The irreducibility of symmetric Yagzhev maps

Abstract:

Let $F:\mathbb {C}^n \rightarrow \mathbb {C}^n$ be a polynomial mapping in Yagzhev form, i.e. \[ F(x_1,\ldots ,x_n)=(x_1+H_1(x_1,\ldots ,x_n),\ldots ,x_n+H_n(x_1,\ldots ,x_n)),\] where $H_i$ are homogeneous polynomials of degree 3. We show that if $\mathrm {Jac}(F) \in \mathbb {C}^*$ and the Jacobian matrix of $F$ is symmetric, then the polynomials $x_i+H_i(x_1,\ldots ,x_n)$ are irreducible as elements of the ring $\mathbb {C}[x_1,\ldots ,x_n]$.