Connect Four and Graph Decomposition

We introduce the standard decomposition, a way of decomposing a labeled graph into a sum of certain labeled subgraphs. We motivate this graph-theoretic concept by relating it to Connect Four decompositions of standard sets. We prove that all standard decompositions can be generated in polynomial time, which implies that all Connect Four decompositions can be generated in polynomial time

The Slice Algorithm For Irreducible Decomposition of Monomial Ideals

Irreducible decomposition of monomial ideals has an increasing number of applications from biology to pure math. This paper presents the Slice Algorithm for computing irreducible decompositions, Alexander duals and socles of monomial ideals. The paper includes experiments showing good performance in practice.Comment: 25 pages, 8 figures. See http://www.broune.com/ for the data use

Signature rewriting in gröbner basis computation

International audienceWe introduce the RB algorithm for Gro ̈bner basis compu- tation, a simpler yet equivalent algorithm to F5GEN. RB contains the original unmodified F5 algorithm as a special case, so it is possible to study and understand F5 by con- sidering the simpler RB. We present simple yet complete proofs of this fact and of F5's termination and correctness. RB is parametrized by a rewrite order and it contains many published algorithms as special cases, including SB. We prove that SB is the best possible instantiation of RB in the following sense. Let X be any instantiation of RB (such as F5). Then the S-pairs reduced by SB are always a subset of the S-pairs reduced by X and the basis computed by SB is always a subset of the basis computed by X

The Parametric Frobenius Problem

Given relatively prime positive integers a(1), ... , a(n), the Frobenius number is the largest integer that cannot be written as a nonnegative integer combination of the a(i). We examine the parametric version of this problem: given a(i) = a(i)(t) as functions of t, compute the Frobenius number as a function of t. A function f : Z(+) -\u3e Z is a quasi-polynomial if there exists a period m and polynomials f(0), ..., f(m-1) such that f(t) = f(t mod m)(t) for all t. We conjecture that, if the a(i)(t) are polynomials (or quasi-polynomials) in t, then the Frobenius number agrees with a quasi-polynomial, for sufficiently large t. We prove this in the case where the a(i)(t) are linear functions, and also prove it in the case where n (the number of generators) is at most 3