Real Algebraic and Analytic Geometry

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196. José F. Fernando, Jesús M. Ruiz, Claus Scheiderer:
Sums of squares of linear forms.

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Submission: 2006, April 22.

Let k be a real field. We show that every non-negative homogeneous quadratic polynomial f(x1,...,xn) with coefficients in the polynomial ring k[t] is a sum of 2n.\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.

Mathematics Subject Classification (2000): 11 E 25, 13 J 15, 14 P 99, 15 A 63.

Keywords and Phrases: sums of squares, quadratic forms, level, Pythagoras numbers, local henselian rings.

Full text, 9p.: dvi 52k, ps.gz 146k, pdf 182k.

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