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Mark van Hoeij

Publications and source records attributed to Mark van Hoeij.

At least 19 recordsLinked to original sources
arXiv2025

Hypergeometric Solutions of Linear Difference Systems

We extend Petkovšek's algorithm for computing hypergeometric solutions of scalar difference equations to the case of difference systems $τ(Y) = M Y$, with $M \in {\rm GL}_n(C(x))$, where $τ$ is the shift operator. Hypergeometric solutions are solutions of the form $γP$ where $P \in C(x)^n$ and $γ$ is a hypergeometric term over $C(x)$, i.e. ${τ(γ)}/γ \in C(x)$. Our contributions concern efficient computation of a set of candidates for ${τ(γ)}/γ$ which we write as $λ= c\frac{A}{B}$ with monic $A, B \in C[x]$, $c \in C^*$. Factors of the denominators of $M^{-1}$ and $M$ give candidates for $A$ and $B$, while another algorithm is needed for $c$. We use the super-reduction algorithm to compute candidates for $c$, as well as other ingredients to reduce the list of candidates for $A/B$. To further reduce the number of candidates $A/B$, we bound the so-called type of $A/B$ by bounding local types. Our algorithm has been implemented in Maple and experiments show that our implementation can handle systems of high dimension, which is useful for factoring operators.

cs.SC↗
arXiv2025

Solving Order 3 Difference Equations

We classify order $3$ linear difference operators over $\mathbb{C}(x)$ that are solvable in terms of lower order difference operators. To prove this result, we introduce the notion of absolute irreducibility for difference modules, and classify (for arbitrary order) modules that are irreducible but not absolutely irreducible.

math.RA↗
arXiv2024

The Modularity of Z.-W. Sun's Conjectural Formulas for $\frac{1}π$

In this work, we establish modular parameterizations for two general formulas for $\frac{1}π$ that subsume conjectural Ramanujan type formulas due to Z.-W. Sun, which have remained open since 2011. As an application of this, in a conceptual way we interpret how Sun's conjectural formulas arise and can be verified, as well as recover other cases that were proved by Cooper, Wan and Zudilin.

math.NT↗
arXiv2024

A hyperelliptic saga on a generating function of the squares of Legendre polynomials

We decompose the generating function $\sum_{n=0}^\infty\binom{2n}nP_n(y)^2z^n$ of the squares of Legendre polynomials as a product of periods of hyperelliptic curves. These periods satisfy a family of $\textit{second}$ order differential equations. This is highly unusual since $\textit{four}$ is the expected order for genus 2. These second order equations are arithmetic and yet, surprisingly, their monodromy group is dense in $\operatorname{SL}_2(\mathbb{R})$. This suggests that they cannot be solved in terms of hypergeometric functions, which is novel for arithmetic second order differential equations that are $\textit{defined over}$ $\mathbb{Q}$, and also novel for a $\textit{family}$ of such equations. We complement our analysis with a recipe for constructing similar examples. Maple's support for the paper is available at https://www.math.fsu.edu/~hoeij/saga/.

math.NT↗
arXiv2024

Solving Third Order Linear Difference Equations in Terms of Second Order Equations

We present two algorithms for computing what we call the absolute factorization of a difference operator. We also give an algorithm to solve third order difference equations in terms of second order equations, together with applications to OEIS sequences. The latter algorithm is similar to existing algorithms for differential equations.

math.AC↗
arXiv2024

Sporadic Cubic Torsion

Let $K$ be a number field, and let $E/K$ be an elliptic curve over $K$. The Mordell--Weil theorem asserts that the $K$-rational points $E(K)$ of $E$ form a finitely generated abelian group. In this work, we complete the classification of the finite groups which appear as the torsion subgroup of $E(K)$ for $K$ a cubic number field. To do so, we determine the cubic points on the modular curves $X_1(N)$ for \[N = 21, 22, 24, 25, 26, 28, 30, 32, 33, 35, 36, 39, 45, 65, 121.\] As part of our analysis, we determine the complete list of $N$ for which $J_0(N)$ (resp., $J_1(N)$, resp., $J_1(2,2N)$) has rank 0. We also provide evidence to a generalized version of a conjecture of Conrad, Edixhoven, and Stein by proving that the torsion on $J_1(N)(\mathbb{Q})$ is generated by $\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$-orbits of cusps of $X_1(N)_{\bar{\mathbb{Q}}}$ for $N\leq 55$, $N \neq 54$.

