arXiv subjects

Bruno Salvy

Publications and source records attributed to Bruno Salvy.

At least 19 recordsLinked to original sources

arXiv2026

Fractions of Recurrence Operators for Generalized Fourier Series in Classical Orthogonal Polynomials

We consider series expansions in bases of classical orthogonal polynomials. When such a series solves a linear differential equation with polynomial coefficients, its coefficients satisfy a linear recurrence equation. We interpret this equation as the numerator of a fraction of linear recurrence operators. This interpretation lets us give a simple and unified view of previous algorithms computing these recurrences, with a noncommutative Euclidean algorithm as the algorithmic engine. Finally, we demonstrate the effectiveness of our approach on various examples.

math.CA↗

arXiv2026

General formulas for a class of Euler sums

Let $H_k = 1 + 1/2 + 1/3 + \cdots + 1/k$ denote the $k$th harmonic number. We present an easy-to-implement algorithm for the computation of explicit closed-form evaluations, in terms of the digamma and polygamma functions, for Euler sums of the form \begin{align} \sum_{k=1}^\infty R(k) H_k, \end{align} where $R(k)$ is a rational function (quotient of two polynomials) whose denominator degree is at least two larger than the numerator degree. We apply the same method to show how the computation of a general formula for Euler sums of the form \begin{align*} \sum_{k=1}^\infty \frac{H_k}{(m_1 k + n_1)^{p_1} (m_2 k + n_2)^{p_2} \cdots (m_r k + n_r)^{p_r}} \end{align*} reduces to partial fraction decomposition. We present explicit formulae for sums with one or two terms in the denominator, with powers $p_i$ ranging up to 3, and with multipliers $m_i$ ranging up to 4. We also include results for related Euler sums such as \begin{align*} \sum_{k=1}^\infty \frac{k^q H_k}{(m k + n)^p}. \end{align*} Computation of Euler sums directly to very high precision enables us to rigorously check the above-mentioned formulas in many specific cases.

math.NT↗

arXiv2026

Faster modular composition using two relation matrices

Modular composition is the problem of computing the composition of two univariate polynomials modulo a third one. For a long time, the fastest algebraic algorithm for this problem was that of Brent and Kung (1978). Recently, we improved Brent and Kung's algorithm by computing and using a polynomial matrix that encodes a certain basis of algebraic relations between the polynomials. This is further improved here by making use of two polynomial matrices of smaller dimension. Under genericity assumptions on the input, this results in an algorithm using $\tilde{O}(n^{(\omega+3)/4})$ arithmetic operations in the base field, where $\omega$ is the exponent of matrix multiplication. With naive matrix multiplication, this is $\tilde{O}(n^{3/2})$, while with the best currently known exponent $\omega$ this is $O(n^{1.343})$, improving upon the previously most efficient algorithms.

cs.SC↗

arXiv2025

Effective Asymptotics of Combinatorial Systems

Analytic combinatorics studies asymptotic properties of families of combinatorial objects using complex analysis on their generating functions. In their reference book on the subject, Flajolet and Sedgewick describe a general approach that allows one to derive precise asymptotic expansions starting from systems of combinatorial equations. In the situation where the combinatorial system involves only cartesian products and disjoint unions, the generating functions satisfy polynomial systems with positivity constraints for which many results and algorithms are known. We extend these results to the general situation. This produces an almost complete algorithmic chain going from combinatorial systems to asymptotic expansions. Thus, it is possible to compute asymptotic expansions of all generating functions produced by the symbolic method of Flajolet and Sedgewick when they have algebraic-logarithmic singularities (which can be decided), under the assumption that Schanuel's conjecture from number theory holds. That conjecture is not needed for systems that do not involve the constructions of sets and cycles.

math.CO↗

arXiv2025

On deciding transcendence of power series

It is well known that algebraic power series are differentially finite (D-finite): they satisfy linear differential equations with polynomial coefficients. The converse problem, whether a given D-finite power series is algebraic or transcendental, is notoriously difficult. We prove that this problem is decidable: we give two theoretical algorithms and a transcendence test that is efficient in practice.

math.NT↗

arXiv2024

Positivity Proofs for Linear Recurrences through Contracted Cones

Deciding the positivity of a sequence defined by a linear recurrence with polynomial coefficients and initial condition is difficult in general. Even in the case of recurrences with constant coefficients, it is known to be decidable only for order up to~5. We consider a large class of linear recurrences of arbitrary order, with polynomial coefficients, for which an algorithm decides positivity for initial conditions outside of a hyperplane. The underlying algorithm constructs a cone, contracted by the recurrence operator, that allows a proof of positivity by induction. The existence and construction of such cones relies on the extension of the classical Perron-Frobenius theory to matrices leaving a cone invariant.

