Positive Univariate Polynomials: SOS certificates, algorithms, bit complexity, and T-systems
We consider certificates of positivity for univariate polynomials with rational coefficients that are positive over (an interval of)~$\mathbb{R}$. Such certificates take the form of weighted sums of squares (SOS) of polynomials with rational coefficients. We build on the algorithm of Chevillard, Harrison, Jolde{\c{s}}, and Lauter~\cite{chml-usos-alg-11}, and we introduce a variant that we refer to as \usos. Given a polynomial of degree~$d$ with maximum coefficient bitsize~$\tau$, we show that \usos computes a rational weighted SOS representation in $\widetilde{\mathcal{O}}_B(d^3 + d^2 \tau)$ bit operations; the resulting certificate of posivitity involves rationals of bitsize $\widetilde{\mathcal{O}}(d^2 \tau)$. This improves the best-known complexity bounds by a factor of~$d$ and completes previous analyses. We also extend these results to certificates of positivity over arbitrary rational intervals, via a simple transformation. In this case as well, our techniques yield a factor-$d$ improvement in the complexity bounds. Along the same line, for univariate polynomials with rational coefficients, we introduce a new class of certificates, which we call \emph{perturbed SOS certificates}. They consist of a sum of two rational squares that approximates the input polynomial closely enough so that nonnegativity of the approximation implies the nonnegativity of the original polynomial. This computation has the same bit complexity and yields certificates of the same bitsize as in the weighted SOS case. We further investigate structural properties of these SOS decompositions. Relying on the classical result that any nonnegative univariate real polynomial is the sum of two squares of real polynomials, we show that the summands form an interlacing pair. Consequently, their real roots correspond to the Karlin points of the original polynomial on~$\mathbb{R}$, establishing a new connection with the T-systems studied by Karlin~\cite{Karlin-repr-pos-63}. This connection enables us to compute such decompositions explicitly. Previously, only existential results were known for T-systems. We obtain analogous results for positivity over $(0, \infty)$, and hence over arbitrary real intervals. Finally, we present our open-source Maple implementation of the \usos algorithm, together with experiments on various data sets demonstrating the efficiency of our approach.