Polynomials that are positive on an interval

Abstract:

This paper discusses representations of polynomials that are positive on intervals of the real line. An elementary and constructive proof of the following is given: If $h(x), p(x) \in \mathbb {R}[x]$ such that $\{ \alpha \in \mathbb {R} \mid h(\alpha ) \geq 0 \} = [-1,1]$ and $p(x) > 0$ on $[-1,1]$, then there exist sums of squares $s(x), t(x) \in \mathbb {R}[x]$ such that $p(x) = s(x) + t(x) h(x)$. Explicit degree bounds for $s$ and $t$ are given, in terms of the degrees of $p$ and $h$ and the location of the roots of $p$. This is a special case of Schmüdgen’s Theorem, and extends classical results on representations of polynomials positive on a compact interval. Polynomials positive on the non-compact interval $[0,\infty )$ are also considered.