Van Hoeij Research Articles

Determination of the modular Jacobian varieties J_1(M,MN) with the Mordell–Weil rank zero

In this paper, we determine all modular Jacobian varieties J_1(M,MN) over the cyclotomic fields {{mathbb {Q}}}(zeta _M) with the Mordell–Weil rank zero assuming the Birch–Swinnerton-Dyer conjecture, following the method of Derickx, Etropolski, van Hoeij, Morrow, and Zureick-Brown.

Symbolic Conversion of Holonomic Functions to Hypergeometric Type Power Series

A term $a_n$ is $m$-fold hypergeometric, for a given positive integer $m$, if the ratio $a_{n+m}/a_n$ is a rational function over a field $K$ of characteristic zero. We establish the structure of holonomic recurrence equation, i.e. linear and homogeneous recurrence equations having polynomial coefficients, that have $m$-fold hypergeometric term solutions over $K$, for any positive integer $m$. Consequently, we describe an algorithm, say $mfoldHyper$, that extends van Hoeij's algorithm (1998) which computes a basis of the subspace of hypergeometric $(m=1)$ term solutions of holonomic recurrence equations to the more general case of $m$-fold hypergeometric terms. We generalize the concept of hypergeometric type power series introduced by Koepf (1992), by considering linear combinations of Laurent-Puiseux series whose coefficients are $m$-fold hypergeometric terms. Thus thanks to $mfoldHyper$, we deduce a complete procedure to compute these power series; indeed, it turns out that every linear combination of power series with $m$-fold hypergeometric term coefficients, for finitely many values of $m$, is detected. On the other hand, we investigate an algorithm to represent power series of non-holonomic functions. The algorithm follows the same steps of Koepf's algorithm, but instead of seeking holonomic differential equations, quadratic differential equations are computed and the Cauchy product rule is used to deduce recurrence equations for the power series coefficients. This algorithm defines a normal function that yields together with enough initial values normal forms for many power series of non-holonomic functions. Therefore, non-trivial identities are automatically proved using this approach. This paper is accompanied by implementations in the Computer Algebra Systems (CAS) Maxima 5.44.0 and Maple 2019.

New Approaches to the General Linearization Problem of Jacobi Polynomials Based on Moments and Connection Formulas

This article deals with the general linearization problem of Jacobi polynomials. We provide two approaches for finding closed analytical forms of the linearization coefficients of these polynomials. The first approach is built on establishing a new formula in which the moments of the shifted Jacobi polynomials are expressed in terms of other shifted Jacobi polynomials. The derived moments formula involves a hypergeometric function of the type 4F3(1), which cannot be summed in general, but for special choices of the involved parameters, it can be summed. The reduced moments formulas lead to establishing new linearization formulas of certain parameters of Jacobi polynomials. Another approach for obtaining other linearization formulas of some Jacobi polynomials depends on making use of the connection formulas between two different Jacobi polynomials. In the two suggested approaches, we utilize some standard reduction formulas for certain hypergeometric functions of the unit argument such as Watson’s and Chu-Vandermonde identities. Furthermore, some symbolic algebraic computations such as the algorithms of Zeilberger, Petkovsek and van Hoeij may be utilized for the same purpose. As an application of some of the derived linearization formulas, we propose a numerical algorithm to solve the non-linear Riccati differential equation based on the application of the spectral tau method.

New Specific and General Linearization Formulas of Some Classes of Jacobi Polynomials

The main purpose of the current article is to develop new specific and general linearization formulas of some classes of Jacobi polynomials. The basic idea behind the derivation of these formulas is based on reducing the linearization coefficients which are represented in terms of the Kampé de Fériet function for some particular choices of the involved parameters. In some cases, the required reduction is performed with the aid of some standard reduction formulas for certain hypergeometric functions of unit argument, while, in other cases, the reduction cannot be done via standard formulas, so we resort to certain symbolic algebraic computation, and specifically the algorithms of Zeilberger, Petkovsek, and van Hoeij. Some new linearization formulas of ultraspherical polynomials and third-and fourth-kinds Chebyshev polynomials are established.

