Project-galaad:Algebraic Geometric Computing

Inria / Raweb 2009
Presentation of the Project galaad

Section: New Results

Algebraic Geometric Computing

The Hilbert scheme of points and its link with border basis

Participants : Jérôme Brachat, Bernard Mourrain.

We give new explicit representations of the Hilbert scheme of $\mu$ points in $Im1 $\#8473 ^r$$ as a projective subvariety of a Grassmanniann variety. This new explicit description of the Hilbert scheme is simpler than the existing ones and global. It involves equations of degree 2. We show how these equations are deduced from the commutation relations characterizing border bases. Next, we consider infinitesimal perturbations of an input system of equations on this Hilbert scheme and describe its tangent space. We propose an effective criterion to test if it is a flat deformation, that is if the perturbed system remains on the Hilbert scheme of the initial equations. In particular, this criterion involves in particular formal reduction with respect to border bases.

A preprint version of this work, done in collaboration with Mariemi Alonso, Departmento de Algebra, UCM and submitted for publication, is available at http://hal.inria.fr/inria-00433127 .

Modular Las Vegas algorithms for polynomial absolute factorization

Participants : André Galligo, Cristina Bertone.

Let $Im2 ${f(X,Y)\#8712 \#8484 [X,Y]}$$ be an irreducible polynomial over $Im3 ${\#8474 [X,Y]}$$ . We study the absolute factorization $Im4 ${f_1\#8943 f_s}$$ of f . First we give a Las Vegas absolute irreducibility test based on a property of the Newton polygon of f . Then thanks to this test we give a new strategy based on modular computations and LLL which gives the absolute factorization.

These results, in collaboration with Guillaume Chèze, were presented at MEGA 2009 and are described in http://hal.inria.fr/inria-00436063/en/ .

Algorithms for irreducible decomposition of curves in $Im5 $\#8450 ^N$$

Participants : André Galligo, Cristina Bertone.

The aim of this work is to construct an effective method for computing the irreducible components of a curve of the affine N -dimensional space $Im5 $\#8450 ^N$$ , eventually starting from the existing algorithms for decomposition of curves in the plane $Im6 $\#8450 ^2$$ , i.e. algorithms for the absolute factorization of polynomials. We developed a strategy using the modular techniques of section 6.2.2 allowing us to compute: the number of irreducible components of a curve, their degrees, multiplicities, the Hilbert function of the “simple” components and the algebraic extensions for the non-rational components.

Subdivision methods for solving polynomial equations

Participant : Bernard Mourrain.

This works presents a new algorithm for solving a system of polynomials in a domain of $Im7 $\#8477 ^n$$ . It can be seen as an improvement of the Interval Projected Polyhedron algorithm proposed by Sherbrooke and Patrikalakis. It uses a powerful reduction strategy based on univariate root finder using Bernstein basis representation and Descarte's rule. We analyse the behavior of the method, from a theoretical point of view, show that for simple roots, it has a local quadratic convergence speed and give new bounds for the complexity of approximating its real roots in a box of $Im7 $\#8477 ^n$$ . The improvement of our approach, compared with classical subdivision methods, is illustrated on geometric modeling applications such as computing intersection points of implicit curves, self-intersection points of rational curves, and on the classical parallel robot benchmark problem. An implementation of this algorithm is available in the module realroot of Mathemagix project.

This work, done in collaboration with Jean-Pascal Pavone, is published in [20] .

Multivariate continued fraction solvers for polynomial equations

Participants : Angelos Mantzaflaris, Bernard Mourrain, Elias Tsigaridas.

We present a new algorithm for isolating the real roots of a system of multivariate polynomials, given in the monomial basis. It is inspired by existing subdivision methods in the Bernstein basis; it can be seen as generalization of the univariate continued fraction algorithm or alternatively as an analog of Bernstein subdivision in the monomial basis. The representation of the subdivided domains is done through homographies, which allows us to use only integer arithmetic and to treat efficiently unbounded regions. We use univariate bounding functions, projection and preconditioning techniques to reduce the domain of search. The resulting boxes have optimized rational coordinates, corresponding to the first terms of the continued fraction expansion of the real roots. An extension of Vincent's theorem to multivariate polynomials is proved and used for the termination of the algorithm. New complexity bounds are provided for a simplified version of the algorithm. Our C++ implementation in done as part of realroot module of the Mathemagix system.

