The irreducibility of symmetric Yagzhev maps

Abstract

Let F : C n → C n F:\mathbb {C}^n \rightarrow \mathbb {C}^n be a polynomial mapping in Yagzhev form, i.e. \[ F ( x 1 , … , x n ) = ( x 1 + H 1 ( x 1 , … , x n ) , … , x n + H n ( x 1 , … , x n ) ) , F(x_1,\ldots ,x_n)=(x_1+H_1(x_1,\ldots ,x_n),\ldots ,x_n+H_n(x_1,\ldots ,x_n)), \] where H i H_i are homogeneous polynomials of degree 3. We show that if J a c ( F ) ∈ C ∗ \mathrm {Jac}(F) \in \mathbb {C}^* and the Jacobian matrix of F F is symmetric, then the polynomials x i + H i ( x 1 , … , x n ) x_i+H_i(x_1,\ldots ,x_n) are irreducible as elements of the ring C [ x 1 , … , x n ] \mathbb {C}[x_1,\ldots ,x_n] .