The Pythagoras number of function fields
Date:
Abstract: The Pythagoras number of a field $K$ is the smallest natural number n such that every sum of squares of elements of $K$ is a sum of $n$ squares of elements of $K$, or infinity, if such a natural number does not exist. Let us denote the Pythagoras number of $K$ by $p(K)$. Any non-zero natural number (and infinity) is the Pythagoras number of some field.
Very little is known about the behaviour of the Pythagoras number under field extensions, in particular how quickly and freely the Pythagoras number can grow. For example, when $L/K$ is a finite field extension, we only in general know that $p(L)$ is bounded by $[L : K]p(K)$ but in practice, we do not know of any example where $p(L) > p(K) + 2$ when $L/K$ is a finite field extension. A related open question is whether, when $p(K)$ is finite, then also $p(K(X))$ is finite, where $K(X)$ is a rational function field over $K$. We also do not know of any example where $p(K(X)) > p(K) + 2$.
In this talk, I discuss joint work with Karim Johannes Becher, David Grimm, Gonzalo Manzano-Flores, and Marco Zaninelli, in which we prove for an arbitrary field $K$ that, if $p(K(X)) = 2$ (the lowest possible value), then p(L) is at most 5 for any finite field extension $L$ of $K(X)$, thereby providing a step in the direction of this open question.
This talk was given at the Seminar on Arithmetic Geometry and Algebraic Groups. You may find the slides for this talk here and the recording here.