This paper is devoted to the study of the separation number of a polynomial system f : C → C defining a zero-dimensional nonsingular variety. The separation number of f is defined by sep(f) = min ‖x− y‖ where the minimum is taken for x 6= y ∈ C with f(x) = f(y) = 0. In the case of a single variable polynomial f(x) = ad ∏d i=1(x − ri) various lower bounds for sep(f) have been proved and can be f...

We present an algorithm to decompose a polynomial system into a finite set of normal ascending sets such that the set of the zeros of the polynomial system is the union of the sets of the regular zeros of the normal ascending sets. If the polynomial system is 0-dimensional, the set of the zeros of the polynomials is the union of the sets of the zeros of the normal ascending sets.

Abstract. The number of real roots of a system of polynomial equations fitting inside a given box can be counted using a vector symmetric polynomial introduced by P. Milne, the volume function. We provide the expansion of Milne’s volume function in the basis of monomial vector symmetric functions, and observe that only monomial functions of a particular kind appear in the expansion, the squaref...

We propose a method for computing a perturbation bound that preserves the number of common zeros in (C×)n of a polynomial system (f1, . . . , fn), where C× = C \ {0} and fj ∈ C[x1, . . . , xn], by using Stetter’s results on the nearest polynomial with a given zero, Bernshtein’s theorem, and minimization techniques for rational functions such as sum of squares (SOS) relaxations.

This report is meant to serve as a working man s introduction to resultants It is aimed at engineers computer scientists and applied mathematicians who encounter systems of polynomial equations in their practice As such it is more of a how to manual than a theoretical study Most of the techniques and results presented here can be found in the classical literature Unfortunately those results are...

Problems of simulation and control in production and economic system (PES) under condition of economy’s modernization are discussed in the paper. The developed control system considers the coordination of three systems – PES, market and taxation institutional system. We take into account the PES features as dynamism, large number of parameters, nonlinearity, nonstationarity, strong interconnect...

We show several estimates on the probability distribution of some data at points in real complete intersection varieties: norms of real affine solutions, condition number of real solution of real systems of multivariate polynomial equations and convergence radius of Newton’s operator for under-determined system of multi-variate polynomial equations.

Polynomial chaos is a powerful technique for propagating uncertainty through ordinary and partial differential equations. Random variables are expanded in terms of orthogonal polynomials and differential equations are derived for the expansion coefficients. Here we study the structure and dynamics of these differential equations when the original system has Hamiltonian structure, has multiple t...

The security of many recently proposed cryptosystems is based on the difficulty of solving large systems of quadratic multivariate polynomial equations. The classical algorithm for solving such a system is Buchberger’s algorithm for constructing Gröbner bases. Another algorithm for solving such a system is XL algorithm. For sparse system, Buchberger’s algorithm benefits from sparsity of the sys...