Related papers: Certifying zeros of polynomial systems using inter…
This paper presents two new constructions related to singular solutions of polynomial systems. The first is a new deflation method for an isolated singular root. This construc-tion uses a single linear differential form defined from the…
In this paper, a geometric resolution of singularities algorithm is developed. This method is elementary in its statement and proof, using explicit coordinate systems as much as possible. Each coordinate change used in the resolution…
In the field of scientific computing, one often finds several alternative software packages (with open or closed source code) for solving a specific problem. These packages sometimes even use alternative methodological approaches, e.g.,…
We show that existence of a global polynomial Lyapunov function for a homogeneous polynomial vector field or a planar polynomial vector field (under a mild condition) implies existence of a polynomial Lyapunov function that is a sum of…
Numerical nonlinear algebra is a computational paradigm that uses numerical analysis to study polynomial equations. Its origins were methods to solve systems of polynomial equations based on the classical theorem of B\'ezout. This was…
We establish the first complexity analysis for Krawczyk-based certified homotopy tracking. It consists of explicit a priori stepsize bounds ensuring the success of the Krawczyk test, and an iteration count bound proportional to the weighted…
Many seminal results in Interactive Proofs (IPs) use algebraic techniques based on low-degree polynomials, the study of which is pervasive in theoretical computer science. Unfortunately, known methods for endowing such proofs with zero…
This paper deals with the algorithmic aspects of solving feasibility problems of semidefinite programming (SDP), aka linear matrix inequalities (LMI). Since in some SDP instances all feasible solutions have irrational entries, numerical…
On the one hand, ordered completion is a fundamental technique in equational theorem proving that is employed by automated tools. On the other hand, their complexity makes such tools inherently error prone. As a remedy to this situation we…
Numerical solutions of differential equations are usually not smooth functions. However, they should resemble the smoothness of the corresponding real solutions in one way or another. In two of our recent papers, a kind of spacial…
We use our recent implementation of a certified homotopy tracking algorithm to search for start systems that minimize the average complexity of finding all roots of a regular system of polynomial equations. While finding optimal start…
A polynomial identity testing algorithm must determine whether a given input polynomial is identically equal to 0. We give a deterministic black-box identity testing algorithm for univariate polynomials of the form $\sum_{j=0}^t c_j…
The goal of this paper is to provide computational tools able to find a solution of a system of polynomial inequalities. The set of inequalities is reformulated as a system of polynomial equations. Three different methods, two of which…
By a numerical continuation method called a diagonal homotopy we can compute the intersection of two positive dimensional solution sets of polynomial systems. This paper proposes to use this diagonal homotopy as the key step in a procedure…
A complete method is proposed to compute a certified, or ambient isotopic, meshing for an implicit algebraic surface with singularities. By certified, we mean a meshing with correct topology and any given geometric precision. We propose a…
We present a certified algorithm that takes a smooth algebraic curve in $\mathbb{R}^n$ and computes an isotopic approximation for a generic projection of the curve into $\mathbb{R}^2$. Our algorithm is designed for curves given implicitly…
We propose a conservative algorithm to test the geometrical validity of simplicial (triangles, tetrahedra), tensor product (quadrilaterals, hexahedra), and mixed (prisms) elements of arbitrary polynomial order as they deform over a…
Smale's alpha-theory certifies that Newton iterations will converge quadratically to a solution of a square system of analytic functions based on the Newton residual and all higher order derivatives at the given point. Shub and Smale…
We propose a general methodology for testing whether a given polynomial with integer coefficients is identically zero. The methodology evaluates the polynomial at efficiently computable approximations of suitable irrational points. In…
While automatically generated polynomial elimination templates have sparked great progress in the field of 3D computer vision, there remain many problems for which the degree of the constraints or the number of unknowns leads to…