Products of arithmetic matroids and quasipolynomial invariants of CW-complexes

Abstract In this note we prove that the product of two arithmetic multiplicity functions on a matroid is again an arithmetic multiplicity function. This allows us to answer a question by Bajo–Burdick–Chmutov [2] , concerning the modified Tutte–Krushkal–Renhardy polynomials defined by these authors. Furthermore, we show that the Tutte quasi-polynomial introduced by Branden and Moci encompasses invariants defined by Beck–Breuer–Godkin–Martin [3] and Duval–Klivans–Martin [11] and can thus be considered as a dichromate for CW complexes.

[1]  Felix Breuer,et al.  Enumerating colorings, tensions and flows in cell complexes , 2012, J. Comb. Theory, Ser. A.

[2]  Matthias Lenz.  Splines, lattice points, and arithmetic matroids , 2014, 1408.4041.

[3]  Luca Moci,et al.  Arithmetic matroids, the Tutte polynomial and toric arrangements , 2011 .

[4]  Luca Moci,et al.  Graph colorings, flows and arithmetic Tutte polynomial , 2013, J. Comb. Theory, Ser. A.

[5]  Emanuele Delucchi,et al.  Group actions on semimatroids , 2015, Adv. Appl. Math..

[6]  Art M. Duval,et al.  Simplicial and Cellular Trees , 2015, 1506.06819.

[7]  Luca Moci,et al.  Ehrhart polynomial and arithmetic Tutte polynomial , 2012, Eur. J. Comb..

[8]  Luca Moci,et al.  The multivariate arithmetic Tutte polynomial , 2012 .

[9]  Luca Moci.  A Tutte polynomial for toric arrangements , 2009 .

[10]  Akimichi Takemura,et al.  Periodicity of hyperplane arrangements with integral coefficients modulo positive integers , 2007, math/0703904.

[11]  Vyacheslav Krushkal,et al.  A Polynomial Invariant and Duality for Triangulations , 2010, Electron. J. Comb..

[12]  Bernt Lindström,et al.  Non-regular simplicial matroids , 1979, Discret. Math..

[13]  Alex Fink,et al.  Matroids Over a Ring , 2012 .

[14]  Gil Kalai,et al.  Enumeration ofQ-acyclic simplicial complexes , 1983 .

[15]  Art M. Duval,et al.  Cuts and flows of cell complexes , 2012, Journal of Algebraic Combinatorics.

[16]  W. T. Tutte,et al.  A Contribution to the Theory of Chromatic Polynomials , 1954, Canadian Journal of Mathematics.

[17]  Yvonne Kemper,et al.  Flows on Simplicial Complexes , 2012 .

[18]  Carlos Bajo,et al.  On the Tutte-Krushkal-Renardy polynomial for cell complexes , 2012, J. Comb. Theory, Ser. A.

[19]  Jim Lawrence,et al.  Characteristic polynomials, Ehrhart quasi-polynomials, and torus groups☆ , 2006 .