Let $K$ be a field. A polynomial $F \in \mathbb{Q}[X_1, \dots, X_r]$ which is weighted homogeneous with respect to the grading $\deg(X_k) = k$ is called numerically non-negative for nef vector bundles if for every 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 Positivity in Algebraic Geometry II 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.

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

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

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 numerically non-negative for nef vector bundles 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 Positivity in Algebraic Geometry II 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.

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

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