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.

Comments are welcome. An error in the stability bound in Theorem 1.2 is corrected thanks to the comments of an anonymous referee