Tolerants
Swechchha Adhikari, Brent Hall, Stephen McKean
TL;DR
Problem: generalize the discriminant via the tolerant tol, a square-normalized duplicant, in the context of motivic and algebraic geometry. Approach: define tol, establish translation/homothety invariances, prove rationality, derive an intrinsic coefficient formula using generalized discriminants and Hasse derivatives, and analyze inversion invariance. Contributions: tol is rational for all polynomials in $k[x]$, has an intrinsic resultant-based formula tol = ± gdisc(f), is not universally inversion invariant, and its inversion-invariant set is multiplicative; it also connects to a conjectural unstable Poincaré–Hopf formula at non-rational points. Significance: provides a tool parallel to the discriminant for non-separable cases and motivates further exploration of unstable local degrees in motivic homotopy theory.
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 Poincaré--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.