Submission: 2006, Apr 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.
2000 Mathematics Subject Classification: primary 11 E 25, secondary 13 J 15, 14 P 99, 15 A 63
Keywords and Phrases: sums of squares, quadratic forms, level, Pythagoras numbers, local henselian rings
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