Tolerants
Fuente:
arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866911328677920768 |
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| author | Adhikari, Swechchha Hall, Brent McKean, Stephen |
| author_facet | Adhikari, Swechchha Hall, Brent McKean, Stephen |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_22897 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tolerants Adhikari, Swechchha Hall, Brent McKean, Stephen Algebraic Geometry Commutative Algebra 13P15 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. |
| title | Tolerants |
| topic | Algebraic Geometry Commutative Algebra 13P15 |
| url | https://arxiv.org/abs/2506.22897 |