Related papers: On The Criteria Of The F5 Algorithm
In bracket algebra, the calculation of invariant division and invariant Gr\"{o}bner basis proposed in \cite{li 2014} rely on straightening algorithm. Until now, there are at least three different types of straightening algorithms, among…
We consider ideals involving the maximal minors of a polynomial matrix. For example, those arising in the computation of the critical values of a polynomial restricted to a variety for polynomial optimisation. Gr\"obner bases are a…
Karger (SIAM Journal on Computing, 1999) developed the first fully-polynomial approximation scheme to estimate the probability that a graph $G$ becomes disconnected, given that its edges are removed independently with probability $p$. This…
Exploiting symmetry in Groebner basis computations is difficult when the symmetry takes the form of a group acting by automorphisms on monomials in finitely many variables. This is largely due to the fact that the group elements, being…
Recent efforts to improve the performance of neural network (NN) accelerators that meet today's application requirements have given rise to a new trend of logic-based NN inference relying on fixed-function combinational logic (FFCL). This…
This paper introduces a number of new techniques in the study of the famous question from numerical linear algebra: what is the largest possible growth factor when performing Gaussian elimination with complete pivoting? This question is…
We develop a very simple compensated scheme for computing very accurate Givens rotations. The approach is significantly more straightforward than the one in \cite{borges2021fast}, and the derivation leads to a very satisfying algorithm…
This work formalizes efficient Fast Fourier-based multiplication algorithms for polynomials in quotient rings such as $\mathbb{Z}_{m}[x]/\left<x^{n}-a\right>$, with $n$ a power of 2 and $m$ a non necessarily prime integer. We also present a…
Computing the critical points of a polynomial function $q\in\mathbb Q[X_1,\ldots,X_n]$ restricted to the vanishing locus $V\subset\mathbb R^n$ of polynomials $f_1,\ldots, f_p\in\mathbb Q[X_1,\ldots, X_n]$ is of first importance in several…
Rewriting for semigroups is a special case of Groebner basis theory for noncommutative polynomial algebras. The fact is a kind of folklore but is not fully recognised. The aim of this paper is to elucidate this relationship, showing that…
Randomized algorithms for low-rank matrix approximation are investigated, with the emphasis on the fixed-precision problem and computational efficiency for handling large matrices. The algorithms are based on the so-called QB factorization,…
We present a provably more efficient implementation of the Minimum Norm Point Algorithm conceived by Fujishige than the one presented in \cite{FUJI06}. The algorithm solves the minimization problem for a class of functions known as…
This manuscript describes a technique for computing partial rank-revealing factorizations, such as, e.g, a partial QR factorization or a partial singular value decomposition. The method takes as input a tolerance $\varepsilon$ and an…
Computation of the spherical harmonic rotation coefficients or elements of Wigner's d-matrix is important in a number of quantum mechanics and mathematical physics applications. Particularly, this is important for the Fast Multipole Methods…
We give new positive and negative results (some conditional) on speeding up computational algebraic geometry over the reals: (1) A new and sharper upper bound on the number of connected components of a semialgebraic set. Our bound is novel…
As the need for more accurate and powerful Convolutional Neural Networks (CNNs) increases, so too does the size, execution time, memory footprint, and power consumption. To overcome this, solutions such as pruning have been proposed with…
Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Gr{\"o}bner bases taking into account the valuation of K. Because of the use of the valuation, the theory of tropical Gr{\"o}bner bases…
Let T(x) in k[x] be a monic non-constant polynomial and write R=k[x] / (T) the quotient ring. Consider two bivariate polynomials a(x, y), b(x, y) in R[y]. In a first part, T = p^e is assumed to be the power of an irreducible polynomial p. A…
We develop efficient randomized algorithms to solve the black-box reconstruction problem for polynomials over finite fields, computable by depth three arithmetic circuits with alternating addition/multiplication gates, such that output gate…
Total degree reverse lexicographic order is currently generally regarded as most often fastest for computing Groebner bases. This article describes an alternate less mysterious algorithm for computing this order using exponent subtotals and…