Guillaume Chèze

Lifting and recombination techniques for absolute factorization

In the vein of recent algorithmic advances in polynomial factorization based on lifting and recombination techniques, we present new faster algorithms for computing the absolute factorization of a bivariate polynomial. The running time of our probabilistic algorithm is less than quadratic in the dense size of the polynomial to be factored.

Four lectures on polynomial absolute factorization

Polynomial factorization is one of the main chapters of Computer Algebra. Recently, significant progress was made on absolute factorization (i.e., over the complex field) of a multivariate polynomial with rational coefficients, with two families of algorithms proposing two different strategies of computation. One is represented by Gao’s algorithm and is explained in Lecture 2. The other is represented by the Galligo-Rupprecht-Cheze algorithm, presented in Lectures 4 and 5. The latter relies on an original use of the monodromy map attached to a generic projection of a plane curve on a line. It also involves zero-sums relations (introduced by Sasaki and his collaborators) with efficient semi-numerical computations to produce a certified exact result.

From an approximate to an exact absolute polynomial factorization

We propose an algorithm for computing an exact absolute factorization of a bivariate polynomial from an approximate one. This algorithm is based on some properties of the algebraic integers over Z and is certified. It relies on a study of the perturbations in a Vandermonde system. We provide a sufficient condition on the precision of the approximate factors, depending only on the height and the degree of the polynomial.

Absolute polynomial factorization in two variables and the knapsack problem

A recent algorithmic procedure for computing the absolute factorization of a polynomial <i>P(X,Y)</i>, after a linear change of coordinates, is via a factorization modulo <i>X</i>³. This was proposed by A. Galligo and D. Rupprecht in [7],[16]. Then absolute factorization is reduced to finding the minimal zero sum relations between a set of approximated numbers <i>b;_i;</i>, <i>i</i> =1 to <i>n</i> such that ∑<i>ⁿ;_{i =1;}</i> <i>b;_i;</i> =0, (see also [17]). Here this problem with an a priori exponential complexity, is efficiently solved for large degrees (<i>n</i>›100). We rely on LLL algorithm, used with a strategy of computation inspired by van Hoeijs treatment in [23]. For that purpose we prove a theorem on bounded integer relations between the numbers <i>b;_i;,</i>, also called linear traces in [19].

A subdivision method for computing nearest gcd with certification

A new subdivision method for computing the nearest univariate gcd is described and analyzed. It is based on an exclusion test and an inclusion test. The exclusion test in a cell exploits Taylor expansion of the polynomial at the center of the cell. The inclusion test uses Smales @a-theorems to certify the existence and unicity of a solution in a cell. Under the condition of simple roots for the distance minimization problem, we analyze the complexity of the algorithm in terms of a condition number, which is the inverse of the distance to the set of degenerate systems. We report on some experimentation on representative examples to illustrate the behavior of the algorithm.

On the total order of reducibility of a pencil of algebraic plane curves

In this paper, the problem of bounding the number of reducible curves in a pencil of algebraic plane curves is addressed. Unlike most of the previous related works, each reducible curve of the pencil is here counted with its appropriate multiplicity. It is proved that this number of reducible curves, counted with multiplicity, is bounded by d^2-1 where d is the degree of the pencil. Then, a sharper bound is given by taking into account the Newtons polygon of the pencil.

Efficient algorithms for computing rational first integrals and Darboux polynomials of planar polynomial vector fields

We present fast algorithms for computing rational first integrals with bounded degree of a planar polynomial vector field. Our approach builds upon a method proposed by Ferragut and Giacomini, whose main ingredients are the calculation of a power series solution of a first order differential equation and the reconstruction of a bivariate polynomial annihilating this power series. We provide explicit bounds on the number of terms needed in the power series. This enables us to transform their method into a certified algorithm computing rational first integrals via systems of linear equations. We then significantly improve upon this first algorithm by building a probabilistic algorithm with arithmetic complexity

A recombination algorithm for the decomposition of multivariate rational functions

\~O(N^{2 \omega})

Computing nearest Gcd with certification

and a deterministic algorithm solving the problem in at most

Existence of a simple and equitable fair division: A short proof

\~O(d^2N^{2 \omega+1})