Stability in the cohomology of the space of complex irreducible polynomials in several variables

Abstract

We prove that the space of complex irreducible polynomials of degree $d$ in $n$ variables satisfies two forms of homological stability: first, its cohomology stabilizes as $d$ increases, and second, its compactly supported cohomology stabilizes as $n$ increases. Our topological results are inspired by counting results over finite fields due to Carlitz and Hyde.