1,025 research outputs found
Euler characteristic reciprocity for chromatic, flow and order polynomials
The Euler characteristic of a semialgebraic set can be considered as a
generalization of the cardinality of a finite set. An advantage of
semialgebraic sets is that we can define "negative sets" to be the sets with
negative Euler characteristics. Applying this idea to posets, we introduce the
notion of semialgebraic posets. Using "negative posets", we establish Stanley's
reciprocity theorems for order polynomials at the level of Euler
characteristics. We also formulate the Euler characteristic reciprocities for
chromatic and flow polynomials.Comment: 16 pages; flow polynomial reciprocity added; title change
Around matrix-tree theorem
Generalizing the classical matrix-tree theorem we provide a formula counting
subgraphs of a given graph with a fixed 2-core. We use this generalization to
obtain an analog of the matrix-tree theorem for the root system (the
classical theorem corresponds to the -case). Several byproducts of the
developed technique, such as a new formula for a specialization of the
multivariate Tutte polynomial, are of independent interest.Comment: 13 pages, no figure
Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs
The chromatic polynomial of a graph G counts the number of proper colorings
of G. We give an affirmative answer to the conjecture of Read and
Rota-Heron-Welsh that the absolute values of the coefficients of the chromatic
polynomial form a log-concave sequence. We define a sequence of numerical
invariants of projective hypersurfaces analogous to the Milnor number of local
analytic hypersurfaces. Then we give a characterization of correspondences
between projective spaces up to a positive integer multiple which includes the
conjecture on the chromatic polynomial as a special case. As a byproduct of our
approach, we obtain an analogue of Kouchnirenko's theorem relating the Milnor
number with the Newton polytope.Comment: Improved readability. Final version, to appear in J. Amer. Math. So
The maximum likelihood degree of a very affine variety
We show that the maximum likelihood degree of a smooth very affine variety is
equal to the signed topological Euler characteristic. This generalizes Orlik
and Terao's solution to Varchenko's conjecture on complements of hyperplane
arrangements to smooth very affine varieties. For very affine varieties
satisfying a genericity condition at infinity, the result is further
strengthened to relate the variety of critical points to the
Chern-Schwartz-MacPherson class. The strengthened version recovers the
geometric deletion-restriction formula of Denham et al. for arrangement
complements, and generalizes Kouchnirenko's theorem on the Newton polytope for
nondegenerate hypersurfaces.Comment: Improved readability. Final version, to appear in Compositio
Mathematic
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