Irreducibility of the symmetric Yagzhev's maps

Abstract

Let $F:\Cn \to \Cn$ be a polynomial mapping in Yagzhev's form,i.e. $$F(x_1,\ld,x_n)=(x_1+H_1(x_1,\ld,x_n),\ld,x_n+H_n(x_1,\ld,x_n)),$$ where $H_i$ are homogenous polynomials of degree 3. In this paper we show that if $\Jac(F) \in \mathbb{C}^*$ and the Jacobian matrix of $F$ is symmetric, then all the polynomials $x_i+H_i(x_1,\ld,x_n)$ are irreducible as elements of the ring $\mathbb{C}[x_1,\ld,x_n]$.