Related papers: On the complexity of class group computations for …
Classes of polynomial differential equations of degree n are considered. An explicit upper bound on the size of the coefficients are given which implies that each equation in the class has exactly n complex periodic solutions. In most of…
We study the complexity of Gr\"obner bases computation, in particular in the generic situation where the variables are in simultaneous Noether position with respect to the system. We give a bound on the number of polynomials of degree $d$…
In previous works, we described algorithms to compute the number field cut out by the mod ell representation attached to a modular form of level N=1. In this article, we explain how these algorithms can be generalised to forms of higher…
Clustering is an unsupervised learning problem that aims to partition unlabelled data points into groups with similar features. Traditional clustering algorithms provide limited insight into the groups they find as their main focus is…
Lenstra and Guruswami described number field analogues of the algebraic geometry codes of Goppa. Recently, the first author and Oggier generalised these constructions to other arithmetic groups: unit groups in number fields and orders in…
Typestate systems ensure many desirable properties of imperative programs, including initialization of object fields and correct use of stateful library interfaces. Abstract sets with cardinality constraints naturally generalize typestate…
In this paper, approximation by means of algebraic polynomials of classes of functions defined by a generalised modulus of smoothness of operators of differentiation of these functions is considered. We give structural characteristics of…
Following Hasse's example, various authors have been deriving divisibility properties of minus class numbers of cyclotomic fields by carefully examining the analytic class number formula. In this paper we will show how to generalize these…
We consider a few modifications of the Big prime modular $\gcd$ algorithm for polynomials in $\Z[x]$. Our modifications are based on bounds of degrees of modular common divisors of polynomials, on estimates of the number of prime divisors…
In this paper we give a framework for describing how abstract systems can be used to compute if no randomness or error is involved. Using this we describe a class of classical "physical" computation systems whose computational capabilities…
In a recent paper, Kim and Kopparty (Theory of Computing, 2017) gave a deterministic algorithm for the unique decoding problem for polynomials of bounded total degree over a general grid. We show that their algorithm can be adapted to solve…
We investigate group coding for arbitrary finite groups acting linearly on a vector space. These yield robust codes based on real or complex matrix groups. We give necessary and sufficient conditions for correct subgroup decoding using…
We first show a simple but striking result in bilevel optimization: unconstrained $C^\infty$ smooth bilevel programming is as hard as general extended-real-valued lower semicontinuous minimization. We then proceed to a worst-case analysis…
By applying interpretable machine learning methods such as decision trees, we study how simple models can classify the Galois groups of Galois extensions over $\mathbb{Q}$ of degrees 4, 6, 8, 9, and 10, using Dedekind zeta coefficients. Our…
Computation of (approximate) polynomials common factors is an important problem in several fields of science, like control theory and signal processing. While the problem has been widely studied for scalar polynomials, the scientific…
We propose and rigorously analyze two randomized algorithms to factor univariate polynomials over finite fields using rank $2$ Drinfeld modules. The first algorithm estimates the degree of an irreducible factor of a polynomial from…
Given a square, nonsingular matrix of univariate polynomials $\mathbf{F}\in\mathbb{K}[x]^{n\times n}$ over a field $\mathbb{K}$, we give a deterministic algorithm for finding the determinant of $\mathbf{F}$. The complexity of the algorithm…
The problem of finding a nontrivial factor of a polynomial f(x) over a finite field F_q has many known efficient, but randomized, algorithms. The deterministic complexity of this problem is a famous open question even assuming the…
We propose new algorithms for computing triangular decompositions of polynomial systems incrementally. With respect to previous works, our improvements are based on a {\em weakened} notion of a polynomial GCD modulo a regular chain, which…
We give a new complexity bound for calculating the complex dimension of an algebraic set. Our algorithm is completely deterministic and approaches the best recent randomized complexity bounds. We also present some new, significantly sharper…