Degree bounds for Putinar's Positivstellensatz on the hypercube
Lorenzo Baldi, Lucas Slot
TL;DR
This work studies Putinar's Positivstellensatz on the hypercube $[-1,1]^n$ and derives explicit degree bounds for Putinar-type representations. It proves an upper bound of the form $f \in \mathcal{Q}(1-x_1^2,\dots,1-x_n^2)_{rn}$ with $r \ge 4\mathfrak{c} d^2 (\log n) \frac{f_{\max}}{f_{\min}} + O\left(\frac{f_{\max}}{f_{\min}}\right)^{1/2}$, together with a matching lower bound $r = \Omega\left(\frac{f_{\max}}{f_{\min}}\right)^{1/8}$, establishing the first Putinar-type lower bound for a hypercube with interior. The approach combines an outer-domain Schmüdgen-type certificate on a scaled hypercube with degree-shift lifting to the Putinar module on $[-1,1]^n$, leveraging Markov-type inequalities and Chebyshev bounds. These results significantly improve prior general bounds (which were exponential or superpolynomial in the key ratio) and have direct implications for the convergence rate of the moment-SOS hierarchy in polynomial optimization on the hypercube. The paper also discusses extensions to descriptions using $1\pm x_i$, and provides constructive examples illustrating the inherent degree growth in Putinar certificates, along with connections to stability theory and potential avenues for improving bounds under local optimality conditions.
Abstract
The Positivstellensätze of Putinar and Schmüdgen 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 Positivstellensätze 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 Schmüdgen-type representations and not for Putinar-type ones. Complementing this upper degree bound, we show a lower degree bound in $Ω(\sqrt[8]{f_{\max}/f_{\min}})$. This is the first lower bound for Putinar-type representations on a semialgebraic set with nonempty interior described by a standard set of inequalities.