Equivariant algebraic and semi-algebraic geometry of infinite affine space

We study Sym ( ∞ ) -orbit closures of non-necessarily closed points in the Zariski spectrum of the infinite polynomial ring C [ x i j : 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 [ x i j : i ∈ N , j ∈ [ n ] ] . For n = 1 we prove a quantifier elimination type result which fails for n > 1 .