math.NT↗
arXiv2024

Submodule approach to creative telescoping

This paper proposes ideas to speed up the process of creative telescoping, particularly when the telescoper is reducible. One can interpret telescoping as computing an annihilator $L \in D$ for an element $m$ in a $D$-module $M$. The main idea is to look for submodules of $M$. If $N$ is a non-trivial submodule of $M$, constructing the minimal operator $R$ of the image of $m$ in $M/N$ gives a right-factor of $L$ in $D$. Then $L = L' R$ where the left-factor $L'$ is the telescoper of $R(m) \in N$. To expedite computing $L'$, compute the action of $D$ on a natural basis of $N$, then obtain $L'$ with a cyclic vector computation. The next main idea is that when $N$ has automorphisms, use them to construct submodules. An automorphism with distinct eigenvalues can be used to decompose $N$ as a direct sum $N_1 \oplus \cdots \oplus N_k$. Then $L'$ is the LCLM (Least Common Left Multiple) of $L_1, \ldots, L_k$ where $L_i$ is the telescoper of the projection of $R(m)$ on $N_i$. An LCLM can greatly increase the degrees of coefficients, so $L'$ and $L$ can be much larger expressions than the factors $L_1,\ldots,L_k$ and $R$. Examples show that computing each factor $L_i$ and $R$ seperately can save a lot of CPU time compared to computing $L$ in expanded form with standard creative telescoping.

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arXiv2022

Desingularization and p-Curvature of Recurrence Operators

Linear recurrence operators in characteristic $p$ are classified by their $p$-curvature. For a recurrence operator $L$, denote by $χ(L)$ the characteristic polynomial of its $p$-curvature. We can obtain information about the factorization of $L$ by factoring $χ(L)$. The main theorem of this paper gives an unexpected relation between $χ(L)$ and the true singularities of $L$. An application is to speed up a fast algorithm for computing $χ(L)$ by desingularizing $L$ first. Another contribution of this paper is faster desingularization.

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arXiv2020

A Family of Denominator Bounds for First Order Linear Recurrence Systems

For linear recurrence systems, the problem of finding rational solutions is reduced to the problem of computing polynomial solutions by computing a content bound or a denominator bound. There are several bounds in the literature. The sharpest bound leads to polynomial solutions of lower degrees, but this advantage need not compensate for the time spent on computing that bound. To strike the best balance between sharpness of the bound versus CPU time spent obtaining it, we will give a family of bounds. The $J$'th member of this family is similar to (Abramov, Barkatou, 1998) when $J=1$, similar to (van Hoeij, 1998) when $J$ is large, and novel for intermediate values of $J$, which give the best balance between sharpness and CPU time. The setting for our content bounds are systems $τ(Y) = MY$ where $τ$ is an automorphism of a UFD, and $M$ is an invertible matrix with entries in its field of fractions. This setting includes the shift case, the $q$-shift case, the multi-basic case and others. We give two versions, a global version, and a version that bounds each entry separately.

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arXiv2019

Computing an order complete basis for $M^{\infty}(N)$ and Applications

This paper gives a quick way to construct all modular functions for the group $Γ_0(N)$ having only a pole at $τ= i \infty$. We assume that we are given two modular functions $f,g$ for $Γ_0(N)$ with poles only at $i \infty$ and coprime pole orders. As an application we obtain two new identities from which one can derive that $p(11n+6)\equiv 0\pmod{11}$, here $p(n)$ is the usual partition function.

math.NT↗
arXiv2018

Stringy Hirzebruch classes of Weierstrass fibrations

A Weierstrass fibration is an elliptic fibration $Y\to B$ whose total space $Y$ may be given by a global Weierstrass equation in a $\mathbb{P}^2$-bundle over $B$. In this note, we compute stringy Hirzebruch classes of singular Weierstrass fibrations associated with constructing non-Abelian gauge theories in $F$-theory. For each Weierstrass fibration $Y\to B$ we then derive a generating function $χ^{\text{str}}_y(Y;t)$, whose degree-$d$ coefficient encodes the stringy $χ_y$-genus of $Y\to B$ over an unspecified base of dimension $d$, solely in terms of invariants of the base. To facilitate our computations, we prove a formula for general characteristic classes of blowups along (possibly singular) complete intersections.

