2015/01/24 by de Bondt, Michiel
#12E05 #12F20 #13B22 #13N15 #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.1501.06046
We classify all rational maps H ∈ K(x)n for which \rm trdegK K(tH1,tH2,…,tHn) ≤ 2, where K is any field and t is another indeterminate. Furthermore, we classify all such maps for which additionally JH ⋅ H = \rm tr JH ⋅ H (where JH is the Jacobian matrix of H), i.e. ∑i=1n Hi (∂)/(∂ xi) Hk = ∑i=1n Hk (∂)/(∂ xi) Hi for all k ≤ n. This generalizes a theorem of Paul Gordan and Max Nöther, in which both sides and the characteristic of K are assumed to be zero. Besides this, we use some of our tools to obtain several results about K-subalgebras R of K(x) for which \rm trdegK L = 1, where L is the fraction field of R. We start with some observations about to what extent, Lüroth's theorem can be generalized.