Related papers: A Curious Case of Curbed Condition
Cylindrical Algebraic Decomposition (CAD) is a key tool in computational algebraic geometry, particularly for quantifier elimination over real-closed fields. However, it can be expensive, with worst case complexity doubly exponential in the…
The classical approach to system identification is based on stochastic assumptions about the measurement error, and provides estimates that have random nature. Worst-case identification, on the other hand, only assumes the knowledge of…
We describe a deterministic algorithm that computes an approximate root of n complex polynomial equations in n unknowns in average polynomial time with respect to the size of the input, in the Blum-Shub-Smale model with square root. It…
In this paper we study the computational feasibility of an algorithm to prove orbifold equivalence between potentials describing Landau-Ginzburg models. Through a comparison with leading results of Groebner basis computations in cryptology,…
We consider the task of verifying the correctness of quantum computation for a restricted class of circuits which contain at most two basis changes. This contains circuits giving rise to the second level of the Fourier Hierarchy, the lowest…
In the paper, we show that the transformations between modified Jacobi and Bernstein bases of the constrained space of polynomials of degree at most $n$ can be performed with the complexity $O(n^2)$. As a result, the algorithm of degree…
We give an elementary introduction to some recent polyhedral techniques for understanding and solving systems of multivariate polynomial equations. We provide numerous concrete examples and illustrations, and assume no background in…
This paper presents a regularization theory for numerical computation of polynomial greatest common divisors and a convergence analysis, along with a detailed description of a blackbox-type algorithm. The root of the ill-posedness in…
In this paper, we construct a family of Bernstein functions using a class of rational parametrization. The new family of rational Bernstein basis on an index $\alpha \in {\left(-\infty \, , \, 0 \right)}\cup {\left(1 \, , \,…
Many high-dimensional uncertainty quantification problems are solved by polynomial dimensional decomposition (PDD), which represents Fourier-like series expansion in terms of random orthonormal polynomials with increasing dimensions. This…
The expectation-maximization (EM) algorithm is an iterative method for finding maximum likelihood estimates when data are incomplete or are treated as being incomplete. The EM algorithm and its variants are commonly used for parameter…
The paper addresses parametric inequality systems described by polynomial functions in finite dimensions, where state-dependent infinite parameter sets are given by finitely many polynomial inequalities and equalities. Such systems can be…
Functions are a fundamental object in mathematics, with countless applications to different fields, and are usually classified based on certain properties, given their domains and images. An important property of a real-valued function is…
We introduce a new algorithm computing the characteristic polynomials of hyperplane arrangements which exploits their underlying symmetry groups. Our algorithm counts the chambers of an arrangement as a byproduct of computing its…
Given a family of rational curves depending on a real parameter, defined by its parametric equations, we provide an algorithm to compute a finite partition of the parameter space (${\Bbb R}$, in general) so that the shape of the family…
From celestial mechanics to quantum theory of atoms and molecules, perturbation theory has played a central role in natural sciences. Particularly in quantum mechanics, the amount of information needed for specifying the state of a…
The Bernstein-B\'ezier form of a polynomial is widely used in the fields of computer aided geometric design, spline approximation theory and, more recently, for high order finite element methods for the solution of partial differential…
Field experiments are often difficult and expensive to make. To bypass these issues, industrial companies have developed computational codes. These codes intend to be representative of the physical system, but come with a certain amount of…
In this paper we introduce a new family of Bernstein-type exponential polynomials on the hypercube $[0, 1]^d$ and study their approximation properties. Such operators fix a multidimensional version of the exponential function and its…
Polynomial system solving is a classical problem in mathematics with a wide range of applications. This makes its complexity a fundamental problem in computer science. Depending on the context, solving has different meanings. In order to…