  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.
