D-finite Functions, Shaoshi Chen, Mark van Hoeij, Manuel Kauers, Christoph Koutschan, Journal of Symbolic Computation, vol 85 p. 108-127 (2018).

Closed Form Solutions. Slides of invited talk at ISSAC'2017, abstract. Also presented at RADCOR'2017.

A hyperelliptic saga on a generating function of the squares of Legendre polynomials. Joint with Duco van Straten and Wadim Zudilin (2023). Accepted, Journal of Experimental Mathematics.

The modularity of Z.-W. Sun's conjectural formulas for 1/pi. Mark van Hoeij, Wei-Lun Tsai and Dongxi Ye. accepted Proc. AMS (2024).

Physics

The Ising model: from elliptic curves to modular forms and Calabi-Yau equations A. Bostan, S. Boukraa, S. Hassani, M. van Hoeij, J.-M. Maillard, J-A. Weil and N. Zenine. J. Phys. A: Math. Theor. 44 045204 (2011).

Diagonal Ising susceptibility: elliptic integrals, modular forms and Calabi-Yau equations M. Assis, S. Boukraa, S. Hassani, M. van Hoeij, J-M. Maillard, B.M. McCoy. J. Phys. A: Math. Theor. 45 075205 (2012) (highlights 2012)

On Hirzebruch invariants of elliptic fibrations, joint with James Fullwood. Proceedings of Symposia in Pure Mathematics, Vol. 85, p. 355-366, (2012). implementation

On stringy invariants of GUT vacua, joint with James Fullwood, Communications for Number Theory in Physics, Vol 07 No. 4, p.551-579 (2013). implementation

The perimeter generating functions of three-choice, imperfect, and 1-punctured staircase polygons M. Assis, M. van Hoeij, J-M. Maillard. J. of Physics A: Math. Theor. 49 214002 (2016).

Iterated Elliptic and Hypergeometric Integrals for Feynman Diagrams, J. Ablinger, J. Bluemlein, A. De Freitas, M. van Hoeij, E. Imamoglu, C.G. Raab, C.-S. Radu, C. Schneider Journal of Mathematical Physics 59, 062305 (2018).

Iterative and Iterative-Noniterative Integral Solutions in 3-Loop Massive QCD Calculations, J. Ablinger, A. Behring, J. Bluemlein, A. De Freitas, E. Imamoglu, M. van Hoeij, A. von Manteuffel, C.G. Raab, C.-S. Radu, C. Schneider. In Proceedings of RADCOR 2017, p. 1-13 (2017).

The rho parameter at three loops and elliptic integrals J. Bluemlein, A. De Freitas, M. van Hoeij, E. Imamoglu, P. Marquard, C. Schneider (2018).

Stringy Hirzebruch classes of Weierstrass fibrations, joint with James Fullwood. Accompanying implementation+data (also on github). Communications in Number Theory and Physics, Vol. 14, No. 3 (2020), pp. 453-485.

Polynomials / Number Theory

Factoring polynomials and the knapsack problem. Journal of Number Theory, 95, 167-189, (2002). ps file. Additional information can be found in the source code of the implementation and in this summary (proceedings of CaLC'2001, Lect. Notes in Comp. Science, 2146, p. 45-50). Some sample inputs, many of which could not be factored before, e.g. M12.

Towards Factoring Bivariate Approximate Polynomials. Rob Corless, Mark Giesbrecht, Mark van Hoeij, Ilias Kotsireas and Stephen Watt, ISSAC '01 proceedings, 85-92 (2001) ps file.

A Modular GCD algorithm over Number Fields presented with Multiple Extensions. Joint work with Michael Monagan. ISSAC'02 proceedings, 109-116, (2002).
      See also: extended version.

Algorithms for Polynomial GCD Computation over Algebraic Function Fields. Some comments. Joint work with Michael Monagan, ISSAC'04 proceedings, 297-304, (2004). ps file.

Factoring polynomials over global fields. Joint work with K. Belabas, J. Klueners, and A. Steel. Preprint arXiv:math/0409510v1, Journal de Theorie des Nombres de Bordeaux, 21, 15-39, (2009).

Generating Subfields, Joint work with J. Klueners. Abstract of a talk at Oberwolfach, July 22, 2005.

