Related papers: Sparse trace tests
We present a probabilistic algorithm to compute the product of two univariate sparse polynomials over a field with a number of bit operations that is quasi-linear in the size of the input and the output. Our algorithm works for any field of…
Polynomial multiplication is known to have quasi-linear complexity in both the dense and the sparse cases. Yet no truly linear algorithm has been given in any case for the problem, and it is not clear whether it is even possible. This…
We propose a new approach to studies on partial Steiner triple systems consisting in determining complete graphs contained in them. We establish the structure which complete graphs yield in a minimal PSTS that contains them. As a by-product…
In this paper, we develop a new deflation technique for refining or verifying the isolated singular zeros of polynomial systems. Starting from a polynomial system with an isolated singular zero, by computing the derivatives of the input…
Motivated by a connection with the factorization of multivariate polynomials, we study integral convex polytopes and their integral decompositions in the sense of the Minkowski sum. We first show that deciding decomposability of integral…
We develop a collection of numerical algorithms which connect ideas from polyhedral geometry and algebraic geometry. The first algorithm we develop functions as a numerical oracle for the Newton polytope of a hypersurface and is based on…
First a formula for the number of zeros of the orthogonal polynomial in the intervals is presented. Then a criteria about the appearance of a zero in a gap is given. Finally a necessary and sufficient condition is derived such that the…
An algorithm is presented for the efficient and accurate computation of the coefficients of the characteristic polynomial of a general square matrix. The algorithm is especially suited for the evaluation of canonical traces in determinant…
We give a survey concerning both very classical and recent results on the electrostatic interpretation of the zeros of some well-known families of polynomials, and the interplay between these models and the asymptotic distribution of their…
We consider the numerical irreducible decomposition of a positive dimensional solution set of a polynomial system into irreducible factors. Path tracking techniques computing loops around singularities connect points on the same irreducible…
Univariate polynomial root-finding is a classical subject, still important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the…
The multivariate resultant is a fundamental tool of computational algebraic geometry. It can in particular be used to decide whether a system of n homogeneous equations in n variables is satisfiable (the resultant is a polynomial in the…
We develop randomized matrix-free algorithms for estimating partial traces, a generalization of the trace arising in quantum physics and chemistry. Our algorithm improves on the typicality-based approach used in [T. Chen and Y-C. Cheng,…
We investigate the problem of safety verification of infinite-state parameterized programs that are formed based on a rich class of topologies. We introduce a new proof system, called parametric proof spaces, which exploits the underlying…
Quantile regression is studied in combination with a penalty which promotes structured (or group) sparsity. A mixed $\ell_{1,\infty}$-norm on the parameter vector is used to impose structured sparsity on the traditional quantile regression…
The pooled data problem asks to identify the unknown labels of a set of items from condensed measurements. More precisely, given $n$ items, assume that each item has a label in $\cbc{0,1,\ldots, d}$, encoded via the ground-truth $\SIGMA$.…
We establish sharp estimates that adapt the polynomial method to arbitrary varieties. These include a partitioning theorem, estimates on polynomials vanishing on fixed sets and bounds for the number of connected components of real algebraic…
Traces and their extension called combined traces (comtraces) are two formal models used in the analysis and verification of concurrent systems. Both models are based on concepts originating in the theory of formal languages, and they are…
Given a straight-line program whose output is a polynomial function of the inputs, we present a new algorithm to compute a concise representation of that unknown function. Our algorithm can handle any case where the unknown function is a…
We develop a method for determining the density of squarefree values taken by certain multivariate integer polynomials that are invariants for the action of an algebraic group on a vector space. The method is shown to apply to the…