Related papers: On Chudnovsky-Based Arithmetic Algorithms in Finit…
A solution for Smale's 17th problem, for the case of systems with bounded degree was recently given. This solution, an algorithm computing approximate zeros of complex polynomial systems in average polynomial time, assumed infinite…
We address the problem of the best uniform approximation by linear combinations of a finite system of functions. If the system is Chebyshev and the problem is unconstrained, then the classical Remez algorithm provides a fast and precise…
We study maximal subalgebras of an arbitrary finite dimensional algebra over a field, and obtain full classification/description results of such algebras. This is done by first obtaining a complete classification in the semisimple case, and…
We propose efficient parallel algorithms and implementations on shared memory architectures of LU factorization over a finite field. Compared to the corresponding numerical routines, we have identified three main difficulties specific to…
Exponential divided differences arise in numerical linear algebra, matrix-function evaluation, and quantum Monte Carlo simulations, where they serve as kernel weights for time evolution and observable estimation. Efficient and numerically…
This paper proposes novel approaches for designing control Lyapunov functions (CLFs) for constrained linear systems. We leverage recent configuration-constrained polyhedral computing techniques to devise piecewise affine convex CLFs.…
When modeling scientific and industrial problems, geometries are typically modeled by explicit boundary representations obtained from computer-aided design software. Unfitted (also known as embedded or immersed) finite element methods offer…
Permutation polynomials have been a subject of study for a long time and have applications in many areas of science and engineering. However, only a small number of specific classes of permutation polynomials are described in the literature…
Planar functions, introduced by Dembowski and Ostrom, are functions from a finite field to itself that give rise to finite projective planes. They exist, however, only for finite fields of odd characteristics. They have attracted much…
We study product sets of finite arithmetic progressions of polynomials over a finite field. We prove a lower bound for the size of the product set, uniform in a wide range of parameters. We apply our results to resolve the function field…
We give upper and lower bounds on the Chevalley-Bass number of a field of characteristic zero, whenever this quantity is well-defined. We also describe an algorithm which computes the Chevalley-Bass number of a field, provided its maximal…
We completely describe the functional graph associated to iterations of Chebyshev polynomials over finite fields. Then, we use our structural results to obtain estimates for the average rho length, average number of connected components and…
Greedy algorithms are a fundamental category of algorithms in mathematics and computer science, characterized by their iterative, locally optimal decision-making approach, which aims to find global optima. In this review, we will discuss…
We present an algorithm to perform a simultaneous modular reduction of several residues. This algorithm is applied fast modular polynomial multiplication. The idea is to convert the $X$-adic representation of modular polynomials, with $X$…
We present a new algorithm for finding isolated zeros of a system of real-valued functions in a bounded interval in $\mathbb{R}^n$. It uses the Chebyshev proxy method combined with a mixture of subdivision, reduction methods, and…
This paper presents an algorithm to simulate Gaussian random vectors whose precision matrix can be expressed as a polynomial of a sparse matrix. This situation arises in particular when simulating Gaussian Markov random fields obtained by…
We present two exact implementations of efficient output-sensitive algorithms that compute Minkowski sums of two convex polyhedra in 3D. We do not assume general position. Namely, we handle degenerate input, and produce exact results. We…
This paper proposes an efficient FDTD technique for determining electromagnetic fields interacting with a finite-sized 2D and 3D periodic structures. The technique combines periodic boundary conditions---modelling fields away from the edges…
We describe here a library aimed at automating the solution of partial differential equations using the finite element method. By employing novel techniques for automated code generation, the library combines a high level of expressiveness…
A general formula is presented for any order derivative of Chebyshev polynomials instead of the existing recursive relationship. Hence, the Chebyshev finite difference method is made applicable not only to second order problems but also to…