Approximate Bivariate Factorization, a Geometric Viewpoint, Joint work with A. Galligo, SNC'2007 Proceedings, p. 1-10 (2007).

Complexity of factoring polynomials with rational number coefficients, slides at a plenary talk at Journees Arithmetique, July 2-6, Edinburgh.
      See also: preprint, the poster at ISSAC'2007, the poster at ECCAD'2008, the Ph.D thesis of Andrew Novocin, and for more recent details see the slides of the ISSAC'2013 tutorial below.

Gradual sub-lattice reduction and a new complexity for factoring polynomials, joint with Andrew Novocin, LATIN 2010, p. 539-553 (2010).
      Also in: Algorithmica 63(3): 616-633 (2012).

Appendix in the paper: The Complete Generating Function for Gessel Walks is Algebraic by Alin Bostan and Manuel Kauers, Proc. Amer. Math. Soc. 138 (2010), p. 3063-3078.

Isomorphisms of Algebraic Number Fields, joint work with Vivek Pal, JNTB Vol 24, Number 2, p. 293-305 (2012).

Generating Subfields, joint work with Juergen Klueners and Andrew Novocin, ISSAC'2011 Proceedings, p. 345-352 (2011). Slides of the talk and a Maple example.
      See also: extended version which appeared in Journal of Symbolic Computation, Vol. 52, p. 17-34, (2013).

Practical Polynomial Factoring in Polynomial Time, joint work with William Hart and Andrew Novocin, ISSAC'2011 Proceedings, p. 163-170 (2011).

The complexity of factoring univariate polynomials over the rationals, Tutorial Abstract and slides, ISSAC'2013.

Groebner basis in Boolean rings is not polynomial-space, three pages (not submitted), 2015.

Solving problems with the LLL algorithm. Slides at AMS Invited Address giving intro to LLL and its applications, 2015.

The Complexity of Computing all Subfields of an Algebraic Number Field, joint with Jonas Szutkoski (2016). implementation and slides. JSC 2018.

Functional Decomposition using Principal Subfields, L. E. Allem, J. Capaverde, M. van Hoeij, J. Szutkoski. Examples and timings. Implementation which is now part of Magma. The slides of the presentation by Jonas at ISSAC 2017.

Ph.D theses of my students:

Andy Novocin, Factoring Univariate Polynomials over the Rationals (April 9, 2008).

Giles Levy, Solutions of Second Order Recurrence Relations (December 4, 2009).

Yongjae Cha, Closed Form Solutions of Linear Difference Equations (December 7, 2010).

Quan Yuan, Finding all Bessel type solutions for Linear Differential Equations with Rational Function Coefficients, see also slides (March 19, 2012). The 2018 PhD thesis of Merlin Mouafo Wouodjie gives a generalization to order 3.

Randy Heaton (I was co-director, with Amod Agashe as director), Algorithms for Computing Congruences Between Modular Forms (June 28, 2012).

Tingting Fang, Solving Linear Differential Equations in Terms of Hypergeometric Functions by 2-Descent, see also slides. (October 16, 2012).

Vijay Kunwar, Hypergeometric Solutions of Linear Differential Equations with Rational Function Coefficients, see also slides. (June 11, 2014).

Xin Li (I became co-director after the tragic passing of Xin's Ph.D advisor, Mika Seppala). Myrberg's numerical uniformatization (April 8, 2015).

Erdal Imamoglu, Algorithms for Solving Linear Differential Equations with Rational Function Coefficients, see also slides (May 25, 2017).

Wen Xu, Third Order A-hypergeometric Functions, see also slides (November 16, 2017).

Jonas Szutkoski, Computing Subfields (December, 2017). Directors: V. Trevisan and L. E. Allem (Jonas visited me at FSU in 2013 and a full year in 2015 as part of his PhD program). Implementations (decomposition is now part of Magma)

Shayea Aldossari, Algorithms for Simplifying Differential Equations (July 6, 2020).

Ben Prather (I became director in 2021). Octonions - Hilbert Spaces, Fibrations and Analysis (October 28, 2021).

Yi Zhou, Algorithms for Factoring Linear Recurrence Relations (April 8, 2022).