2009/04/22 by Miguel Ángel Marco-Buzunariz, Marco-Buzunariz, Miguel Ángel, Jorge Martín-Morales +1
Mathematics · #13N15 #32S25 #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #msc:13N15 #msc:32S25
paper · pdf · doi:10.48550/arxiv.0904.3465
arxiv created 2009/04/22 · arxiv updated 2009/12/01
Let Q∈ \K[x1,...,xn] = S be a homogeneous polynomial of degree d. The freeness of the logarithmic derivation module, D(Q), and of its natural generalizations, has been widely studied. In the free case, D(Q) ≃ \bigoplusi=1n S(-di) where the di's are the exponents of the module; and as a direct consequence of the Saito-Ziegler criterion, the formula d = ∑i di holds. In this paper we give a generalization of this formula in the non-free case. Moreover, we show that an equivalent formula is also true in the quasi-homogeneous case, and show to what extent it can be generalized for arbitrary polynomials.