Mohamed Elkadi

Generalized resultants over unirational algebraic varieties

In this paper, we propose a new method, based on Bezoutian matrices, for computing a nontrivial multiple of the resultant over a projective variety X, which is described on an open subset by a parameterization. This construction, which generalizes the classical and toric one, also applies for instance to blowing up varieties and to residual intersection problems. We recall the classical notion of resultant over a variety X. Then we extend it to varieties which are parameterized on a dense open subset and give new conditions for the existence of the resultant over these varieties. We prove that any maximal nonzero minor of the corresponding Bezoutian matrix yields a nontrivial multiple of the resultant. We end with some experiments.

Resultant over the residual of a complete intersection

In this article, we study the residual resultant which is the necessary and sufficient condition for a polynomial system F to have a solution in the residual of a variety, defined here by a complete intersection G. We show that it corresponds to an irreducible divisor and give an explicit formula for its degree in the coefficients of each polynomial. Using the resolution of the ideal (F:G) and computing its regularity, we give a method for computing the residual resultant using a matrix which involves a Macaulay and a Bezout part. In particular, we show that this resultant is the gcd of all the maximal minors of this matrix. We illustrate our approach for the residual of points and end by some explicit examples.

Algorithms for residues and Lojasiewicz exponents

Abstract In this article, we investigate some problems of effectivity, related to algebraic residue theory. We show how matrix techniques based on Bezoutian formulations, enable us to derive new algorithms, as well as new bounds for the polynomials involved in these computations. More precisely, we focus on the computation of relations of algebraic dependency between n +1 polynomials in n variables and show how to deduce the residue of n polynomials in n variables. Applications for testing the properness of a polynomial map, for computing the Lojasiewicz exponent, and for inverting polynomial maps are also considered.

A new algorithm for the geometric decomposition of a variety

In this article, we present a new met.hod for computing the deconiposition of a variety into irreducible componc:nts. It. is bawd on a property of Bczoutia.n mat.ricc:s; which allows us to c:omput,e a multiple of t.lio Chow form of t.he isolat,cil points of the varict.y and to deduce a rational representation of thcsc points. This t,ools is used recursively to compute t,he irreduc.iblc components from the lowest to the highest. dimension. The asymptotic complexity is of the same order t.han the best complesity bound known for this problem. Our approach provides a subst,nntial simplification of the previous methods and yields bounds on the height of polynomials involved in these representations. =\n iml.‘lelllrnt.ation in MAPLE of t.his algorithm is described at, thr end.

Algebraic geometry and geometric modeling

Algebraic Geometry provides an impressive theory targeting the understanding of geometric objects defined algebraically. Geometric Modeling uses every day, in order to solve practical and difficult problems, digital shapes based on algebraic models. In this book, we have collected articles bridging these two areas. The confrontation of the different points of view results in a better analysis of what the key challenges are and how they can be met. We focus on the following important classes of problems: implicitization, classification, and intersection. The combination of illustrative pictures, explicit computations and review articles will help the reader to handle these subjects

Towards toric absolute factorization

This article presents an algorithmic approach to study and compute the absolute factorization of a bivariate polynomial, taking into account the geometry of its monomials. It is based on algebraic criterions inherited from algebraic interpolation and toric geometry.

Intersection and self-intersection of surfaces by means of Bezoutian matrices

The computation of intersection and self-intersection loci of parametrized surfaces is an important task in Computer Aided Geometric Design. We address these problems via four resultants with separated variables; two of them are specializations of general multivariate resultants and the two others are specializations of determinantal resultants. We give a rigorous study in these four cases and provide new formulas in terms of Bezoutian matrix.

Symbolic-numeric methods for solving polynomial equations and applications

This tutorial gives an introductory presentation of algebraic and geometric methods to solve a polynomial system ƒ1 = ⋯ = ƒm = 0. The algebraic methods are based on the study of the quotient algebra A of the polynomial ring modulo the ideal I = (ƒ1,..., ƒm). We show how to deduce the geometry of solutions from the structure of A and in particular, how solving polynomial equations reduces to eigenvalue and eigenvector computations of multiplication operators in A. We give two approaches for computing the normal form of elements in A, used to obtain a representation of multiplication operators. We also present the duality theory and its application to solving systems of algebraic equations. The geometric methods are based on projection operations which are closely related to resultant theory. We present different constructions of resultants and different methods for solving systems of polynomial equations based on these formulations. Finally, we illustrate these tools on problems coming from applications in computer-aided geometric design, computer vision, robotics, computational biology and signal processing.

A computational study of ruled surfaces

We study rational ruled surfaces and @m-bases which were recently considered in a series of articles by Chen and coworkers. We give short and conceptual proofs with geometric insights and efficient algorithms. In particular, we provide a method to reparameterize an improper parameterization and we also briefly explain how to deal with approximate input data. Finally we provide an algorithmic description of self-intersection loci.

Parametrized surfaces in huge P 3 of bidegree (1,2)

Parametrized surfaces of low degrees are very useful in applications, specially in Computer Aided Geometric Design and Geometric Modeling. The precise description of their geometry is not easy in general. Here we study surfaces of bidegree (1,2). We show that, generically up to linear changes of coordinates, they are classified by two continuous parameters (modulus). We present an elegant combinatorial description where these modulus appear as cross ratios. We provide compact implicit equations for these surfaces and for their singular locus together with a geometric interpretation.