A degree-linear upper bound for the Pólya exponent of forms with fixed arity
Abstract
Let P = ∑ω aωxω be a degree d homogeneous polynomial in x0, …, xn with real coefficients. Pólya showed that, if P is strictly positive on the standard n-simplex, then (x0 + ⋯ + xn)M P has strictly positive coefficients for all M ≫ 0. The least such M is known as the \emph{Pólya exponent} of P. de Klerk-Laurent-Parrilo showed that the Pólya exponent is O(d2) P(0)max/λ(P), where P(0)max := maxωaω/binomdω and λ(P) is the infimum of P on the standard simplex. Pólya's Positivstellensatz is a consequence of the classical approximation of the multivariate hypergeometric distribution by the multinomial distribution, as was already implicitly known to de Klerk-Laurent-Sun. By applying known estimates of this approximation from probability theory, we obtain a different On(d) L(P)/λ(P) upper bound of the Pólya exponent, where L(P) := max |aω|/binomdω. An example shows that On(d) is tight.
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BibTeX
@misc{li2026degreelinear,
title = {{A degree-linear upper bound for the Pólya exponent of forms with fixed arity}},
author = {Yulong Li and Colin Tan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14535},
eprint = {2610.14535},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14535},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - A degree-linear upper bound for the Pólya exponent of forms with fixed arity
AU - Yulong Li
AU - Colin Tan
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.14535
UR - https://arxiv.org/abs/2610.14535
AB - Let P = ∑_ω a_ωx^ω be a degree d homogeneous polynomial in x_0, …, x_n with real coefficients. Pólya showed that, if P is strictly positive on the standard n-simplex, then (x_0 + ⋯ + x_n)^M P has strictly positive coefficients for all M ≫ 0. The least such M is known as the \emph{Pólya exponent} of P. de Klerk-Laurent-Parrilo showed that the Pólya exponent is O(d^2) P^(0)_max/λ(P), where P^(0)_max := max_ωa_ω/binomdω and λ(P) is the infimum of P on the standard simplex. Pólya's Positivstellensatz is a consequence of the classical approximation of the multivariate hypergeometric distribution by the multinomial distribution, as was already implicitly known to de Klerk-Laurent-Sun. By applying known estimates of this approximation from probability theory, we obtain a different O_n(d) L(P)/λ(P) upper bound of the Pólya exponent, where L(P) := max |a_ω|/binomdω. An example shows that O_n(d) is tight.
ER -