Irreducibility of the symmetric Yagzhev's maps

TL;DR

This paper proves that for symmetric Jacobian polynomial maps of Yagzhev's form with nonzero Jacobian determinant, each component polynomial is irreducible over the complex polynomial ring.

Contribution

It establishes the irreducibility of component polynomials in symmetric Jacobian Yagzhev maps with nonzero Jacobian, extending understanding of their algebraic structure.

Findings

All component polynomials are irreducible in the complex polynomial ring.

The Jacobian determinant being nonzero is crucial for irreducibility.

Symmetry of the Jacobian matrix implies algebraic irreducibility of the map components.

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 $\mathbb{C}[x_1,\ld,x_n]$.