Wednesday, 25 May 2011

nt.number theory - Nonnegative polynomial in two variables

The following theorem of Artin -- his solution of Hilbert's 17th problem, but in a stronger form than Hilbert himself asked for -- answers the question.



Theorem (Artin, 1927): Let F be a subfield of mathbbR that has a unique ordering, and let f(t)=f(t1,ldots,tn)inF(t1,ldots,tn) be a rational function such that
f(a)geq0 for all a=(a1,ldots,an)inFn for which f is defined. Then f is a sum of squares of rational functions with coefficients in F.



A proof can be found in Jacobson, Basic Algebra II, Section 11.4.



Note that the tempting strengthening -- that if f is a polynomial, it is a sum of squares of polynomials -- is false, as Hilbert himself showed.

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