Euler characteristic of the space of real multivariate irreducible polynomials

TL;DR

This paper calculates the Euler characteristic of the space of real irreducible polynomials in multiple variables, revealing a surprising connection to the balanced binary expansion of the number of variables.

Contribution

It introduces a novel link between algebraic geometry and binary number representations by computing the Euler characteristic for these polynomial spaces.

Findings

Euler characteristic values correspond to digits in the balanced binary expansion of n

Provides explicit formulas for the Euler characteristic in terms of binary expansion

Establishes a new connection between topology and number theory in polynomial spaces

Abstract

We compute the compactly supported Euler characteristic of the space of degree $d$ irreducible polynomials in $n$ variables with real coefficients and show that the values are given by the digits in the so-called balanced binary expansion of the number of variables $n$.