cs.SC↗

arXiv2024

Effective Quadratic Error Bounds for Floating-Point Algorithms Computing the Hypotenuse Function

We provide tools to help automate the error analysis of algorithms that evaluate simple functions over the floating-point numbers. The aim is to obtain tight relative error bounds for these algorithms, expressed as a function of the unit round-off. Due to the discrete nature of the set of floating-point numbers, the largest errors are often intrinsically "arithmetic" in the sense that their appearance may depend on specific bit patterns in the binary representations of intermediate variables, which may be present only for some precisions. We focus on generic (i.e., parameterized by the precision) and analytic over-estimations that still capture the correlations between the errors made at each step of the algorithms. Using methods from computer algebra, which we adapt to the particular structure of the polynomial systems that encode the errors, we obtain bounds with a linear term in the unit round-off that is sharp in manycases. An explicit quadratic bound is given, rather than the $O()$-estimate that is more common in this area. This is particularly important when using low precision formats, which are increasingly common in modern processors. Using this approach, we compare five algorithms for computing the hypotenuse function, ranging from elementary to quite challenging.

math.NA↗

arXiv2024

Optimal Experimental Design for Partially Observable Pure Birth Processes

We develop an efficient algorithm to find optimal observation times by maximizing the Fisher information for the birth rate of a partially observable pure birth process involving $n$ observations. Partially observable implies that at each of the $n$ observation time points for counting the number of individuals present in the pure birth process, each individual is observed independently with a fixed probability $p$, modeling detection difficulties or constraints on resources. We apply concepts and techniques from generating functions, using a combination of symbolic and numeric computation, to establish a recursion for evaluating and optimizing the Fisher information. Our numerical results reveal the efficacy of this new method. An implementation of the algorithm is available publicly.

math.ST↗

arXiv2023

Reduction-Based Creative Telescoping for Definite Summation of D-finite Functions

Creative telescoping is an algorithmic method initiated by Zeilberger to compute definite sums by synthesizing summands that telescope, called certificates. We describe a creative telescoping algorithm that computes telescopers for definite sums of D-finite functions as well as the associated certificates in a compact form. The algorithm relies on a discrete analogue of the generalized Hermite reduction, or equivalently, a generalization of the Abramov-Petkovšek reduction. We provide a Maple implementation with good timings on a variety of examples.

cs.SC↗

arXiv2023

Positivity certificates for linear recurrences

We consider linear recurrences with polynomial coefficients of Poincaré type and with a unique simple dominant eigenvalue. We give an algorithm that proves or disproves positivity of solutions provided the initial conditions satisfy a precisely defined genericity condition. For positive sequences, the algorithm produces a certificate of positivity that is a data-structure for a proof by induction. This induction works by showing that an explicitly computed cone is contracted by the iteration of the recurrence.

cs.SC↗

arXiv2023

Explicit maximal totally real embeddings

This article is the continuation of the first named author work "On maximal totally real embeddings". For real analytic compact manifolds equipped with a covariant derivative operator acting on the real analytic sections of its tangent bundle, a construction of canonical maximal totally real embeddings is known from previous works by Guillemin-Stenzel, Lempert, Lempert-Sz{ö}ke, Sz{ö}ke and Bielawski. The construction is based on the use of Jacobi fields, which are far from being explicit. As a consequence, the form of the corresponding complex structure has been a mystery since the very beginning. A quite simple recursive expression for such complex structures has been provided in the above cited first named author work. In our series of papers we always focus on the torsion free case. In the present paper we provide a fiberwise Taylor expansion of the canonical complex structure which is expressed in terms of symmetrization of curvature monomials and a rather simple and explicit expression of the coefficients of the expansion. Our main argument applies to far more generals settings that can be useful for the study of open questions in the theory of the embeddings in consideration. In this paper we provide also evidence for some remarkable canonical vanishing of some of the integrability equations in general settings.

