### Math ClubHilbert's 14th ProblemMihai Fulger (UConn)

Wednesday, September 25, 2019
5:45pm – 6:35pm

Storrs Campus
Monteith 226

A polynomial $$f(x,y)$$ in $$x$$ and $$y$$ is called symmetric if it is unchanged by swapping $$x$$ and $$y$$: $$f(y,x) = f(x,y)$$. Examples include the sum and product $$x+y$$ and $$xy$$, and it turns out that every symmetric polynomial in $$x$$ and $$y$$ is a polynomial in $$x+y$$ and $$xy$$. For example, $$x^4+y^4$$ is symmetric and here it is in terms of $$x+y$$ and $$xy$$: $x^4 + y^4 = (x+y)^4 - 4(xy)(x+y)^2 + 2(xy)^2.$ We say the set of all symmetric polynomials in $$x$$ and $$y$$ is finitely generated, with generators $$x+y$$ and $$xy$$.

Hilbert's 14th problem essentially asks whether something like this (finite generatedness) is true for the polynomials in $$n$$ variables (not just $$n = 2$$) that are unchanged by any set of invertible transformations (not just swapping $$x$$ and $$y$$).

The answer to this question is sometimes yes, but in general is no. We will describe some important cases where the answer is yes and some counterexamples for the general case.

Note: Free pizza and drinks!

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