Equivariant algebraic and semi-algebraic geometry of infinite affine space

TL;DR

This paper explores the structure of orbit closures under the infinite symmetric group in algebraic and semi-algebraic geometry, characterizing invariant prime ideals and analyzing projections of orbit-defined sets.

Contribution

It provides a characterization of invariant prime ideals and studies projections of equivariant semi-algebraic sets, including a quantifier elimination result for the case n=1.

Findings

Characterization of invariant prime ideals in the infinite polynomial ring.

Analysis of projections of equivariant semi-algebraic sets.

Quantifier elimination result for the case n=1.

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 $n>1$.