2019/11/03 by Alexander Levin, Levin, Alexander
Mathematics · #12H05 #Commutative Algebra (math.AC) #FOS: Mathematics #Rings and Algebras (math.RA) #math.AC #math.RA #msc:12H05
paper · pdf · doi:10.48550/arxiv.1911.00875
arxiv created 2019/11/03 · arxiv updated 2019/11/05
Let K be an inversive difference-differential field and L a (not necessarily inversive) finitely generated difference-differential field extension of K. We consider the natural filtration of the extension L/K associated with a finite system η of its difference-differential generators and prove that for any intermediate difference-differential field F, the transcendence degrees of the components of the induced filtration of F are expressed by a certain numerical polynomial χK, F,η(t). This polynomial is closely connected with the dimension Hilbert-type polynomial of a submodule of the module of Kähler differentials ΩL∗|K where L∗ is the inversive closure of L. We prove some properties of polynomials χK, F,η(t) and use them for the study of the Krull-type dimension of the extension L/K. In the last part of the paper, we present a generalization of the obtained results to multidimensional filtrations of L/K associated with partitions of the sets of basic derivations and translations.