Finiteness of Pythagoras numbers of finitely generated real algebras

Abstract

In this paper, we establish two finiteness results and propose a conjecture concerning the Pythagoras number $P(A)$ of a finitely generated real algebra $A$. Let $X\hookrightarrow {\mathbb{P}}^n$ be an integral projective surface over $\mathbb{R}$, let $\tilde{X}$ be the normalization of $X$, and let $s\in \Gamma (X,{\mathcal{O}}_X(1))$ be a nonzero section such that $({\tilde{X}}_{s=0}{)}^{\mathrm{red}}$ is formally real. We prove $P(\Gamma (X_{s\ne 0}))=\infty$. As a corollary, the Pythagoras numbers of integral smooth affine curves over $\mathbb{R}$ are shown to be unbounded. For any finitely generated $\mathbb{R}$-algebra $A$, if the Zariski closure of the real points of $\mathrm{Spec}(A)$ has dimension less than two, we demonstrate $P(A)<\infty$.

20 pages, comments welcome!