This work has been published in [28] .

Computing nearest GCD with certification

Participants : André Galligo, Bernard Mourrain.

A bisection method, based on exclusion and inclusion tests, is used to address the nearest univariate GCD problem formulated as a bivariate real minimization problem of a rational fraction.

The paper [22] presents an algorithm, a first implementation and complexity analysis relying on Smale's $\alpha$ -theory. We report its behavior on an illustrative example.

This work was done in collaboration with Guillaume Chèze, Univ. Toulouse, Jean-Claude Yakoubsohn, Univ. Toulouse.

Continuation and monodromy on random Riemann surfaces

Participants : André Galligo, Adrien Poteaux.

The main motivation is to analyze and develop further factorization algorithms for bivariate polynomials in $Im8 ${\#8450 [x,y]}$$ , which proceed by continuation methods.

We consider a Riemann surface X defined by a polynomial f(x, y) of degree d , the coefficients of which are chosen randomly. In this way we can suppose that X is smooth, that the discriminant $\delta$ (x) of f has d(d-1) simple roots and also that $\delta$ (0) $\ne$ 0 , that is to say that the corresponding fiber has d distinct points {y₁, ..., y_d} . When we lift a loop $Im9 ${0\#8712 \#947 \#8834 \#8450 -\#916 }$$ by a continuation method, we get d paths in X connecting {y₁, ..., y_d} , hence defining a permutation of that set. This is called monodromy.

We present experimentations in Maple to get statistics on the distribution of transpositions corresponding to the loops turning around each point of $\upper_delta$ . Multiplying families of “neighbor” transpositions, we construct permutations then subgroups of the symmetric group. This allows us to establish and study experimentally some conjectures on the distribution of these transpositions then on transitivity of the generated subgroups.

These results provide interesting insights on the structure of such Riemann surfaces; hence of their union. They are used to develop fast algorithms for absolute multivariate polynomial factorization, under some genericity hypothesis: we assume that the factors behave like random polynomials the coefficients of which follow uniform distributions.

This work is published in [24] .

Curve/surface intersection problem by means of matrix representations

Participants : Laurent Busé, Thang Luu Ba, Bernard Mourrain.

We introduce matrix representations of plane algebraic curves and space algebraic surfaces for Computer Aided Geometric Design (CAGD). The idea of using matrix representations in CAGD is quite old. The novelty of our contribution is to enable non square matrices, extension which is motivated by recent research in this topic. We show how to manipulate these representations by proposing a dedicated algorithm to address the curve/surface intersection problem by means of numerical linear algebra techniques.

This work has been published in the proceedings of the SNC'09 conference [27] .

Bernstein Bezoutian and some intersection problems

Participants : Elimane Ba, Mohamed Elkadi.

We study the Bézier curve-surface and Bézier surface-surface intersection problems avoiding the well-know unstable conversion between Bernstein basis and power basis. These varieties are given by parameterizations in Bernstein bases and all computations are performed in that form. We construct an adapted resultant for generic Bernstein polynomial systems with a special shape appearing in the intersection problems. This work has been submitted to the journal CAGD.

Isotopic triangulation of a real algebraic surface

Participant : Bernard Mourrain.

We present a new algorithm for computing the topology of a real algebraic surface S in a ball B , even in singular cases. We use algorithms for 2D and 3D algebraic curves and show how one can compute a topological complex equivalent to S , and even a simplicial complex isotopic to S by exploiting properties of the contour curve of S . The correctness proof of the algorithm is based on results from stratification theory. We construct an explicit Whitney stratification of S , by resultant computation. Using Thom's isotopy lemma, we show how to deduce the topology of S from a finite number of characteristic points on the surface. An analysis of the complexity of the algorithm and effectiveness issues are also studied.

This work, done in collaboration with Lionel Alberti and Jean-Pierre Técourt is published in [12] .

Towards toric absolute factorization

Participants : André Galligo, Mohamed Elkadi.

We present 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 criteria inherited from algebraic interpolation and toric geometry.

This work, done in collaboration with Martin Weimann, Univ. Barcelona, is published in [17] .

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