Math ClubHilbert's 17th ProblemAnthony Rizzie(UConn)

Wednesday, October 2, 2019
5:45pm – 6:35pm

Storrs Campus
Monteith 226

A real number that is $$\geq 0$$ has to be a square of some real number, but a one-variable polynomial whose values are always $$\geq 0$$ need not be the square of a polynomial, e.g., $$x^2 - x + 1 \geq 0$$ for all $$x$$ but we can't write $$x^2 - x + 1 = f(x)^2$$ for a polynomial $$f(x)$$. However, $$x^2 - x + 1$$ is a sum of squares of polynomials:

$x^2 - x + 1 = (x-1/2)^2 + 3/4 = (x-1/2)^2 + (\sqrt{3}/2)^2.$

A sum of squares of polynomials has values that are always $$\geq 0$$. Conversely, if a polynomial has all of its values $$\geq 0$$, must it in fact be a sum of squares of polynomials? And what about polynomials in two or more variables? This question is the setting of Hilbert's 17th problem.

Note: Free pizza and drinks!

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