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Degree and algebraic properties of lattice and matrix ideals
O'Carroll, Liam; Planas Vilanova, Francesc d'Assís; Villarreal Rodríguez, Rafael Heraclio
Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada I; Universitat Politècnica de Catalunya. EGSA - Equacions Diferencials, Geometria, Sistemes Dinàmics i de Control, i Aplicacions
We study the degree of nonhomogeneous lattice ideals over arbitrary fields, and give formulas to compute the degree in terms of the torsion of certain factor groups of Z(s) and in terms of relative volumes of lattice polytopes. We also study primary decompositions of lattice ideals over an arbitrary field using the Eisenbud-Sturmfels theory of binomial ideals over algebraically closed fields. We then use these results to study certain families of integer matrices (positive critical binomial (PCB), generalized positive critical binomial (GPCB), critical binomial (CB), and generalized critical binomial (GCB) matrices) and the algebra of their corresponding matrix ideals. In particular, the family of GPCB matrices is shown to be closed under transposition, and previous results for PCB ideals are extended to GPCB ideals. Then, more particularly, we give some applications to the theory of 1-dimensional binomial ideals. If G is a connected graph, we show as a further application that the order of its sandpile group is the degree of the Laplacian ideal and the degree of the toppling ideal. We also use our earlier results to give a structure theorem for graded lattice ideals of dimension 1 in 3 variables and for homogeneous lattices in Z(3) in terms of CB ideals and CB matrices, respectively, thus complementing a well-known theorem of Herzog on the toric ideal of a monomial space curve.
Peer Reviewed
Àrees temàtiques de la UPC::Matemàtiques i estadística
lattice ideal
graded binomial ideal
primary decomposition
PCB ideal
Classificació AMS::13 Commutative rings and algebras::13F Arithmetic rings and other special rings
Classificació AMS::13 Commutative rings and algebras::13A General commutative ring theory
Classificació AMS::13 Commutative rings and algebras::13H Local rings and semilocal rings
Classificació AMS::13 Commutative rings and algebras::13P Computational aspects of commutative algebra
Attribution-NonCommercial-NoDerivs 3.0 Spain

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