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Manuel Kauers

Publications and source records attributed to Manuel Kauers.

90 records · Page 5Linked to original sources
arXiv2013

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.

cs.SC↗
arXiv2012

Proof of George Andrews's and David Robbins's q-TSPP Conjecture

The conjecture that the orbit-counting generating function for totally symmetric plane partitions can be written as an explicit product formula, has been stated independently by George Andrews and David Robbins around 1983. We present a proof of this long-standing conjecture.

math.CO↗
arXiv2012

A method for determining the mod-$2^k$ behaviour of recursive sequences, with applications to subgroup counting

We present a method to obtain congruences modulo powers of 2 for sequences given by recurrences of finite depth with polynomial coefficients. We apply this method to Catalan numbers, Fuß-Catalan numbers, and to subgroup counting functions associated with Hecke groups and their lifts. This leads to numerous new results, including many extensions of known results to higher powers of 2.

math.CO↗
arXiv2012

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.

cs.SC↗
arXiv2012

Telescopers for Rational and Algebraic Functions via Residues

We show that the problem of constructing telescopers for functions of m variables is equivalent to the problem of constructing telescopers for algebraic functions of m -1 variables and 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.

cs.SC↗
arXiv2012

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.

cs.SC↗
arXiv2011

A Refined Denominator Bounding Algorithm for Multivariate Linear Difference Equations

We continue to investigate which polynomials can possibly occur as factors in the denominators of rational solutions of a given partial linear difference equation. In an earlier article we had introduced the distinction between periodic and aperiodic factors in the denominator, and we gave an algorithm for predicting the aperiodic ones. Now we extend this technique towards the periodic case and present a refined algorithm which also finds most of the periodic factors.

cs.SC↗
arXiv2010

The computational challenge of enumerating high-dimensional rook walks

We provide guessed recurrence equations for the counting sequences of rook paths on d-dimensional chess boards starting at (0..0) and ending at (n..n), where d=2,3,...,12. Our recurrences suggest refined asymptotic formulas of these sequences. Rigorous proofs of the guessed recurrences as well as the suggested asymptotic forms are posed as challenges to the reader.

math.CO↗
arXiv2010

Dominance in the family of Sugeno-Weber t-norms

The dominance relationship between two members of the family of Sugeno Weber t-norms is proven by using a quantifer elimination algorithm. Further it is shown that dominance is a transitive, and therefore also an order relation, on this family of t-norms.

math.FA↗
arXiv2010

When can we decide that a P-finite sequence is positive?

We consider two algorithms which can be used for proving positivity of sequences that are defined by a linear recurrence equation with polynomial coefficients (P-finite sequences). Both algorithms have in common that while they do succeed on a great many examples, there is no guarantee for them to terminate, and they do in fact not terminate for every input. For some restricted classes of P-finite recurrence equations of order up to three we provide a priori criteria that assert the termination of the algorithms.

cs.SC↗
arXiv2010

Partial Denominator Bounds for Partial Linear Difference Equations

We investigate which polynomials can possibly occur as factors in the denominators of rational solutions of a given partial linear difference equation (PLDE). Two kinds of polynomials are to be distinguished, we call them /periodic/ and /aperiodic/. The main result is a generalization of a well-known denominator bounding technique for univariate equations to PLDEs. This generalization is able to find all the aperiodic factors of the denominators for a given PLDE.

cs.SC↗
arXiv2009

The complete Generating Function for Gessel Walks is Algebraic

Gessel walks are lattice walks in the quarter plane $\set N^2$ which start at the origin $(0,0)\in\set N^2$ and consist only of steps chosen from the set $\{\leftarrow,\swarrow,\nearrow,\to\}$. We prove that if $g(n;i,j)$ denotes the number of Gessel walks of length $n$ which end at the point $(i,j)\in\set N^2$, then the trivariate generating series $G(t;x,y)=\sum_{n,i,j\geq 0} g(n;i,j)x^i y^j t^n$ is an algebraic function.

math.CO↗
arXiv2009

A Non-Holonomic Systems Approach to Special Function Identities

We extend Zeilberger's approach to special function identities to cases that are not holonomic. The method of creative telescoping is thus applied to definite sums or integrals involving Stirling or Bernoulli numbers, incomplete Gamma function or polylogarithms, which are not covered by the holonomic framework. The basic idea is to take into account the dimension of appropriate ideals in Ore algebras. This unifies several earlier extensions and provides algorithms for summation and integration in classes that had not been accessible to computer algebra before.

cs.SC↗
arXiv2009

A Proof of George Andrews' and Dave Robbins' q-TSPP Conjecture (modulo a finite amount of routine calculations)

In the historic conference Combinatoire Enumerative[LL] wonderfully organized by Gilbert Labelle and Pierre Leroux there were many stimulating lectures, including a very interesting one by Pierre Leroux himself, who talked about his joint work with Xavier Viennot[LV], on solving differential equations combinatorially! During the problem session of that very same colloque, chaired by Pierre Leroux, Richard Stanley raised some intriguing problems about the enumeration of plane partitions, that he later expanded into a fascinating article[Sta1]. Most of these problems concerned the enumeration of symmetry classes of plane partitions, that were discussed in more detail in another article of Stanley[Sta2]. All of the conjectures in the latter article have since been proved (see Dave Bressoud's modern classic[B]), except one, that, so far, resisted the efforts of the greatest minds in enumerative combinatorics. It concerns the proof of an explicit formula for the q-enumeration of totally symmetric plane partitions, conjectured independently by George Andrews and Dave Robbins([Sta2],[Sta1](conj. 7), [B](conj. 13)). In this tribute to Pierre Leroux, we describe how to prove that last stronghold.

math.CO↗
arXiv2008

The Quasi-Holonomic Ansatz and Restricted Lattice Walks

The great enumerator Germain Kreweras empirically discovered this intriguing fact, and then needed lots of pages[K], and lots of human ingenuity, to prove it. Other great enumerators, for example, Heinrich Niederhausen[N], Ira Gessel[G1], and Mireille Bousquet-Mélou[B], found other ingenious, ``simpler'' proofs. Yet none of them is as simple as ours! Our proof (with the generous help of our faithful computers) is ``ugly'' in the traditional sense, since it would be painful for a lowly human to follow all the steps. But according to our humble aesthetic taste, this proof is much more elegant, since it is (conceptually) one-line. So what if that line is rather long (a huge partial-recurrence equation satisfied by the general counting function), it occupies less storage than a very low-resolution photograph.

math.CO↗
arXiv2008

Proof of Ira Gessel's Lattice Path Conjecture

We present a computer-aided, yet fully rigorous, proof of Ira Gessel's tantalizingly simply-stated conjecture that the number of ways of walking $2n$ steps in the region $x+y \geq 0, y \geq 0$ of the square-lattice with unit steps in the east, west, north, and south directions, that start and end at the origin, equals $16^n\frac{(5/6)_n(1/2)_n}{(5/3)_n(2)_n}$ .

math.CO↗
arXiv2007

Experiments with a Positivity Preserving Operator

We consider some multivariate rational functions which have (or are conjectured to have) only positive coefficients in their series expansion. We consider an operator that preserves positivity of series coefficients, and apply the inverse of this operator to the rational functions. We obtain new rational functions which seem to have only positive coefficients, whose positivity would imply positivity of the original series, and which, in a certain sense, cannot be improved any further.

math.CO↗