Related papers: Groebner bases in Java with applications in comput…
The theme of symbolic computation in algebraic categories has become of utmost importance in the last decade since it enables the automatic modeling of modern algebra theories. On this theoretical background, the present paper reveals the…
In this article we establish an exponential lower bound on the Graver complexity of integer programs. This provides new type of evidence supporting the presumable intractability of integer programming. Specifically, we show that the Graver…
In this article we discuss a data structure, which combines advantages of two different ways for representing graphs: adjacency matrix and collection of adjacency lists. This data structure can fast add and search edges (advantages of…
This paper develops an algorithm that identifies and decomposes a median graph of a triangulation of a 2-dimensional (2D) oriented bordered surface and in addition restores all corresponding triangulation whenever they exist. The algorithm…
We report on the computation of invariants, covariants, and contravariants of cubic surfaces. All algorithms are implemented in the computer algebra system magma.
The complexity of Gr\"{o}bner computations has inspired many improvements to Buchberger's algorithm over the years. Looking for further insights into the algorithm's performance, we offer a threaded implementation of classical Buchberger's…
This work studies certain aspects of graphs embedded on surfaces. Initially, a colored graph model for a map of a graph on a surface is developed. Then, a concept analogous to (and extending) planar graph is introduced in the same spirit as…
Two fundamental questions in the theory of Groebner bases are decision ("Is a basis G of a polynomial ideal a Groebner basis?") and transformation ("If it is not, how do we transform it into a Groebner basis?") This paper considers the…
In this paper we present the first-ever computer formalization of the theory of Gr\"obner bases in reduction rings, which is an important theory in computational commutative algebra, in Theorema. Not only the formalization, but also the…
We develop numerical homotopy algorithms for solving systems of polynomial equations arising from the classical Schubert calculus. These homotopies are optimal in that generically no paths diverge. For problems defined by hypersurface…
We apply the theory of Groebner bases to the computation of free resolutions over a polynomial ring, the defining equations of a canonically embedded curve, and the unirationality of the moduli space of curves of a fixed genus.
Computing the diameter of the intersection graphs of objects is a basic problem in computational geometry. Previous works showed that the complexity of computing the diameter mainly depends on the object types: for unit disks and squares in…
In modeling physical systems it is sometimes useful to construct border bases of 0-dimensional polynomial ideals which are contained in the ideal generated by a given set of polynomials. We define and construct such subideal border bases,…
In this work we present the notion of greyscale of a graph as a colouring of its vertices that uses colours from the real interval [0,1]. Any greyscale induces another colouring by assigning to each edge the non-negative difference between…
We describe a general algorithm for computing intersection pairings on arithmetic surfaces. We have implemented our algorithm for curves over $\mathbb Q$, and we show how to use it to compute regulators for a number of Jacobians of smooth…
The boundary of the convex hull of a compact algebraic curve in real 3-space defines a real algebraic surface. For general curves, that boundary surface is reducible, consisting of tritangent planes and a scroll of stationary bisecants. We…
To integer programming problems, computational algebraic approaches using Grobner bases or standard pairs via the discreteness of toric ideals have been studied in recent years. Although these approaches have not given improved time…
The problem of mathematical modeling in geography is one of the most important strategies in order to establish the evolution and the prevision of geographical phenomena. Models must have a simplified structure, to reflect essential…
We detail a procedure for the computation of the polynomial form of an electronic combinational circuit from the design equations in a truth table. The method uses the Buchberger algorithm rather than current traditional methods based on…
The article proposes a new method for finding the triangle-triangle intersection in 3D space, based on the use of computer graphics algorithms -- cutting off segments on the plane when moving and rotating the beginning of the coordinate…