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

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.