Positivity of rational functions and their diagonals

Abstract

The problem to decide whether a given rational function in several variables is positive, in the sense that all its Taylor coefficients are positive, goes back to Szeg\H{o} as well as Askey and Gasper, who inspired more recent work. It is well known that the diagonal coefficients of rational functions are $D$-finite. This note is motivated by the observation that, for several of the rational functions whose positivity has received special attention, the diagonal terms in fact have arithmetic significance and arise from differential equations that have modular parametrization. In each of these cases, this allows us to conclude that the diagonal is positive. Further inspired by a result of Gillis, Reznick and Zeilberger, we investigate the relation between positivity of a rational function and the positivity of its diagonal.Comment: 16 page

Multivariate Apéry numbers and supercongruences of rational functions

“…The Askey-Gasper rational function (31), whose positivity is proved in [AG77] and [GRZ83], is an interesting instance of a rational function on the boundary of positivity (if the 4 is replaced by 4 + ε, for any ε > 0, then the resulting rational function is not positive). The present work was, in part, motivated by the observation [SZ14] that for several of the rational functions, which have been shown or conjectured to be on the boundary of positivity, the diagonal coefficients are arithmetically interesting sequences with links to modular forms. Note that the Askey-Gasper rational function (31) corresponds to the choice λ = (3) and α = −4 in Theorem 3.1, which makes its Taylor coefficients G(n) = A (3),−4 (n) explicit.…”

Diagonal asymptotics for symmetric rational functions via ACSV

“…In the cases d = 4, 5, 6, Kauers [13] proved nonnegativity of these diagonal coefficients by applying cylindrical algebraic decomposition (CAD) to the respective recurrences. On the other hand, it is suggested in [25] that the diagonal coefficients are eventually positive if c < (d − 1) d−1 .…”

“…To be precise, (iii) ⇒ (ii) ⇒ (i) is trivial (look at δ 1 ); nonnegativity of all coefficients of F c,d holds for some interval c ∈ [0, c max ], therefore the conjecture comes down to nonnegativity of [11] but omitted from the paper due to length. This question is generalized in [25] to all of M d . More specifically, with nonnegativity in place of positivity, the authors of that paper wonder whether positivity of F is equivalent to positivity of diag F together with positivity of F (x 1 , .…”

Minimization and Positivity of the Tensorial Rational Bernstein Form

“…The problem to decide whether a given rational function in n−variables is positive, in the sense that all its Bernstein coefficients are positive, goes back to [23,24]. The same problem was addressed over different domains by other authors in [2] and [21].…”