Euler characteristic of the space of real multivariate irreducible polynomials

Abstract

Given a field K and integers d, n ≥ 1, let Poly d,n (K) denote the set of all monic total degree d polynomials in n variables with coefficients in K (see Definition 2.1.) Let Irr d,n (K) ⊆ Poly d,n (K) denote the subset of all polynomials which are irreducible over K. Our main result expresses the compactly supported Euler characteristic of Irr d,n (R) in terms of the so-called balanced binary expansion of the number of variables n. The balanced binary expansion of an integer n ≥ 1 is the unique expressionwhere 0 ≤ k 0 < k 1 < . . . < k 2m is a strictly increasing sequence of natural numbers of even length and the signs on the right hand side alternate. Theorem 1.1. Let n ≥ 1 be an integer with balanced binary expansion n = ℓ k=0 b k 2 k and let χ c denote the compactly supported Euler characteristic. Then χ c (Irr d,n (R)) = b k if d = 2 k , 0 otherwise. Example 1.2. The balanced binary expansion of n = 13 is 13 = 2 4 − 2 2 + 2 1 − 1. Thus Theorem 1.1 implies that χ c (Irr d,13 (R)) =      1 if d = 2 or 16, −1 if d = 1 or 4, 0 otherwise.