Patrizia M. Gianni

Efficient Computation of Zero-dimensional Gröbner Bases by Change of Ordering

We present an efficient algorithm for the transformation of a Grobner basis of a zero-dimensional ideal with respect to any given ordering into a Grobner basis with respect to any other ordering. This algorithm is polynomial in the degree of the ideal. In particular the lexicographical Grobner basis can be obtained by applying this algorithm after a total degree Grobner basis computation: it is usually much faster to compute the basis this way than with a direct application of Buchbergers algorithm.

Gröbner bases and primary decomposition of polynomial ideals

We present an algorithm to compute the primary decomposition of any ideal in a polynomialring over a factorially closed algorithmic principal ideal domain R. This means that the ring R is a constructive PID and that we are given an algorithm to factor polynomials over fields which are finitely generated over R or residue fields of R. We show how basic ideal theoretic operations can be performed using Grobner bases and we exploit these constructions to inductively reduce the problem to zero dimensional ideals. Here we again exploit the structure of Grobner bases to directly compute the primary decomposition using polynomial factorization. We also show how the reduction process can be applied to computing radicals and testing ideals for primality.

The singular value decomposition for polynomial systems

This paper introduces singular value decomposition (SVD) algorithms for some standard polynomial computations, in the case where the coecients are inexact or imperfectly known. We first give an algorithm for computing univariate GCD’s which gives exact results for interesting nearby problems, and give ecient algorithms for computing precisely how nearby. We generalize this to multivariate GCD computation. Next, we adapt Lazard’s u-resultant algorithm for the solution of overdetermined systems of polynomial equations to the inexact-coecient case. We also briefly discuss an application of the modified Lazard’s method to the location of singular points on approximately known projections of algebraic curves.

A reordered Schur factorization method for zero-dimensional polynomial systems with multiple roots

The technique of solving systems of multivariate polynomial equations via rigenproblems has become a topic of active research (with applications in computer-aided design and {untrul theory, for example) at least since the papers [2, 6, 9]. one may approach the problem via various resultant formulations ,x by Grijbner bases. As more understanding is gained, it is becoming clearer that eigenvalue problems are the “weakly nonlinear nucleus to which the original, strongly nonlinear task may be reduced’ [13], Earl,v works mmcemt,rat,ed on the case of simple roots. An example of such was, the paper [5], which used a numerical adaptation of il resultant technique due to Lazard to attack the problem directly, without, reference to Grobner bases.

Properties of Gröbner bases under specializations

In this paper we prove some properties of Grobner bases under specialization maps. In particular we state sufficient conditions for the image of a Grobner basis to be a Grobner basis. We apply these results to the resolution of systems of polynomial equations. In particular we show that, if the system has a finite number of solutions, (in an algebraic closure of the base field K), the problem is totally reduced to a single Grobner basis computation (w.r.t. purely lexicographical ordering), followed by a search for the roots of univariate polynomials and a “few” evaluations in suitable algebraic extensions of K.

Algorithms to compute the topology of orientable real algebraic surfaces

We present constructive algorithms to determine the topological type of a non-singular orientable real algebraic projective surface S in the real projective space, starting from a polynomial equation with rational coefficients for S. We address this question when there exists a line in RP3 not intersecting the surface, which is a decidable problem; in the case of quartic surfaces, when this condition is always fulfilled, we give a procedure to find a line disjoint from the surface. Our algorithm computes the homology of the various connected components of the surface in a finite number of steps, using as a basic tool Morse theory. The entire procedure has been implemented in Axiom.

Scratchpad's view of algebra II: A categorical view of factorization

This paper explains how Scratchpad solves the problem of presenting a categorical view of factorization in unique factorization domains, i.e. a view which can be propagated by functors such as SparseUnivari atepolynomial or Fraction. This is not easy, es the constructive version of the classical concept of UniqueFactorizationDomain cannot be so propagated. The solution adopted is based largely on Seidenberg’s conditions (F) and (P), but there are several additional points that have to be borne in mind to produce reasonably efficient algorithms in the required generality. The consequence of the algorithms and interfaces presented in this paper is that Scratchpad can factorize in any extension of the integers or finite fields by any combination of polynomial, fraction and algebraic extensions: a capability far more general than any other computer algebra system possesses. The solution is not perfect: for example we cannot use these general constructions to factorize polynomials in Z [~] [z] since the domain Z[~ is not a unique factorization domain, even though Z[~] is, since it is a field. Of course, we can factor polynomials in Z?[@?] [z].

Riemann Surfaces, Plane Algebraic Curves and Their Period Matrices

The aim of this paper is to present theoretical basis for computing a representation of a compact Riemann surface as an algebraic plane curve and to compute a numerical approximation for its period matrix. We will describe a program C ars (Semmler et al., 1996) that can be used to define Riemann surfaces for computations. C ars allows one also to perform the Fenchel?Nielsen twist and other deformations on Riemann surfaces.Almost all theoretical results presented here are well known in classical complex analysis and algebraic geometry. The contribution of the present paper is the design of an algorithm which is based on the classical results and computes first an approximation of a polynomial representing a given compact Riemann surface as a plane algebraic curve and further computes an approximation for a period matrix of this curve. This algorithm thus solves an important problem in the general case. This problem was first solved, in the case of symmetric Riemann surfaces, in Seppala (1994).

Degree reduction under specialization

We examine the degree relationship between the elements of an ideal I ⊆ R[x] and the elements of ’(I ) where ’ : R → R is a ring homomorphism. When R is a multivariate polynomial ring over a 3eld, we use this relationship to show that the image of a Gr4 obner basis remains a Gr4 obner basis if we specialize all the variables but one, with no requirement on the dimension of I . As a corollary we obtain the GCD for a collection of parametric univariate polynomials. We also apply this result to solve parametric systems of polynomial equations and to reexamine the extension theoremfor such system s. c

Algorithmical determination of the topology of a real algebraic surface

We present an algorithm to compute the topology of a non-singular real algebraic surface S in RP^3, that is the number of its connected components and a topological model for each of them. Our strategy consists in computing the Euler characteristic of each connected component by means of a Morse-type investigation of S or of a suitably constructed compact affine surface. This procedure can be used to determine the topological type of an arbitrary non-singular surface; in particular it extends an existing algorithm applicable only to surfaces disjoint from a line.