Quadratic polynomial maps with Jacobian rank two

Abstract

Let K be any field and x=(x1,x2,…,xn). We classify all matrices M∈Matm,n(K[x]) whose entries are polynomials of degree at most 1, for which rkM≤2. As a special case, we describe all such matrices M, which are the Jacobian matrix JH (the matrix of partial derivatives) of a polynomial map H from Kn to Km.Among other things, we show that up to composition with linear maps over K, M=JH has only two nonzero columns or only three nonzero rows in this case. In addition, we show that trdegKK(H)=rkJH for quadratic polynomial maps H over K such that 12∈K and rkJH≤2.Furthermore, we prove that up to conjugation with linear maps over K, nilpotent Jacobian matrices N of quadratic polynomial maps, for which rkN≤2, are triangular (with zeroes on the diagonal), regardless of the characteristic of K. This generalizes several results by others.In addition, we prove the same result for Jacobian matrices N of quadratic polynomial maps, for which N2=0. This generalizes a result by others, namely the case where 12∈K and N(0)=0.