A generalization of weight polynomials to matroids

Generalizing polynomials previously studied in the context of linear codes, we define weight polynomials and an enumerator for a matroid $M$. Our main result is that these polynomials are determined by Betti numbers associated with graded minimal free resolutions of the Stanley-Reisner ideals of $M$ and so-called elongations of $M$. Generalizing Greene's theorem from coding theory, we show that the enumerator of a matroid is equivalent to its Tutte polynomial.Comment: 21 page

A generalization of Kung's theorem

We give a generalization of Kung's theorem on critical exponents of linear codes over a finite field, in terms of sums of extended weight polynomials of linear codes. For all i=k+1,...,n, we give an upper bound on the smallest integer m such that there exist m codewords whose union of supports has cardinality at least i

Zero-free regions for multivariate Tutte polynomials (alias Potts-model partition functions) of graphs and matroids

The chromatic polynomial P_G(q) of a loopless graph G is known to be nonzero (with explicitly known sign) on the intervals (-\infty,0), (0,1) and (1,32/27]. Analogous theorems hold for the flow polynomial of bridgeless graphs and for the characteristic polynomial of loopless matroids. Here we exhibit all these results as special cases of more general theorems on real zero-free regions of the multivariate Tutte polynomial Z_G(q,v). The proofs are quite simple, and employ deletion-contraction together with parallel and series reduction. In particular, they shed light on the origin of the curious number 32/27.Comment: LaTeX2e, 49 pages, includes 5 Postscript figure

Some combinatorial invariants determined by Betti numbers of Stanley-Reisner ideals

The papers in this thesis are not available in Munin: Paper 1: Trygve Johnsen, Jan Roksvold, Hugues Verdure (2014): 'Betti numbers associated to the facet ideal of a matroid', available in Bulletin of the Brazilian Mathematical Society 45 no. 4, 727-744 Paper 2: Trygve Johnsen, Jan Roksvold, Hugues Verdure (2014): 'A generalization of weight polynomials to matroids', manuscript Paper 3: Jan Roksvold, Hugues Verdure (2015): 'Betti numbers of skeletons', manuscriptThe thesis contains new results on the connection between the algebraic properties of certain ideals of a polynomial ring and properties of error-correcting linear codes, matroids and simplicial complexes. We demonstrate that the graded Betti numbers of the facet ideal of a matroid are determined by the Betti numbers of the blocks of the matroid. The extended weight enumerator of coding theory is generalized to matroids. We show that this generalization is equivalent to the Tutte-polynomial, and that the coefficients of this polynomial is determined by Betti numbers of the Stanley-Reisner ideal of the matroid and its elongations. The Betti numbers of the Stanley-Reisner ring of a skeleton of a simplicial complex is demonstrated to be an integral linear combination of the Betti numbers associated to the original complex

Higher weight spectra of Veronese codes

We study q-ary linear codes C obtained from Veronese surfaces over finite fields. We show how one can find the higher weight spectra of these codes, or equivalently, the weight distribution of all extension codes of C over all field extensions of the field with q elements. Our methods will be a study of the Stanley-Reisner rings of a series of matroids associated to each code CComment: 14 page