We study algebraic algorithms for expressing the number of non-negative integer solutions to a unimodular system of linear equations as a function of the right hand side. Our methods include Todd classes of toric varieties via Gröbner bases, and rational generating functions as in Barvinok's algorithm. We report polyhedral and computational results for two special cases: counting contingency tables and Kostant's partition function.
Imperial College London（帝国理工学院）；Technische Universität Berlin（柏林工业大学）；Max-Planck Institute for Mathematics in the Sciences, Leipzig（莱比锡马克斯·普朗克科学计量学研究所）；University of Bayreuth（拜罗伊特大学）
The set of all m-tuples of compatible full conditional distributions on discrete random variables is an algebraic set whose defining ideal is a unimodular toric ideal. We identify the defining polynomials of these ideals with closed walks on a bipartite graph. Our algebraic characterization provides a natural generalization of the requirement that compatible conditionals have identical odds ratios and holds regardless of the patterns of zeros in the conditional arrays.
Helmke et al. have recently given a formula for the number of reachable pairs of matrices over a finite field. We give a new and elementary proof of the same formula by solving the equivalent problem of determining the number of so called zero kernel pairs over a finite field. We show that the problem is equivalent to certain other enumeration problems and outline a connection with some recent results of Guo and Yang on the natural density of rectangular unimodular matrices over $\Fq[x]$. We also propose a new conjecture on the density of unimodular matrix polynomials.
Unimodular triangulations of lattice polytopes arise in algebraic geometry, commutative algebra, integer programming and, of course, combinatorics. In this article, we review several classes of polytopes that do have unimodular triangulations and constructions that preserve their existence. We include, in particular, the first effective proof of the classical result by Knudsen-Mumford-Waterman stating that every lattice polytope has a dilation that admits a unimodular triangulation. Our proof yields an explicit (although doubly exponential) bound for the dilation factor.
Tropical geometry is a piecewise linear "shadow" of algebraic geometry. It allows for the computation of several cohomological invariants of an algebraic variety. In particular, its application to enumerative algebraic geometry led to significant progress. In this survey, we give an introduction to tropical geometry techniques for algebraic curve counting problems. We also survey some recent developments, with a particular emphasis on the computation of the degree of the Severi varieties of the complex projective plane and other toric surfaces as well as Hurwitz numbers and applications to rea…
A lattice polytope is "free" (or "empty") if its vertices are the only lattice points it contains. In the context of valuation theory, Klain (1999) proposed to study the functions $α_i(P;n)$ that count the number of free polytopes in $nP$ with $i$ vertices. For $i=1$, this is the famous Ehrhart polynomial. For $i > 3$, the computation is likely impossible and for $i=2,3$ computationally challenging. In this paper, we develop a theory of coprime Ehrhart functions, that count lattice points with relatively prime coordinates, and use it to compute $α_2(P;n)$ for unimodular simplices. We show that…
We discuss the problem to count, or, more modestly, to estimate the number f(m,n) of unimodular triangulations of the planar grid of size $m\times n$. Among other tools, we employ recursions that allow one to compute the (huge) number of triangulations for small m and rather large n by dynamic programming; we show that this computation can be done in polynomial time if m is fixed, and present computational results from our implementation of this approach. We also present new upper and lower bounds for large m and n, and we report about results obtained from a computer simulation of the random …
We consider weighted small step walks in the positive quadrant, and provide algebraicity and differential transcendence results for the underlying generating functions: we prove that depending on the probabilities of allowed steps, certain of the generating functions are algebraic over the field of rational functions, while some others do not satisfy any algebraic differential equation with rational function coefficients. Our techniques involve differential Galois theory for difference equations as well as complex analysis (Weierstrass parameterization of elliptic curves). We also extend to th…
We prove that $n$ plane algebraic curves determine $O(n^{(k+2)/(k+1)})$ points of $k$-th order tangency. This generalizes an earlier result of Ellenberg, Solymosi, and Zahl on the number of (first order) tangencies determined by $n$ plane algebraic curves.
