Systems of polynomial equations are a common occurrence in problem formulations in
engineering, science, and mathematics. Solution sets of such systems, ie, algebraic sets, are …

Since a real univariate polynomial does not always have real roots, a very natural
algorithmic problem, is to design a method to count the number of real roots of a given …

We present a new algorithm for solving basic parametric constructible or semi-algebraic
systems of the form C={x∈ Cn, p1 (x)= 0,…, ps (x)= 0, f1 (x)≠ 0,…, fl (x)≠ 0} or S={x∈ Rn …

Let f1, ldots, fs be polynomials in Q [X1,..., Xn] that generate a radical ideal and let V be their
complex zero-set. Suppose that V is smooth and equidimensional; then we show that …

McCallum's projection operator for cylindrical algebraic decomposition (CAD) represented a
huge step forward for the practical utility of the CAD algorithm. This paper presents a simple …

A roadmap for a semi-algebraic set S is a curve which has a non-empty and connected
intersection with all connected components of S. Hence, this kind of object, introduced by …

We present a hybrid symbolic-numeric algorithm for certifying a polynomial or rational
function with rational coefficients to be non-negative for all real values of the variables by …

This is the first in a series of two tutorial articles devoted to triangulation-decomposition
algorithms. The value of these notes resides in the uniform presentation of triangulation …

Given a polynomial system f, a fundamental question is to determine if f has real roots. Many
algorithms involving the use of infinitesimal deformations have been proposed to answer …

We initiate a study of the Euclidean distance degree in the context of sparse polynomials.
Specifically, we consider a hypersurface f= 0 f= 0 defined by a polynomial f that is general …