Indecomposability of polynomials via Jacobian matrix (Q984987)

A multivariate polynomial is called decomposable if it can be written as a composition of a univariate polynomial of degree \(\geq 2\) and a multivariate polynomial. In the first section, deomposability is characterized by a rank condition of the so-called Jacobi-matrix of the polynomial. For characteristic zero, this criterion can be found in [\textit{Y. Stein}, Isr. J. Math. 68, No. 1, 109--122 (1989; Zbl 0716.12001)]. Here the result is generalized to positive characteristics. Based on this criterion, some classes of polynomials are shown to be indecomposable.