2016/01/04 by de Bondt, Michiel
#12E05 #12F20 #14R05 #14R10 #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.1601.00579
Let K be any field and x = (x1,x2,…,xn). We classify all matrices M ∈ \rm Matm,n(K[x]) whose entries are polynomials of degree at most 1, for which \rm rk M ≤ 2. As a special case, we describe all such matrices M, which are the Jacobian matrix J H (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 = J H has only two nonzero columns or only three nonzero rows in this case. In addition, we show that \rm trdegK K(H) = \rm rk J H for quadratic polynomial maps H over K such that \frac12 ∈ K and \rm rk J H ≤ 2. Furthermore, we prove that up to conjugation with linear maps over K, nilpotent Jacobian matrices N of quadratic polynomial maps, for which \rm rk N ≤ 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 \frac12 ∈ K and N(0) = 0.