Let $K$ be a field. A polynomial $F \in \mathbb{Q}[X_1, \dots, X_r]$ which is weighted homogeneous of degree $n$ with respect to the grading $\deg(X_k) = k$ is called <i>numerically non-negative for nef vector bundles</i> if for every $n$-dimensional projective $K$-variety $X$ and every nef vector bundle $E$ of rank $r$ on $X$ we have $\int_X F(c_1(E), \dots, c_r(E)) \ge 0$. According to Example 8.3.10 in Lazarsfeld's book <i>Positivity in Algebraic Geometry II</i> a polynomial is numerically non-negative for nef vector bundles if and only if it is a non-negative linear combination of the Schur polynomials. However Lazarsfeld only works over the complex ground field.

<b>Question:</b> Is there a citeable reference where the fact stated above is stated (and proved) over an arbitrary ground field?

Note that the paper <i>Positive Polynomials for Ample Vector Bundles</i> by Fulton and Lazarsfeld allows an arbitrary ground field, but there only positive polyomials are considered which have a similar description.