Tolerants

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.