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arXiv · 2610.14506

Characterization of $\mathbb{A}^1$-Milnor numbers of small rank

C-Y. Jean Chan·Thomas Hagedorn·Joel Louwsma·Michael Wijaya
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Abstract

The $\mathbb{A}^1$-Milnor number of a polynomial map is a quadratic enrichment of the classical Milnor number, defined as the local $\mathbb{A}^1$-Brouwer degree of the gradient. We study which elements of the Grothendieck-Witt ring of nondegenerate symmetric bilinear forms arise as $\mathbb{A}^1$-Milnor numbers. Over fields of characteristic not $2$, $3$, or $5$, we show that every element of the Grothendieck-Witt ring of rank at most $7$ that is a local $\mathbb{A}^1$-Brouwer degree is also an $\mathbb{A}^1$-Milnor number. Unlike for local $\mathbb{A}^1$-Brouwer degrees, however, in general not all $\mathbb{A}^1$-Milnor numbers can be obtained from two-variable polynomials. Over an arbitrary field of characteristic not $2$, $3$, $5$, or $7$, we characterize the elements of the Grothendieck-Witt ring of rank at most $8$ that are $\mathbb{A}^1$-Milnor numbers of two-variable polynomials. Over subfields of $\mathbb{R}$, this implies that there are local $\mathbb{A}^1$-Brouwer degrees of ranks $4$, $6$, and $8$ that are not the $\mathbb{A}^1$-Milnor number of a two-variable polynomial. We also prove that under a mild hypothesis one can reduce the number of variables of a polynomial while preserving the $\mathbb{A}^1$-Milnor number up to multiplication by a unit.

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BibTeXRIS

C-Y. Jean Chan, Thomas Hagedorn, Joel Louwsma, Michael Wijaya. 2026-10-10. Characterization of $\mathbb{A}^1$-Milnor numbers of small rank. https://arxiv.org/abs/2610.14506

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