Characteristic integrals in 3D and linear degeneracy

Conservation laws vanishing along characteristic directions of a given system of PDEs are known as characteristic conservation laws, or characteristic integrals. In 2D, they play an important role in the theory of Darboux-integrable equations. In this paper we discuss characteristic integrals in 3D and demonstrate that, for a class of second order linearly degenerate dispersionless integrable PDEs, the corresponding characteristic integrals are parametrised by points on the Veronese variety

On proving the absence of oscillations in models of genetic circuits

International audienceUsing computer algebra methods to prove that gene regulatory networks cannot oscillate appears to be easier than expected. We illustrate this claim on a family of models related to historical examples

Decision Problems for Second-Order Holonomic Recurrences

We study decision problems for sequences which obey a second-order holonomic recurrence of the form f(n + 2) = P(n) f(n + 1) + Q(n) f(n) with rational polynomial coefficients, where P is non-constant, Q is non-zero, and the degree of Q is smaller than or equal to that of P. We show that existence of infinitely many zeroes is decidable. We give partial algorithms for deciding the existence of a zero, positivity of all sequence terms, and positivity of all but finitely many sequence terms. If Q does not have a positive integer zero then our algorithms halt on almost all initial values (f(1), f(2)) for the recurrence. We identify a class of recurrences for which our algorithms halt for all initial values. We further identify a class of recurrences for which our algorithms can be extended to total ones

On the problem of proper reparametrization for rational curves and surfaces

A rational parametrization of an algebraic curve (resp. surface) establishes a rational correspondence of this curve (resp. surface) with the affine or projective line (resp. affine or projective plane). This correspondence is a birational equivalence if the parametrization is proper. So, intuitively speaking, a rational proper parametrization trace the curve or surface once. We consider the problem of computing a proper rational parametrization from a given improper one. For the case of curves we generalize, improve and reinterpret some previous results. For surfaces, we solve the problem for some special surface's parametrizations

A partial solution to the problem of proper reparametrization for rational surfaces

Given an algebraically closed field K, and a rational parametrization P of an algebraic surface V ⊂ K3 , we consider the problem of computing a proper rational parametrization Q from P (reparametrization problem). More precisely, we present an algorithm that computes a rational parametrization Q of V such that the degree of the rational map induced by Q is less than the degree induced by P. The properness of the output parametrization Q is analyzed. In particular, if the degree of the map induced by Q is one, then Q is proper and the reparametrization problem is solved. The algorithm works if at least one of two auxiliary parametrizations defined from P is not proper

A univariate resultant based implicitation algorithm for surfaces

In this paper, we present a new algorithm for computing the implicit equation of a rational surface V from a rational parametrization P(t). The algorithm is valid independent of the existence of base points, and is based on the computation of polynomial gcds and univariate resultants. Moreover, we prove that the resultant-based formula provides a power of the implicit equation. In addition, performing a suitable linear change of parameters, we prove that this power is indeed the degree of the rational map induced by the parametrization. We also present formulas for computing the partial degrees of the implicit equation.Ministerio de Educación y cienci

Implementing the Thull-Yap algorithm for computing Euclidean remainder sequences

International audienceThere are two types of integer gcd algorithms: those which compute the sequence of remainders of Euclid's algorithm and those which build different sequences. The former are more difficult to validate and analyse, whereas the latter are simpler and more efficient. When one wants the euclidean remainders (for instance if one wants to compute continued fractions), only the former can be used. Our main focus is the subquadratic time Thull-Yap GCD algorithm, and in fact on its core computing a half gcd (TYHGCD). This algorithm is tricky due to the difficulty in correcting the remainder sequence that comes back from a recursive call. The aim of this work is to revise TYHGCD in order to implement it using GMP. We clarify some points of the algorithm, in particular the stopping conditions that are always difficult to set correctly. We add a base case to speed up the whole algorithm, using Jebelean's quadratic algorithm with a stopping condition. We give our own modified version and add the proofs where needed. We insist on the test phase for this algorithm, giving families of hard cases for all branches, some of which are rarely activated. We give some details on our implementation in GMP using low-level functions, adding some remarks on the use of fast multiplications techniques. We pay attention to the data structure needed to store partial quotients, enabling to navigate rapidly back and forth in the sequence of Euclidean remainders. Benchmarks are provided. Some comments are made on Lichtblau's algorithm, which is close in spirit to the Thull-Yap algorithm

Fast Fourier Transforms over Prime Fields of Large Characteristic and their Implementation on Graphics Processing Units

Prime field arithmetic plays a central role in computer algebra and supports computation in Galois fields which are essential to coding theory and cryptography algorithms. The prime fields that are used in computer algebra systems, in particular in the implementation of modular methods, are often of small characteristic, that is, based on prime numbers that fit on a machine word. Increasing precision beyond the machine word size can be done via the Chinese Remaindering Theorem or Hensel Lemma. In this thesis, we consider prime fields of large characteristic, typically fitting on n machine words, where n is a power of 2. When the characteristic of these fields is restricted to a subclass of the generalized Fermat numbers, we show that arithmetic operations in such fields offer attractive performance both in terms of algebraic complexity and parallelism. In particular, these operations can be vectorized, leading to efficient implementation of fast Fourier transforms on graphics processing units