Improved effective estimates of P\'olya's Theorem for quadratic forms

Following de Loera and Santos, the P'olya exponent of a $n$-ary real form (i.e. a homogeneous polynomial in $n$ variables with real coefficients) $f$ is the infimum of the upward closed set of nonnegative integers $m$ such that $(x_1+\cdots +x_n{)}^{m}f$ strictly has positive coefficients. By a theorem of P'olya, a form assumes only positive values over the standard $(n-1)$-simplex in Euclidean $n$-space if and only if its P'olya exponent is finite. In this note, we compute an upper bound of the P'olya exponent of a quadratic form $f$ that assumes only positive values over the standard simplex. Our bound improves a previous upper bound due to de Klerk, Laurent and Parrilo. For example, for the binary quadratic form $f_{\kappa }={\lambda }^2{x}_1^2-2\kappa \lambda x_1{x}_2+x_2^2$, which assumes only positive values over the standard $1$-simplex whenever $0\le \kappa <1<\lambda$, our upper bound of its P'olya’s exponent is $O(1\mathrm{/}\lambda )$ times that of de Klerk, Laurent and Parrilo’s as $\lambda$ tends to infinity.