Shaoshi Chen

Hermite reduction and creative telescoping for hyperexponential functions

We present a new reduction algorithm that simultaneously extends Hermites reduction for rational functions and the Hermite-like reduction for hyperexponential functions. It yields a unique additive decomposition that allows to decide hyperexponential integrability. Based on this reduction algorithm, we design a new algorithm to compute minimal telescopers for bivariate hyperexponential functions. One of its main features is that it can avoid the costly computation of certificates. Its implementation outperforms Maples function DEtools[Zeilberger]. We also derive an order bound on minimal telescopers that is tighter than the known ones.

Complexity of creative telescoping for bivariate rational functions

The long-term goal initiated in this work is to obtain fast algorithms and implementations for definite integration in Almkvist and Zeilbergers framework of (differential) creative telescoping. Our complexity-driven approach is to obtain tight degree bounds on the various expressions involved in the method. To make the problem more tractable, we restrict to bivariate rational functions. By considering this constrained class of inputs, we are able to blend the general method of creative telescoping with the well-known Hermite reduction. We then use our new method to compute diagonals of rational power series arising from combinatorics.

Telescopers for rational and algebraic functions via residues

We show that the problem of constructing telescopers for rational functions of m + 1 variables is equivalent to the problem of constructing telescopers for algebraic functions of m variables and we present a new algorithm to construct telescopers for algebraic functions of two variables. These considerations are based on analyzing the residues of the input. According to experiments, the resulting algorithm for rational functions of three variables is faster than known algorithms, at least in some examples of combinatorial interest. The algorithm for algebraic functions implies a new bound on the order of the telescopers.

Order-degree curves for hypergeometric creative telescoping

Creative telescoping applied to a bivariate proper hypergeometric term produces linear recurrence operators with polynomial coefficients, called telescopers. We provide bounds for the degrees of the polynomials appearing in these operators. Our bounds are expressed as curves in the (r, d)-plane which assign to every order r a bound on the degree d of the telescopers. These curves are hyperbolas, which reflect the phenomenon that higher order telescopers tend to have lower degree, and vice versa.

Trading order for degree in creative telescoping

We analyze the differential equations produced by the method of creative telescoping applied to a hyperexponential term in two variables. We show that equations of low order have high degree, and that higher order equations have lower degree. More precisely, we derive degree bounding formulas which allow to estimate the degree of the output equations from creative telescoping as a function of the order. As an application, we show how the knowledge of these formulas can be used to improve, at least in principle, the performance of creative telescoping implementations, and we deduce bounds on the asymptotic complexity of creative telescoping for hyperexponential terms.

Desingularization explains order-degree curves for ore operators

Desingularization is the problem of finding a left multiple of a given Ore operator in which some factor of the leading coefficient of the original operator is removed. An order-degree curve for a given Ore operator is a curve in the (r,d)-plane such that for all points (r,d) above this curve, there exists a left multiple of order r and degree d of the given operator. We give a new proof of a desingularization result by Abramov and van Hoeij for the shift case, and show how desingularization implies order-degree curves which are extremely accurate in examples.

On the existence of telescopers for mixed hypergeometric terms

We present a criterion for the existence of telescopers for mixed hypergeometric terms, which is based on additive and multiplicative decompositions. The criterion enables us to determine the termination of Zeilbergers algorithms for mixed hypergeometric inputs, and to verify that certain indefinite sums do not satisfy any polynomial differential equation.

A generalized Apagodu-Zeilberger algorithm

The Apagodu-Zeilberger algorithm can be used for computing annihilating operators for definite sums over hypergeometric terms, or for definite integrals over hyperexponential functions. In this paper, we propose a generalization of this algorithm which is applicable to arbitrary Δ-finite functions. In analogy to the hypergeometric case, we introduce the notion of proper Δ-finite functions. We show that the algorithm always succeeds for these functions, and we give a tight a priori bound for the order of the output operator.

A Modified Abramov-Petkovsek Reduction and Creative Telescoping for Hypergeometric Terms

The Abramov-Petkovsek reduction computes an additive decomposition of a hypergeometric term,which extends the functionality of the Gosper algorithm for indefinite hypergeometric summation. We modify the Abramov-Petkovsek reduction so as to decompose a hypergeometric term as the sum of a summable term and a non-summable one. The outputs of the Abramov-Petkovsek reduction and our modified version share the same required properties. The modified reduction does not solve any auxiliary linear difference equation explicitly. It is also more efficient than the original reduction according to computational experiments. Based on this reduction, we design a new algorithm to compute minimal telescopers for bivariate hypergeometric terms. The new algorithm can avoid the costly computation of certificates.

Reduction-Based Creative Telescoping for Algebraic Functions

Continuing a series of articles in the past few years on creative telescoping using reductions, we develop a new algorithm to construct minimal telescopers for algebraic functions. This algorithm is based on Tragers Hermite reduction and on polynomial reduction, which was originally designed for hyperexponential functions and extended to the algebraic case in this paper.