Finiteness of Pythagoras numbers of finitely generated real algebras
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917008304504832 |
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| author | Ouyang, Yi Song, Qimin Zhang, Chenhao |
| author_facet | Ouyang, Yi Song, Qimin Zhang, Chenhao |
| contents | 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 $P(A)<\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_17751 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finiteness of Pythagoras numbers of finitely generated real algebras Ouyang, Yi Song, Qimin Zhang, Chenhao Number Theory Rings and Algebras 14P05, 14P25, 26C05, 26C99 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 $P(A)<\infty$. |
| title | Finiteness of Pythagoras numbers of finitely generated real algebras |
| topic | Number Theory Rings and Algebras 14P05, 14P25, 26C05, 26C99 |
| url | https://arxiv.org/abs/2503.17751 |