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arXiv · 2203.11921

Equivariant algebraic and semi-algebraic geometry of infinite affine space

Mario Kummer·Cordian Riener
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Abstract

We study $\textrm{Sym}(\infty)$-orbit closures of not necessarily closed points in the Zariski spectrum of the infinite polynomial ring $\mathbb{C}[x_{ij}:\, i\in\mathbb{N},\,j\in[n]]$. Among others, we characterize invariant prime ideals in this ring. Furthermore, we study projections of basic equivariant semi-algebraic sets defined by $\textrm{Sym}(\infty)$ orbits of polynomials in $\mathbb{R}[x_{ij}:\, i\in\mathbb{N},\,j\in[n]]$. For $n=1$ we prove a quantifier elimination type result which fails for $n>1$.

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Mario Kummer, Cordian Riener. 2022-03-22. Equivariant algebraic and semi-algebraic geometry of infinite affine space. https://doi.org/10.1016/j.jalgebra.2024.11.016

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