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 $O(1/\lambda)$ times that of de Klerk, Laurent and Parrilo's as $\lambda$ tends to infinity.