Tolerants

Swechchha Adhikari; Brent Hall; Stephen McKean

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arXiv · 2506.22897

Tolerants

Swechchha Adhikari·Brent Hall·Stephen McKean

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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.

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BibTeX RIS

Swechchha Adhikari, Brent Hall, Stephen McKean. 2025-06-28. Tolerants. https://arxiv.org/abs/2506.22897

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