Related papers: Some Ideas to Test if a Polyhedron is Empty
We describe, study, and experiment with an algorithm for finding all solutions of systems of polynomial equations using homotopy continuation and monodromy. This algorithm follows a framework developed in previous work and can operate in…
This note presents some numerical examples worked out in order to show the reader how to implement, within a widely accessible computational setting, the methodology for achieving zero cancellation in linear multivariable systems discussed…
Effective methods are introduced for testing zero-dimensionality of varieties at a point. The motivation of this paper is to compute and analyze deformations of isolated hypersurface singularities. As an application, methods for computing…
It is proved that any polynomial vector field in two complex variables which is complete on a non-algebraic trajectory is complete.
A simple iteration methodology for the solution of a set of a linear algebraic equations is presented. The explanation of this method is based on a pure geometrical interpretation and pictorial representation. Convergence using this method…
Conjecturing and theorem proving are activities at the center of mathematical practice and are difficult to separate. In this paper, we propose a framework for completing incomplete conjectures and incomplete proofs. The framework can turn…
In this paper we describe a methodology for the identification of symmetric quadrature rules inside of quadrilaterals, triangles, tetrahedra, prisms, pyramids, and hexahedra. The methodology is free from manual intervention and is capable…
The analysis of solutions to algebraic equations is further simplified. A couple of functions and their analytic continuation or root findings are required.
This paper addresses the symbolic representation of non-convex real polyhedra, i.e., sets of real vectors satisfying arbitrary Boolean combinations of linear constraints. We develop an original data structure for representing such sets,…
The assumption of elliptical symmetry has an important role in many theoretical developments and applications, hence it is of primary importance to be able to test whether that assumption actually holds true or not. Various tests have been…
This short paper presents saturation-based algorithms for homogenization and elimination. This algorithm can compute elimination ideals by using syzygies and ideal membership test, hence it works with any} monomial order, in particular…
We consider the billiard map in a convex polyhedron of $\mathbb{R}^3$, and we prove that it is of zero topological entropy.
We give an incremental polynomial time algorithm for enumerating the vertices of any polyhedron $\mathcal{P}(A,\mathbf{1})=\{x\in\RR^n \mid Ax\geq \b1,~x\geq \b0\}$, when $A$ is a totally unimodular matrix. Our algorithm is based on…
We present a formalization of convex polyhedra in the proof assistant Coq. The cornerstone of our work is a complete implementation of the simplex method, together with the proof of its correctness and termination. This allows us to define…
We give a strongly polynomial time algorithm which determines whether or not a bivariate polynomial is real stable. As a corollary, this implies an algorithm for testing whether a given linear transformation on univariate polynomials…
We present simple, self-contained proofs of correctness for algorithms for linearity testing and program checking of linear functions on finite subsets of integers represented as n-bit numbers. In addition we explore a generalization of…
We prove that the Jacobian conjecture is false if and only if there exists a solution to a certain system of polynomial equations. We analyse the solution set of this system. In particular we prove that it is zero dimensional.
The polariton system is studied by a concise approach using a simple model. A new ground state with negative energy is obtained and found to exhibit the symmetry breaking.
We present a complete methodology for testing the performances of quantum tomography protocols. The theory is validated by several numerical examples and by the comparison with experimental results achieved with various protocols for whole…
This paper presents a new algorithm for the convex hull problem, which is based on a reduction to a combinatorial decision problem POLYTOPE-COMPLETENESS-COMBINATORIAL, which in turn can be solved by a simplicial homology computation. Like…