Given a polynomial of degree $2n$ over $\mathbb{Q}$, how to represent it as a linear combination (with rational coefficients) of squares of polynomials of degree at most $n$ over $\mathbb{Q}$ such that the number of polynomials is minimal?

In particular, when it is possible to represent a given polynomial of degree $2n$ as a linear combination of two squares of polynomials of degree at most $n$?