Computing integral bases via localization and Hensel lifting

We present a new algorithm for computing integral bases in algebraic function fields of one variable, or equivalently for constructing the normalization of a plane curve. Our basic strategy makes use of the concepts of localization and completion, together with the Chinese remainder theorem, to reduce the problem to the task of finding integral bases for the branches of each singularity of the curve. To solve the latter task, in turn, we work with suitably truncated Puiseux expansions. In contrast to van Hoeij's algorithm (van Hoeij, 1994), which also relies on Puiseux expansions (but pursues a different strategy), we use Hensel's lemma as a key ingredient. This allows us at some steps of the algorithm to compute factors corresponding to conjugacy classes of Puiseux expansions, without actually computing the individual expansions. In this way, we make substantially less use of the Newton-Puiseux algorithm. In addition, our algorithm is inherently parallel. As a result, it outperforms in most cases any other algorithm known to us by far. Typical applications are the computation of adjoint ideals and, based on this, the computation of Riemann-Roch spaces and the parametrization of rational curves.

Asymptotics of lattice walks via analytic combinatorics in several variables

We consider the enumeration of walks on the two-dimensional non-negative integer lattice with steps defined by a finite set S ⊆ {±1, 0}2 . Up to isomorphism there are 79 unique two-dimensional models to consider, and previous work in this area has used the kernel method, along with a rigorous computer algebra approach, to show that 23 of the 79 models admit D-finite generating functions. In 2009, Bostan and Kauers used Pade ́-Hermite approximants to guess differential equations which these 23 generating functions satisfy, in the process guessing asymptotics of their coefficient sequences. In this article we provide, for the first time, a complete rigorous verification of these guesses. Our technique is to use the kernel method to express 19 of the 23 generating functions as diagonals of tri-variate rational functions and apply the methods of analytic combinatorics in several variables (the remaining 4 models have algebraic generating functions and can thus be handled by univariate techniques). This approach also shows the link between combinatorial properties of the models and features of its asymptotics such as asymptotic and polynomial growth factors. In addition, we give expressions for the number of walks returning to the x-axis, the y-axis, and the origin, proving recently conjectured asymptotics of Bostan, Chyzak, van Hoeij, Kauers, and Pech.

Ring extensions of length two

We characterize extensions of commutative rings [Formula: see text] such that [Formula: see text] is minimal for each [Formula: see text]-subalgebra [Formula: see text] of [Formula: see text] with [Formula: see text]. This property is equivalent to [Formula: see text] has length 2. Such extensions are either pointwise minimal or simple. We are able to compute the number of subextensions of [Formula: see text]. Besides commutative algebra considerations, our main result uses the concept of principal subfields of a finite separable field extension, which was recently introduced by van Hoeij et al. As a corollary of this paper, we get that simple extensions of length 2 have FIP.

Computing subfields of number fields and applications to Galois group computations

A polynomial time algorithm to find generators of the lattice of all subfields of a given number field was given in van Hoeij et al. (2013).This article reports on a massive speedup of this algorithm. This is primary achieved by our new concept of Galois-generating subfields. In general this is a very small set of subfields that determine all other subfields in a group-theoretic way. We compute them by targeted calls to the method from van Hoeij et al. (2013). For an early termination of these calls, we give a list of criteria that imply that further calls will not result in additional subfields.Finally, we explain how we use subfields to get a good starting group for the computation of Galois groups.

New formulae between Jacobi polynomials and some fractional Jacobi functions generalizing some connection formulae

In this paper, a new formula relating Jacobi polynomials of arbitrary parameters with the squares of certain fractional Jacobi functions is derived. The derived formula is expressed in terms of a certain terminating hypergeometric function of the type $$_4F_{3}(1)$$ . With the aid of some standard reduction formulae such as Pfaff-Saalschutz’s and Watson’s identities, the derived formula can be reduced in simple forms which are free of any hypergeometric functions for certain choices of the involved parameters of the Jacobi polynomials and the Jacobi functions. Some other simplified formulae are obtained via employing some computer algebra algorithms such as the algorithms of Zeilberger, Petkovsek and van Hoeij. Some connection formulae between some Jacobi polynomials are deduced. From these connection formulae, some other linearization formulae of Chebyshev polynomials are obtained. As an application to some of the introduced formulae, a numerical algorithm for solving nonlinear Riccati differential equation is presented and implemented by applying a suitable spectral method.

Linearization coefficients of some particular Jacobi polynomials via hypergeometric functions

The main purpose of the present paper is to establish two new linearization formulas for certain Jacobi polynomials. The new established formulas are expressed in terms of terminating hypergeometric functions of the type ${}_{4}F_{3}(1)$ . In virtue of the well-known Pfaff-Saalschutz identity, or by using some computer algebra algorithms, and in particular, the algorithms of Zeilberger, Petkovsek and van Hoeij, the resulting ${}_{4}F_{3}(1)$ can be reduced for particular choices of the involved parameters. This reduction leads to obtaining several simple linearization formulas of some particular Jacobi polynomials free of any hypergeometric functions.

