Specifying the bidegrees (n,m) of mixed polynomials P(z,z¯) of the single complex variable z, with complex coefficients, allows to investigate interesting roots structures and counting; intermediate between complex and real algebra. Multivariate mixed polynomials appeared in recent papers dealing with Milnor fibrations, but in this paper we focus on the univariate case and m=1, which is closely related to the important subject of harmonic maps. Here we adapt, to this setting, two algorithms of computer algebra: Vandermonde interpolation and a bissection-exclusion method for root isolation. Implemented in Maple, they are used to explore some interesting classes of examples.

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.

This is a special issue of Theoretical Computer Sciences journal which is devoted to Computational Algebraic Geometry and its Applications.

We describe an efficient algorithm and an implementation for computing an absolute factorization of a bivariate polynomial with a given bidegree. Results of experimentation and an illustrative example are provided. This algorithm is a generalization of the previous one by Rupprecht-Galligo-Chèze which works after a generic change of coordinates. It relies on a general algorithmic approach based on a study of the curve defined by the polynomial to factorize in a toric surface.

Parametrized surfaces of low degrees are very useful in applications, especially in Computer Aided Geometric Design and Geometric Modeling. The precise description of their geometry is not easy in general. Here we study some of the corresponding projective complex surfaces of low implicit degree (i.e., smaller than 12). We show that, generically up to linear changes of coordinates, they are classified by a few number of continuous parameters (called moduli). We present normal forms and provide compact implicit equations for these surfaces and for their singular locus together with a geometric interpretation.

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

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.

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.

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.