Tolerants

TL;DR

This paper introduces the tolerant, a generalization of the polynomial discriminant, explores its properties, formulates a conjecture related to Poincaré–Hopf, and compares it to the classical discriminant.

Contribution

It defines the tolerant, derives a rational formula for it, and investigates its properties and potential applications in motivic homotopy theory.

Findings

The tolerant is rational and expressible via discriminants.

It shares many properties with the discriminant, except for inversion invariance.

A conjectural unstable Poincaré–Hopf formula is proposed.

Abstract

We study a generalization of the discriminant of a polynomial, which we call the tolerant. The tolerant differs by multiplication by a square from the duplicant, which was discovered in recent work on $\mathbb{P}^1$-loop spaces in motivic homotopy theory. We show that the tolerant is rational by deriving a formula in terms of discriminants. This allows us to formulate a conjectural unstable Poincar\'e--Hopf formula over an arbitrary locus of points. We also show that the tolerant satisfies many of the same properties as the discriminant. A notable difference between the two is that the discriminant is inversion invariant for all polynomials, whereas the tolerant is only inversion invariant on a proper multiplicative subset of polynomials.