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 of real numbers is called {\em totally positive}\/ if every minor of is nonnegative. Gantmakher and Krein showed in 1937 that a Hankel matrix of real numbers is totally positive if and only if the underlying sequence is a Stieltjes moment sequence, i.e.~the moments of a positive measure on . Moreover, this holds if and only if the ordinary generating function 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 of polynomials (in some set of indeterminates) will be called {\em coefficientwise totally positive}\/ if every minor of is a polynomial with nonnegative coefficients. And a sequence of polynomials will be called {\em coefficientwise Hankel-totally positive}\/ if the Hankel matrix associated to 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.