math.AG↗
arXiv2018

Classifying (almost)-Belyi maps with Five Exceptional Points

We classify all rational functions whose branching pattern above {0, 1, infinity} satisfy a certain regularity condition with precisely d=5 exceptions. This work is motivated by solving second order linear differential equations, with d=5 true singularities, in terms of hypergeometric functions. A similar problem was solved for d=4 by Vidunas and Filipuk.

math.CO↗
arXiv2017

The Complexity of Computing all Subfields of an Algebraic Number Field

For a finite separable field extension K/k, all subfields can be obtained by intersecting so-called principal subfields of K/k. In this work we present a way to quickly compute these intersections. If the number of subfields is high, then this leads to faster run times and an improved complexity.

cs.SC↗
arXiv2017

Functional Decomposition using Principal Subfields

Let $f\in K(t)$ be a univariate rational function. It is well known that any non-trivial decomposition $g \circ h$, with $g,h\in K(t)$, corresponds to a non-trivial subfield $K(f(t))\subsetneq L \subsetneq K(t)$ and vice-versa. In this paper we use the idea of principal subfields and fast subfield-intersection techniques to compute the subfield lattice of $K(t)/K(f(t))$. This yields a Las Vegas type algorithm with improved complexity and better run times for finding all non-equivalent complete decompositions of $f$.

cs.SC↗
arXiv2016

Reduction-Based Creative Telescoping for Fuchsian D-finite Functions

Continuing a series of articles in the past few years on creative telescoping using reductions, we adapt Trager's Hermite reduction for algebraic functions to fuchsian D-finite functions and develop a reduction-based creative telescoping algorithm for this class of functions, thereby generalizing our recent reduction-based algorithm for algebraic functions, presented at ISSAC 2016.

cs.SC↗
arXiv2016

Hypergeometric Expressions for Generating Functions of Walks with Small Steps in the Quarter Plane

We study nearest-neighbors walks on the two-dimensional square lattice, that is, models of walks on $\mathbb{Z}^2$ defined by a fixed step set that is a subset of the non-zero vectors with coordinates 0, 1 or $-1$. We concern ourselves with the enumeration of such walks starting at the origin and constrained to remain in the quarter plane $\mathbb{N}^2$, counted by their length and by the position of their ending point. Bousquet-Mélou and Mishna [Contemp. Math., pp. 1--39, Amer. Math. Soc., 2010] identified 19 models of walks that possess a D-finite generating function; linear differential equations have then been guessed in these cases by Bostan and Kauers [FPSAC 2009, Discrete Math. Theor. Comput. Sci. Proc., pp. 201--215, 2009]. We give here the first proof that these equations are indeed satisfied by the corresponding generating functions. As a first corollary, we prove that all these 19 generating functions can be expressed in terms of Gauss' hypergeometric functions that are intimately related to elliptic integrals. As a second corollary, we show that all the 19 generating functions are transcendental, and that among their $19 \times 4$ combinatorially meaningful specializations only four are algebraic functions.

math.CO↗
arXiv2016

Computing Hypergeometric Solutions of Second Order Linear Differential Equations using Quotients of Formal Solutions and Integral Bases

We present two algorithms for computing hypergeometric solutions of second order linear differential operators with rational function coefficients. Our first algorithm searches for solutions of the form \[ \exp(\int r \, dx)\cdot{_{2}F_1}(a_1,a_2;b_1;f) \] where $r,f \in \overline{\mathbb{Q}(x)}$, and $a_1,a_2,b_1 \in \mathbb{Q}$. It uses modular reduction and Hensel lifting. Our second algorithm tries to find solutions in the form \[ \exp(\int r \, dx)\cdot \left( r_0 \cdot{_{2}F_1}(a_1,a_2;b_1;f) + r_1 \cdot{_{2}F_1}'(a_1,a_2;b_1;f) \right) \] where $r_0, r_1 \in \overline{\mathbb{Q}(x)}$, as follows: It tries to transform the input equation to another equation with solutions of the first type, and then uses the first algorithm.

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