We provide a short combinatorial proof of Cayley's formula by means of a bijective map to an outcome space of an urn-drawing problem. Furthermore we introduce an algebraic structure on the set of labeled trees, which provides a more standard approach to Cayley's formula. Moreover, this algebraic structure sheds light on the problem of counting the unlabeled trees. In particular, it indicates how counting the number of unlabeled trees on $n$ vertices is connected to finding the number of partitions of $n-2$
For a finite abstract simplicial complex G with n sets, define the n x n matrix K(x,y) which is the number of subsimplices in the intersection of x and y. We call it the counting matrix of G. Similarly as the connection matrix L which is L(x,y)=1 if x and y intersect and 0 else, the counting matrix K is unimodular. Actually, K is always in SL(n,Z). The inverse of K has the Green function entries K^(-1)(x,y)=w(x) w(y) |W^+(x) intersected W^+y|, where W^+(x) is the star of x, the sets in G which contain x and w(x)=(-1)^dim(x). The matrix K is always positive definite. The spectra of K and K^(-1)…
In this paper, we study the basic problem of counting independent sets in a graph and, in particular, the problem of counting antichains in a finite poset, from an algebraic perspective. We show that neither independence polynomials of bipartite Cohen-Macaulay graphs nor Hilbert series of initial ideals of radical zero-dimensional complete intersections ideals, can be evaluated in polynomial time, unless #P=P. Moreover, we present a family of radical zero-dimensional complete intersection ideals J_P associated to a finite poset P, for which we describe a universal Gröbner basis. This implies t…
We study the problem of counting the total number of affine solutions of a system of n binomials in n variables over an algebraically closed field of characteristic zero. We show that we may decide in polynomial time if that number is finite. We give a combinatorial formula for computing the total number of affine solutions (with or without multiplicity) from which we deduce that this counting problem is #P-complete. We discuss special cases in which this formula may be computed in polynomial time; in particular, this is true for generic exponent vectors.
The Ehrhart polynomial $L_P$ of an integral polytope $P$ counts the number of integer points in integral dilates of $P$. Ehrhart polynomials of polytopes are often described in terms of their Ehrhart $h^*$-vector (aka Ehrhart $δ$-vector), which is the vector of coefficients of $L_P$ with respect to a certain binomial basis and which coincides with the $h$-vector of a regular unimodular triangulation of $P$ (if one exists). One important result by Stanley about $h^*$-vectors of polytopes is that their entries are always non-negative. However, recent combinatorial applications of Ehrhart theory …
We give estimates on the number of combinatorial designs, which prove (and generalise) a conjecture of Wilson from 1974 on the number of Steiner Triple Systems. This paper also serves as an expository treatment of our recently developed method of Randomised Algebraic Construction: we give a simpler proof of a special case of our result on clique decompositions of hypergraphs, namely triangle decompositions of quasirandom graphs.
The slope variety of a graph is an algebraic set whose points correspond to drawings of a graph. A complement-reducible graph (or cograph) is a graph without an induced four-vertex path. We construct a bijection between the zeroes of the slope variety of the complete graph on $n$ vertices over $\mathbb{F}_2$, and the complement-reducible graphs on $n$ vertices.
We study the problem #IndSub(P) of counting all induced subgraphs of size k in a graph G that satisfy the property P. This problem was introduced by Jerrum and Meeks and shown to be #W[1]-hard when parameterized by k for some families of properties P including, among others, connectivity [JCSS 15] and even- or oddness of the number of edges [Combinatorica 17]. Very recently [IPEC 18], two of the authors introduced a novel technique for the complexity analysis of #IndSub(P), inspired by the "topological approach to evasiveness" of Kahn, Saks and Sturtevant [FOCS 83] and the framework of graph m…
We show that, for any fixed genus $g$, the ordinary generating function for the genus $g$ partitions of an $n$-element set into $k$ blocks is algebraic. The proof involves showing that each such partition may be reduced in a unique way to a primitive partition and that the number of primitive partitions of a given genus is finite. We illustrate our method by finding the generating function for genus $2$ partitions, after identifying all genus $2$ primitive partitions, using a computer-assisted search.
A recent paper of Church, Ellenberg, and Farb uses topology and representation theory of the symmetric group to prove enumerative results about square-free polynomials and F-stable maximal tori of the general linear group over the algebraic closure of F_q. In this note, we use generating functions to give elementary proofs of some of their results, and some extensions.
