Equivariant algebraic and semi-algebraic geometry of infinite affine space

Abstract

We study $\text{Sym}(\infty )$-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 $\text{Sym}(\infty )$ 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 $n>1$.