Finiteness of Pythagoras numbers of finitely generated real algebras
Yi Ouyang, Qimin Song, Chenhao Zhang
TL;DR
The paper investigates when finitely generated real algebras have finite Pythagoras numbers, introducing a dimension-based criterion tied to the real locus. Using a real Bertini-type approach on integral real surfaces, it proves that certain open complements have infinite Pythagoras numbers, while deducing finiteness results when the real locus is small in dimension. It also derives consequences for unboundedness along integral curves and formulates a conjecture that $P(A)$ is finite precisely when the Zariski closure of the real spectrum has dimension $<2$, with multiple equivalent formulations in the two-dimensional setting. Overall, the work provides a geometric framework linking sums of squares in real algebras to the topology and dimension of real points, offering a clear dichotomy guiding finiteness versus infinity of $P(A)$ in terms of $W$.
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 $\widetilde{X}$ be the normalization of $X$, and let $s \in Γ(X,\mathcal{O}_X(1))$ be a nonzero section such that $\bigl(\widetilde{X}_{s=0}\bigr)^{\mathrm{red}}$ is formally real. We prove $P\bigl(Γ(X_{s\neq 0})\bigr)=\infty$. 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 $P(A)<\infty$.