Equivariant algebraic and semi-algebraic geometry of infinite affine space

Abstract

We study Sym(∞)-orbit closures of non-necessarily closed points in the Zariski spectrum of the infinite polynomial ring C[xij : i ∈ N, j ∈ [n]]. Among others, we characterize invariant prime ideals in this ring. Furthermore, we study projections of basic equivariant semi-algebraic sets defined by Sym(∞) orbits of polynomials in R[xij : i ∈ N, j ∈ [n]]. For n = 1 we prove a quantifier elimination type result which fails for n > 1.