27 novembre 2023 à 11 décembre 2023
Institut Henri Poincaré
Fuseau horaire Europe/Paris

Some problems I’d like solved, from a user of computer algebra by Alan Sokal

6 déc. 2023, 10:00
1h
Amphithéâtre Hermite / Darboux (Institut Henri Poincaré)

Amphithéâtre Hermite / Darboux

Institut Henri Poincaré

11 rue Pierre et Marie Curie 75005 Paris

Description

Abstract. A matrix M of real numbers is called {\em totally positive}\/ if every minor of M is nonnegative. Gantmakher and Krein showed in 1937 that a Hankel matrix H=(ai+j)i,j0 of real numbers is totally positive if and only if the underlying sequence (an)n0 is a Stieltjes moment sequence, i.e.~the moments of a positive measure on [0,). Moreover, this holds if and only if the ordinary generating function n=0antn can be expanded as a Stieltjes-type continued fraction with nonnegative coefficients. So totally positive Hankel matrices are closely connected with the Stieltjes moment problem and with continued fractions. Here I will introduce a generalization: a matrix M of polynomials (in some set of indeterminates) will be called {\em coefficientwise totally positive}\/ if every minor of M is a polynomial with nonnegative coefficients. And a sequence (an)n0 of polynomials will be called {\em coefficientwise Hankel-totally positive}\/ if the Hankel matrix H=(ai+j)i,j0 associated to (an) is coefficientwise totally positive. It turns out that many sequences of polynomials arising naturally in enumerative combinatorics are (empirically) coefficientwise Hankel-totally positive. In some cases this can be proven using continued fractions, by either combinatorial or algebraic methods; in other cases this can be done using a more general algebraic method called {\em production matrices}\/. However, in a very large number of other cases it remains an open problem. Along the way I will mention some problems in computer algebra, the solution of which would be helpful to this research.

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