Descartes Rule of Signs and Linear Programming

Abstract

Let $\Delta=\{(x,y):x+y=1, x,y \geq 0\}$ be the 1-simplex and for $m\geq 2$ consider the (binary) form

$$F(x,y)=u_n x^n +u_0 y^n -\sum_{\begin{array}{c} i, j\geq 1 \\ i+j=n\end{array}} u_i x^i y^j.$$

Using linear programming and a little known refinement of Descartes’ rule of signs due to Laguerre, it is shown that if all $u_i\geq 0$ and F is nonzero and nonnegative on $\Delta,$ then it assumes there exactly one global minimum. The investigation is motivated by a question concerning sum of squares representation.