Characterization of 𝔸^1-Milnor numbers of small rank

Abstract

The 𝔸¹-Milnor number of a polynomial map is a quadratic enrichment of the classical Milnor number, defined as the local 𝔸¹-Brouwer degree of the gradient. We study which elements of the Grothendieck-Witt ring of nondegenerate symmetric bilinear forms arise as 𝔸¹-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 𝔸¹-Brouwer degree is also an 𝔸¹-Milnor number. Unlike for local 𝔸¹-Brouwer degrees, however, in general not all 𝔸¹-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 𝔸¹-Milnor numbers of two-variable polynomials. Over subfields of ℝ, this implies that there are local 𝔸¹-Brouwer degrees of ranks $4$, $6$, and $8$ that are not the 𝔸¹-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 𝔸¹-Milnor number up to multiplication by a unit.

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BibTeX RIS

BibTeX

@misc{chan2026characterization,
  title = {{Characterization of $\mathbbA^1$-Milnor numbers of small rank}},
  author = {C-Y. Jean Chan and Thomas Hagedorn and Joel Louwsma and Michael Wijaya},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.14506},
  eprint = {2610.14506},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.14506},
}

RIS

TY  - UNPB
TI  - Characterization of 𝔸^1-Milnor numbers of small rank
AU  - C-Y. Jean Chan
AU  - Thomas Hagedorn
AU  - Joel Louwsma
AU  - Michael Wijaya
PY  - 2026
DA  - 2026/10/10
PB  - arXiv
DO  - 10.48550/arxiv.2610.14506
UR  - https://arxiv.org/abs/2610.14506
AB  - The 𝔸^1-Milnor number of a polynomial map is a quadratic enrichment of the classical Milnor number, defined as the local 𝔸^1-Brouwer degree of the gradient. We study which elements of the Grothendieck-Witt ring of nondegenerate symmetric bilinear forms arise as 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 𝔸^1-Brouwer degree is also an 𝔸^1-Milnor number. Unlike for local 𝔸^1-Brouwer degrees, however, in general not all 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 𝔸^1-Milnor numbers of two-variable polynomials. Over subfields of ℝ, this implies that there are local 𝔸^1-Brouwer degrees of ranks $4$, $6$, and $8$ that are not the 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 𝔸^1-Milnor number up to multiplication by a unit.
ER  -

Characterization of 𝔸¹-Milnor numbers of small rank

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