math.CV↗

arXiv2023

Faster Modular Composition

A new Las Vegas algorithm is presented for the composition of two polynomials modulo a third one, over an arbitrary field. When the degrees of these polynomials are bounded by $n$, the algorithm uses $O(n^{1.43})$ field operations, breaking through the $3/2$ barrier in the exponent for the first time. The previous fastest algebraic algorithms, due to Brent and Kung in 1978, require $O(n^{1.63})$ field operations in general, and ${n^{3/2+o(1)}}$ field operations in the special case of power series over a field of large enough characteristic. If cubic-time matrix multiplication is used, the new algorithm runs in ${n^{5/3+o(1)}}$ operations, while previous ones run in $O(n^2)$ operations. Our approach relies on the computation of a matrix of algebraic relations that is typically of small size. Randomization is used to reduce arbitrary input to this favorable situation.

cs.SC↗

arXiv2023

Minimization of differential equations and algebraic values of $E$-functions

A power series being given as the solution of a linear differential equation with appropriate initial conditions, minimization consists in finding a non-trivial linear differential equation of minimal order having this power series as a solution. This problem exists in both homogeneous and inhomogeneous variants; it is distinct from, but related to, the classical problem of factorization of differential operators. Recently, minimization has found applications in Transcendental Number Theory, more specifically in the computation of non-zero algebraic points where Siegel's $E$-functions take algebraic values. We present algorithms and implementations for these questions, and discuss examples and experiments.

cs.SC↗

arXiv2022

Differential-Difference Properties of Hypergeometric Series

Six families of generalized hypergeometric series in a variable $x$ and an arbitrary number of parameters are considered. Each of them is indexed by an integer $n$. Linear recurrence relations in $n$ relate these functions and their product by the variable $x$. We give explicit factorizations of these equations as products of first order recurrence operators. Related recurrences are also derived for the derivative with respect to $x$. These formulas generalize well-known properties of the classical orthogonal polynomials.

math.CA↗

arXiv2020

Computation of Tight Enclosures for Laplacian Eigenvalues

Recently, there has been interest in high-precision approximations of the first eigenvalue of the Laplace-Beltrami operator on spherical triangles for combinatorial purposes. We compute improved and certified enclosures to these eigenvalues. This is achieved by applying the method of particular solutions in high precision, the enclosure being obtained by a combination of interval arithmetic and Taylor models. The index of the eigenvalue is certified by exploiting the monotonicity of the eigenvalue with respect to the domain. The classically troublesome case of singular corners is handled by combining expansions at all corners and an expansion from an interior point. In particular, this allows us to compute 100 digits of the fundamental eigenvalue for the 3D Kreweras model that has been the object of previous efforts.

math.NA↗

arXiv2020

Explicit degree bounds for right factors of linear differential operators

If a linear differential operator with rational function coefficients is reducible, its factors may have coefficients with numerators and denominatorsof very high degree. When the base field is $\mathbb C$, we give a completely explicit bound for the degrees of the monic right factors in terms of the degree and the order of the original operator, as well as the largest modulus of the local exponents at all its singularities. As a consequence, if a differential operator $L$ has rational function coefficients over a number field, we get degree bounds for its monic right factors in terms of the degree, the order and the height of $L$, and of the degree of the number field.

math.CA↗

arXiv2020

Effective Coefficient Asymptotics of Multivariate Rational Functions via Semi-Numerical Algorithms for Polynomial Systems

The coefficient sequences of multivariate rational functions appear in many areas of combinatorics. Their diagonal coefficient sequences enjoy nice arithmetic and asymptotic properties, and the field of analytic combinatorics in several variables (ACSV) makes it possible to compute asymptotic expansions. We consider these methods from the point of view of effectivity. In particular, given a rational function, ACSV requires one to determine a (generically) finite collection of points that are called critical and minimal. Criticality is an algebraic condition, meaning it is well treated by classical methods in computer algebra, while minimality is a semi-algebraic condition describing points on the boundary of the domain of convergence of a multivariate power series. We show how to obtain dominant asymptotics for the diagonal coefficient sequence of multivariate rational functions under some genericity assumptions using symbolic-numeric techniques. To our knowledge, this is the first completely automatic treatment and complexity analysis for the asymptotic enumeration of rational functions in an arbitrary number of variables.

cs.SC↗

arXiv2019

Absolute root separation

The absolute separation of a polynomial is the minimum nonzero difference between the absolute values of its roots. In the case of polynomials with integer coefficients, it can be bounded from below in terms of the degree and the height (the maximum absolute value of the coefficients) of the polynomial. We improve the known bounds for this problem and related ones. Then we report on extensive experiments in low degrees, suggesting that the current bounds are still very pessimistic.

math.CA↗