2019/09/01 by Alessio Borzì, Borzì, Alessio, Ivan Martino +1
Mathematics · #05B35 #13F05 #13F55 #14N20 #32S22 #52B40 #52C30 #52C35 #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1909.00332
openalex publication_date 2019/09/01 · openalex created_date 2019/09/12 · openalex updated_date 2026/07/28
In this work, we study matroids over a domain and several classical combinatorial and algebraic invariants related. We define their Grothendieck-Tutte polynomial TM(x,y), extending the definition given by Fink and Moci in 2016, and we show that such polynomial has the classical deletion-contraction property. Moreover, we study the set of independencies for a realizable matroid over a domain, generalizing the definition of poset of torsions Gr(M) given by the second author in 2017. This is a union of identical simplicial posets as for (quasi-)arithmetic matroids. The new notions harmonize naturally through the face module NM of the matroid over a domain. Whenever Gr(M) is a finite poset, the Hilbert series NM(t) of its face module is a specialization of the Tutte polynomial TM(x,y). Further, for arrangements of codimension-one abelian subvarities of an elliptic curve admitting complex multiplication, we extend certain results of Bibby and we provide an algebraic interpretation of the elliptic Tutte polynomial.