Finiteness of Pythagoras numbers of finitely generated real algebras

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 $P(A)<\infty$.