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We give an optimal necessary and sufficient condition for the quotient polynomial and remainder in the division algorithm to have positive coefficients.
We generalize Sylvester single sums to multisets (sets with repeated elements), and show that these sums compute subresultants of two univariate polyomials as a function of their roots independently of their multiplicity structure. This is…
We apply a symbolic approach of the general quadratic decomposition of polynomial sequences - presented in a previous article referenced herein - to polynomial sequences fulfilling specific orthogonal conditions towards two given…
A general class of polynomial remainder codes is considered. Such codes are very flexible in rate and length and include Reed-Solomon codes as a special case. As an extension of previous work, two joint error-and-erasure decoding approaches…
In this paper, we present a new algorithm for computing the linear recurrence relations of multi-dimensional sequences. Existing algorithms for computing these relations arise in computational algebra and include constructing structured…
We study the growth behaviour of rational linear recurrence sequences. We show that for low-order sequences, divergence is decidable in polynomial time. We also exhibit a polynomial-time algorithm which takes as input a divergent rational…
The extended Euclidean algorithm (EEA) for polynomial greatest common divisors is commonly used in solving the key equation in the decoding of Reed-Solomon (RS) codes, and more generally in BCH decoding. For this particular application, the…
We consider two kinds of problems: the computation of polynomial and rational solutions of linear recurrences with coefficients that are polynomials with integer coefficients; indefinite and definite summation of sequences that are…
The purpose of this paper is twofold. An immediate practical use of the presented algorithm is its applicability to the parametric solution of underdetermined linear ordinary differential equations (ODEs) with coefficients that are…
Let $(p_n)_n$ be either the $q$-Meixner or the $q$-Laguerre polynomials. We form a new sequence of polynomials $(q_n)_n$ by considering a linear combination of two consecutive $p_n$: $q_n=p_n+\beta_np_{n-1}$, $\beta_n\in \RR$. Using the…
Motivated by a question of van der Poorten about the existence of infinite chain of prime numbers (with respect to some base), in this paper we advance the study of sequences of consecutive polynomials whose coefficients are chosen…
Multiple orthogonal polynomials satisfy a number of recurrence relations, in particular there is a $(r+2)$-term recurrence relation connecting the type II multiple orthogonal polynomials near the diagonal (the so-called step-line recurrence…
We say that the polynomial sequence $(Q^{(\lambda)}_n)$ is a semiclassical Sobolev polynomial sequence when it is orthogonal with respect to the inner product $$ <p, r>_S=<{{\bf u}} ,{p\, r}> +\lambda <{{\bf u}}, {{\mathscr D}p \,{\mathscr…
This is a companion paper to arXiv:2312.10772. We deduce an equidistribution theorem for periodic nilsequences and use this theorem to give two applications in arithmetic combinatorics. The first application is quasi-polynomial bounds for a…
We present an algorithm to solve a system of diagonal polynomial equations over finite fields when the number of variables is greater than some fixed polynomial of the number of equations whose degree depends only on the degree of the…
This paper presents a method to determine a set of basis polynomials from the extended Euclidean algorithm that allows Generalized Minimum Distance decoding of Reed-Solomon codes with a complexity of O(nd).
Polynomial multiplication is a fundamental problem in symbolic computation. There are efficient methods for the multiplication of two univariate polynomials. However, there is rarely efficiently nontrivial method for the multiplication of…
We develop a fast algorithm for computing the bound of an Ore polynomial over a skew field, under mild conditions. As an application, we state a criterion for deciding whether a bounded Ore polynomial is irreducible, and we discuss a…
It is classical that univariate algebraic functions satisfy linear differential equations with polynomial coefficients. Linear recurrences follow for the coefficients of their power series expansions. We show that the linear differential…
Consider an algorithm computing in a differential field with several commuting derivations such that the only operations it performs with the elements of the field are arithmetic operations, differentiation, and zero testing. We show that,…