New Formulae of Squares of Some Jacobi Polynomials Via Hypergeometric Functions

In this article, a new formula expressing explicitly the squares of Jacobi polynomials of certain parameters in terms of Jacobi polynomials of arbitrary parameters is derived. The derived formula is given in terms of ceratin terminating hypergeometric function of the type $_4F_3(1)$. In some cases, this $_4F_3(1)$ can be reduced by using some well-known reduction formulae in literature such as Watson's and Pfa-Saalschutz's identities. In some other cases, this $_4F_3(1)$ can be reduced by means of symbolic computation, and in particular Zeilberger's, Petkovsek's and van Hoeij's algorithms. Hence, some new squares formulae for Jacobi polynomials of special parameters can be deduced in reduced forms which are free of any hypergeometric functions.

New formulas for the linearization coefficients of some nonsymmetric Jacobi polynomials

The main aim of this paper is to develop four innovative linearization formulas for some nonsymmetric Jacobi polynomials. This means that we find the coefficients of the products of Jacobi polynomials of certain parameters. In general, these coefficients are expressed in terms of certain hypergeometric functions of the unit argument. We employ some symbolic algebraic computations such as the algorithms of Zeilberger, Petkovsek and van Hoeij for reducing such coefficients. Moreover, and based on a certain Whipple transformation, two new closed formulas for summing certain terminating hypergeometric functions of the unit argument are deduced. New formulas for some definite integrals are given with the aid of the derived linearization formulas.

M-Fold Hypergeometric Solutions of Linear Recurrence Equations Revisited

We present two algorithms to compute m-fold hypergeometric solutions of linear recurrence equations for the classical shift case and for the q-case, respectively. The first is an m-fold generalization and q-generalization of the algorithm by van Hoeij (Appl Algebra Eng Commun Comput 17:83–115, 2005; J. Pure Appl Algebra 139:109–131, 1998) for recurrence equations. The second is a combination of an improved version of the algorithms by Petkovšek (Discrete Math 180:3–22, 1998; J Symb Comput 14(2–3):243–264, 1992) for recurrence and q-recurrence equations and the m-fold algorithm from Petkovšek and Salvy (ISSAC 1993 Proceedings, pp 27–33, 1993) for recurrence equations. We will refer to the classical algorithms as van Hoeij or Petkovšek respectively. To formulate our ideas, we first need to introduce an adapted version of an m-fold Newton polygon and its characteristic polynomials for the classical case and q-case, and to prove the important properties in this case. Using the data from the Newton polygon, we are able to present efficient m-fold versions of the van Hoeij and Petkovšek algorithms for the classical shift case and for the q-case, respectively. Furthermore, we show how one can use the Newton polygon and our characteristic polynomials to conclude for which \({m\in \mathbb{N}}\) there might be an m-fold hypergeometric solution at all. Again by using the information obtained from the Newton polygon, the presentation of the q-Petkovšek algorithm can be simplified and streamlined. Finally, we give timings for the ‘classical’ q-Petkovšek, our q-van Hoeij and our modified q-Petkovšek algorithm on some classes of problems and we present a Maple implementation of the m-fold algorithms for the q-case.

Valuations of rational solutions of linear difference equations at irreducible polynomials

We discuss two algorithms which, given a linear difference equation with rational function coefficients over a field k of characteristic 0, construct a finite set M of polynomials, irreducible in k [ x ] , such that if the given equation has a solution F ( x ) ∈ k ( x ) and val p ( x ) F ( x ) < 0 for an irreducible p ( x ) , then p ( x ) ∈ M . After this for each p ( x ) ∈ M the algorithms compute a lower bound for val p ( x ) F ( x ) , which is valid for any rational function solution F ( x ) of the initial equation. The algorithms are applicable to scalar linear equations of arbitrary orders as well as to linear systems of first-order equations. The algorithms are based on a combination of renewed approaches used in earlier algorithms for finding a universal denominator (Abramov and Barkatou (1998) [6]), and on a denominator bound (van Hoeij (1998) [12]). A complexity analysis of the two proposed algorithms is presented.

