# Degree bounds for Putinar's Positivstellensatz on the hypercube

## TL;DR

This paper establishes both upper and lower degree bounds for Putinar's Positivstellensatz representations of polynomials positive on the hypercube, advancing understanding of the complexity of polynomial positivity certificates.

## Contribution

It provides the first known degree bounds for Putinar-type representations on the hypercube, linking the degree to the ratio of maximum and minimum of the polynomial.

## Key findings

- Upper degree bound of O(f_max / f_min) for Putinar representations.
- Lower degree bound of Ω((f_max / f_min)^{1/8}) for these representations.
- First bounds of this kind for Putinar representations on a set with nonempty interior.

## Abstract

The Positivstellens\"atze of Putinar and Schm\"udgen show that any polynomial $f$ positive on a compact semialgebraic set can be represented using sums of squares. Recently, there has been large interest in proving effective versions of these results, namely to show bounds on the required degree of the sums of squares in such representations. These effective Positivstellens\"atze have direct implications for the convergence rate of the celebrated moment-SOS hierarchy in polynomial optimization. In this paper, we restrict to the fundamental case of the hypercube $\mathrm{B}^{n} = [-1, 1]^n$. We show an upper degree bound for Putinar-type representations on $\mathrm{B}^{n}$ of the order $O(f_{\max}/f_{\min})$, where $f_{\max}$, $f_{\min}$ are the maximum and minimum of $f$ on $\mathrm{B}^{n}$, respectively. Previously, specialized results of this kind were available only for Schm\"udgen-type representations and not for Putinar-type ones. Complementing this upper degree bound, we show a lower degree bound in $\Omega(\sqrt[8]{f_{\max}/f_{\min}})$. This is the first lower bound for Putinar-type representations on a semialgebraic set with nonempty interior described by a standard set of inequalities.