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# Mathematical Research Letters

## Volume 13 (2006)

### Number 6

### Sums of squares of linear forms

Pages: 947 – 956

DOI: https://dx.doi.org/10.4310/MRL.2006.v13.n6.a9

#### Authors

#### Abstract

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.

Published 1 January 2006