Factoring univariate polynomials over the rationals

We present an algorithm for factoring a polynomial, f , in one variable with rational coefficients. Our algorithm is a variant of the Belabas [Belabas] version of the van Hoeij [van Hoeij] factoring algorithm. Our algorithm not only contains a practical speed-up over Belabas' but it also allows us to prove a new complexity result for factoring polynomials. The van Hoeij algorithm follows Zassenhaus' [Zass] approach by factoring f mod a prime number and Hensel Lifting the local factors. The practical speed-up in van Hoeij's algorithm comes from using the LLL [LLL] algorithm to solve the exponential recombination problem of decicing which combination of local factors form the true factors. However, the cost of LLL in van Hoeij's approach has been difficult to bound (in fact [van Hoeij] only proves termination and makes no attempt at finding a complexity). Belabas found a fine-tuning [Belabas] of van Hoeij's algorithm which nearly optimizes the practical running times, but still does not provide a good bound for the LLL costs. In [BHKS] both variants were shown to have polynomial complexity, but these bounds are still unsatisfactorily large and do little to illuminate the behavior of these algorithms. Our algorithm improves Belabas' approach in the following ways: We make a practical improvement which ensures that Hensel Lifting is always minimized. We include a decision making process which allows us to bound the total number of LLL switches in the algorithm by O( r 3 ), where r is the number of local factors. This is independent of both degree and coefficient size of f . . In this poster we present an overview of the new algorithm and give a brief look at the style of our proofs. For the full details of the switch bound O( r 3 ) see [Novocin]. Using the Floating-Point LLL [L2] and some minor changes we can show the LLL costs are O( r 7 ). We are still writing down the details of a proof that the new total complexity of our algorithm is O( r 7 ).

Fast arithmetic for triangular sets: From theory to practice

We study arithmetic operations for triangular families of polynomials, concentrating on multiplication in dimension zero. By a suitable extension of fast univariate Euclidean division, we obtain theoretical and practical improvements over a direct recursive approach; for a family of special cases, we reach quasi-linear complexity. The main outcome we have in mind is the acceleration of higher-level algorithms, by interfacing our low-level implementation with languages such as AXIOM or Maple. We show the potential for huge speed-ups, by comparing two AXIOM implementations of van Hoeij and Monagan’s modular GCD algorithm.

Factoring univariate polynomials over the rationals (abstract only)

This poster presents an algorithm for factoring polynomials over the rationals which follows the approach of the van Hoeij algorithm. The key theoretical novelty in our approach is that it is set up in a way that will make it possible to prove a new complexity result. Another difference of this algorithm from prior algorithms is the practical improvement which we call early termination. Our algorithm should outperform prior algorithms in many common classes of polynomials (including irreducibles).

A relative van Hoeij algorithm over number fields

van Hoeij’s algorithm for factoring univariate polynomials over the rational integers rests on the same principle as the Berlekamp–Zassenhaus algorithm, but uses lattice basis reduction to improve drastically on the recombination phase. His ideas give rise to a collection of algorithms, differing greatly in their efficiency. We present two deterministic variants, one of which achieves excellent overall performance. We then generalize these ideas to factor polynomials over number fields.

Implementation of an Algorithm for Finding Analyticity Conditions for a Solution of a Difference Equation

A linear difference operator L with polynomial coefficients and a function F(x) satisfying the equation LF(x) e 0 are considered. The function is assumed to be analytic in the interval (–σ, d), where σ > 0. In the paper, an implementation of an algorithm suggested by S.A. Abramov and M. van Hoeij for finding conditions that guarantee analyticity of F(x) on the entire real axis \mathbb{R} is presented. The analyticity conditions are linear relations for values of F(x) and its derivatives at a given point belonging to the half-interval l0, d). A procedure for computing values of F(x) and its derivatives up to a prescribed order at a given point x0 e \mathbb{R} is also implemented. Examples illustrating the program operation are presented.

Liouvillian Solutions of Linear Differential Equations of Order Three and Higher

Singer and Ulmer (1997) gave an algorithm to compute Liouvillian (“closed-form") solutions of homogeneous linear differential equations. However, there were several efficiency problems that made computations often not practical. In this paper we address these problems. We extend the algorithm in van Hoeij and Weil (1997) to compute semi-invariants and a theorem in Singer and Ulmer (1997)in such a way that, by computing one semi-invariant that factors into linear forms, one gets all coefficients of the minimal polynomial of an algebraic solution of the Riccati equation, instead of only one coefficient. These coefficients come “for free" as a byproduct of our algorithm for computing semi-invariants. We specifically detail the algorithm in the cases of equations of order three (order two equations are handled by the algorithm of Kovacic, 1986,see also Ulmer and Weil, 1996 or Fakler, 1997. In the Appendix, we present several methods to decide when a multivariate polynomial depending on parameters can admit linear factors, which is a necessary ingredient in the algorithm.