Sums of squares of linear forms

Let $k$ be a real field. We show that every non-negative homogeneous quadratic polynomial $f(x_1,\dots,x_n)$ with coefficients in the polynomial ring $k[t]$ is a sum of $2n\cdot\tau(k)$ squares of linear forms, where $\tau(k)$ is the supremum of the levels of the finite non-real field extensions of $k$. From this result we deduce bounds for the Pythagoras numbers of affine curves over fields, and of excellent two-dimensional local henselian rings.

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Published 1 January 2006