The main aim of this paper is to give classes of irreducible infinite dimensional representations and of irreducible *-representations of the q-deformed algebra U ′ q (so 2,2) which is a real form of the non-standard deformation U ′ q (so 4) of the universal enveloping algebra U (so(4, C)). These representations are described by two complex parameters and are obtained by " analytical continuati...

DEMiCs is a software package written in C++ for computing the mixed volume of the Newton polytopes of a general semi-mixed polynomial system through dynamic enumeration of all mixed cells. The underlying mixed cells play an essential role for computing all isolated zeros of a polynomial system by polyhedral homotopy continuation method. A notable feature of DEMiCs is in the construction of a dy...

We present a new algorithmic framework which utilizes tropical geometry and homotopy continuation for solving systems of polynomial equations where some of the polynomials are generic elements in linear subspaces of the polynomial ring. This approach generalizes the polyhedral homotopies by Huber and Sturmfels.

The problem of maximizing a non-negative generalized polynomial of degree at most p on the l p-sphere is shown to be equivalent to a concave one. Arguments where the maximum is attained are characterized in connection with the irreducible decomposition of the polynomial, and an application to the labelling problem is presented where these results are used to select the initial guess of a contin...

CMPSc is a C++ program for homotopy continuation methods to find all isolated solutions of a system of polynomial equations. The numerical methods for solving polynomial systems employed in CMPSc are equivalent to the ones used in a matlab program CMPSm. The focus of CMPSc is on implementation of the cheater’s homotopy. The software CMPSc, this manual and some numerical examples are available a...

PHoM is a software package in C++ for finding all isolated solutions of polynomial systems using a polyhedral homotopy continuation method. Among three modules constituting the package, the first module StartSystem constructs a family of polyhedrallinear homotopy functions, based on the polyhedral homotopy theory, from input data for a given system of polynomial equations f(x) = 0. The second m...

Numerical homotopy continuation of solutions to polynomial equations is the foundation for numerical algebraic geometry, whose development has been driven by applications of mathematics. We use numerical homotopy continuation to investigate the problem in pure mathematics of determining Galois groups in the Schubert calculus. For example, we show by direct computation that the Galois group of t...

Systems of polynomial equations arise throughout mathematics, engineering, and the sciences. It is therefore a fundamental problem both in mathematics and in application areas to find the solution sets of polynomial systems. The focus of this paper is to compare two fundamentally different approaches to computing and representing the solutions of polynomial systems: numerical homotopy continuat...

All isolated solutions of the cyclic-n polynomial equations are not known for larger dimensions than 8 except the dimensions 10 and 11. We exploit two types of symmetric structures in the cyclic-n polynomial to compute all isolated nonsingular solutions of the equations efficiently by the polyhedral homotopy continuation method and to verify the correctness of the generated approximate solution...

CMPSc is a C++ program for homotopy continuation methods to find all isolated solutions of a system of polynomial equations. The numerical methods for solving polynomial systems employed in CMPSc are equivalent to the ones used in a matlab program CMPSm. The focus of CMPSc is on implementation of the cheater’s homotopy. The software CMPSc, this manual and some numerical examples are available a...