A parallel Buchberger algorithm for multigraded ideals

We demonstrate a method to parallelize the computation of a Gr\"obner basis for a homogenous ideal in a multigraded polynomial ring. Our method uses anti-chains in the lattice $\mathbb N^k$ to separate mutually independent S-polynomials for reduction.Comment: 8 pages, 6 figure

Presentations of Finitely Generated Cancellative Commutative Monoids and Nonnegative Solutions of Systems of Linear Equations

Varying methods exist for computing a presentation of a finitely generated commutative cancellative monoid. We use an algorithm of Contejean and Devie [An efficient incremental algorithm for solving systems of linear diophantine equations, Inform. and Comput. 113 (1994) 143–172] to show how these presentations can be obtained from the nonnegative integer solutions to a linear system of equations. We later introduce an alternate algorithm to show how such a presentation can be efficiently computed from an integer basis

Effective Power Series Computations

International audienc

Ideals modulo p

The main focus of this paper is on the problem of relating an ideal I in the polynomial ring Q[x_1,..., x_n] to a corresponding ideal in F_p[x_1, ..., x_n] where p is a prime number; in other words, the reduction modulo p of I. We define a new notion of sigma-good prime for I which depends on the term ordering sigma, and show that all but finitely many primes are good for all term orderings. We relate our notion of sigma-good primes to some other similar notions already in the literature. One characteristic of our approach is that enables us to detect some bad primes, a distinct advantage when using modular methods

Koszul algebras and Donaldson-Thomas invariants

For a given symmetric quiver $Q$, we define a supercommutative quadratic algebra $\mathcal{A}_Q$ whose Poincar\'e series is related to the motivic generating function of $Q$ by a simple change of variables. The Koszul duality between supercommutative algebras and Lie superalgebras assigns to the algebra $\mathcal{A}_Q$ its Koszul dual Lie superalgebra $\mathfrak{g}_Q$. We prove that the motivic Donaldson-Thomas invariants of the quiver $Q$ may be computed using the Poincar\'e series of a certain Lie subalgebra of $\mathfrak{g}_Q$ that can be described, using an action of the first Weyl algebra on $\mathfrak{g}_Q$, as the kernel of the operator $\partial_t$. This gives a new proof of positivity for motivic Donaldson--Thomas invariants. In addition, we prove that the algebra $\mathcal{A}_Q$ is numerically Koszul for every symmetric quiver $Q$ and conjecture that it is in fact Koszul; we also prove this conjecture for quivers of a certain class.Comment: 25 pages, the main result on DT invariants of symmetric quivers is now not conditional on Koszulnes

Veronese and Segre morphisms between non-commutative projective spaces

Number theory, Algebra and Geometr

数学と計算機科学の相互作用：計算代数とHaskellにおける安全性と拡張性

筑波大学 (University of Tsukuba)201