In this paper, a linear univariate representation for the roots of a zero-dimensional polynomial equation system is presented, where the roots of the equation system are represented as linear combinations of roots of several univariate polynomial equations. The main advantage of this representation is that the precision of the roots can be easily controlled. In fact, based on the linear univari...

For any polynomial P ∈ C[X1, X2, ..., Xn], we describe a C-vector space F (P ) of solutions of a linear system of equations coming from some algebraic partial differential equations such that the dimension of F (P ) is the number of irreducible factors of P . Moreover, the knowledge of F (P ) gives a complete factorization of the polynomial P by taking gcd’s. This generalizes previous results b...

Accessible set functors can be presented by signatures and equations as quotients of polynomial functors. We determine how preservation of pullbacks and other related properties (often applied in coalgebra) are re ected in the structure of the system of equations.

An algorithm is designed which tests solvability of a system of k polynomial equations in n variables with degrees d within complexity polynomial in nd 3k . If a systems is solvable then the algorithm yields one of its solutions. Thus, for fixed d, k the complexity of the algorithm is polynomial.

In this paper, we suggest and analyze some iterative methods for solving nonlinear equations using the homotopy perturbation technique coupled with system of equations. In addition, we show that the homotopy polynomial is exactly the the Adomian polynomial and this establishes the equivalence between the homotopy perturbation and decomposition techniques. Our method of establishing the equivale...

We consider chemical reaction networks taken with mass-action kinetics. The steady states of such a system are solutions to a system of polynomial equations. Even for small systems the task of finding the solutions is daunting. We develop an algebraic framework and procedure for linear elimination of variables. The procedure reduces the variables in the system to a set of “core” variables by el...

A lot of dynamical properties of a linear system can be naturally expressed in terms of the positions of its poles. That is why the problem of pole placement has become so popular in control. In scalar linear systems, it is sufficient to assign just the characteristic polynomial. In multi-input multi-output systems, however, one must assign separately all the invariant polynomials (and not mere...

This article is divided in two parts. In the first part we review some recent results concerning the expected number of real roots of random system of polynomial equations. In the second part we deal with a different problem, namely, the distribution of the roots of certain complex random polynomials. We discuss a recent result in this direction, which shows that the associated points in the sp...

By a numerical continuation method called a diagonal homotopy, one can compute the intersection of two irreducible positive dimensional solution sets of polynomial systems. This paper proposes to use this diagonal homotopy as the key step in a procedure to intersect general solution sets that are not necessarily irreducible or even equidimensional. Of particular interest is the special case whe...

The problem of decoding up to error correcting capacity of arbitrary linear codes with the use of Gröbner bases is addressed. A new method is proposed, which is based on reducing an initial decoding problem to solving some system of polynomial equations over a finite field. The peculiarity of this system is that, when we want to decode up to half the minimum distance, it has a unique solution e...