Henri Lombardi

Certified approximate univariate GCDs

Abstract We study the approximate GCD of two univariate polynomials given with limited accuracy or, equivalently, the exact GCD of the perturbed polynomials within some prescribed tolerance. A perturbed polynomial is regarded as a family of polynomials in a classification space, which leads to an accurate analysis of the computation. Considering only the Sylvester matrix singular values, as is frequently suggested in the literature, does not suffice to solve the problem completely, even when the extended euclidean algorithm is also used. We provide a counterexample that illustrates this claim and indicates the problems hardness. SVD computations on subresultant matrices lead to upper bounds on the degree of the approximate GCD. Further use of the subresultant matrices singular values yields an approximate syzygy of the given polynomials, which is used to establish a gap theorem on certain singular values that certifies the maximum-degree approximate GCD. This approach leads directly to an algorithm for computing the approximate GCD polynomial. Lastly, we suggest the use of weighted norms in order to sharpen the theorems conditions in a more intrinsic context.

Sturm-Habicht sequence

Introduction Formal computations with inequalities is a subject of general interest in computer algebra. In particular it is fiindamental in the parallelisation of basic algorithms and quantifier elimination for real closed fields ([BKR] , NW). In

Sturm—Habicht Sequences, Determinants and Real Roots of Univariate Polynomials

1 we give a generalisation of Sturm theorem essentially due to Sylvester which is the key for formal computations with inequalities. Our result is an improvment of previously known results (see [BKR]) since no hypotheses have to be made on the polynomials. In

Dynamical method in algebra: effective Nullstellensätze

11 we study the subrcsultant sequence. WC prccisc some of the classical definitions in order to avoid some .problems appearing in the paper by Loos ([L]) and study specialisation properties in detail. In

New structure theorem for subresultants

111 we introduce the Sturm-Habicht scqucnce, which generalizes Habicht’s work ([H]).This new scqucncc obtained automatically from a subrcsultant scqucncc hns some remarkable properties: -it gives the same information than Ihc Sturm sccpcncc, and this information may be recovcrcd by looking only at its principal coefficients,

A logical approach to abstract algebra

The real root counting problem is one of the main computational problems in Real Algebraic Geometry. It is the following: Let \(\mathbb{D}\) be an ordered domain and \(\mathbb{B}\) a real closed field containing \(\mathbb{D}\). Find algorithms which for every P ∈ \(\mathbb{D}\) [x] compute the number of roots of P in \(\mathbb{B}\). More precisely we shall study the following problem.

The Berlekamp-Massey Algorithm revisited

We give a general method for producing various effective Null and Positivstellensatze, and getting new Positivstellensatze in algebraically closed valued fields and ordered groups. These various effective Nullstellensatze produce algebraic identities certifying that some geometric conditions cannot be simultaneously satisfied. We produce also constructive versions of abstract classical results of algebra based on Zorns lemma in several cases where such constructive version did not exist. For example, the fact that a real field can be totally ordered, or the fact that a field can be embedded in an algebraically closed field. Our results are based on the concepts we develop of dynamical proofs and simultaneous collapse.

Dynamic Galois Theory

We give a new structure theorem for subresultants precising their gap structure and derive from it a new algorithm for computing them. If

Generalizing Cramer's Rule: Solving Uniformly Linear Systems of Equations

d

Generalized Budan-Fourier theorem and virtual roots

is a bound on the degrees and