Equivariant algebraic and semi-algebraic geometry of infinite affine space

Abstract

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