Real Algebraic and Analytic Geometry |

Sums of squares of linear forms.

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homepages: http://www.uam.es/josefrancisco.fernando,
http://www.mat.ucm.es/~jesusr,
http://www.math.uni-konstanz.de/~scheider/index.html

Submission: 2006, April 22.

*Abstract:
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.

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