Elisabetta Fortuna

Elisabetta Fortuna; Patrizia M. Gianni; Paola Parenti; Carlo Traverso

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.

Elisabetta Fortuna; Patrizia M. Gianni; Barry M. Trager

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

Elisabetta Fortuna; Patrizia M. Gianni; Domenico Luminati

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.

Elisabetta Fortuna; Patrizia M. Gianni; Paola Parenti; Carlo Traverso

We present constructive algorithms to determine the topological type of a non-singular real algebraic projective surface S in the real projective space; we address this question when there exists a line in RP3 not intersecting the surface. Starting from a polynomial equation with rational coefficients for S, our algorithm computes the homology of the various connected components of the surface. We reconstruct the homology in a finite number of steps, using as a basic tool Morse theory. The entire procedure has been implemented in Axiom.

Massimo Ferrarotti; Elisabetta Fortuna; Leslie Wilson

Abstract. Two subanalytic subsets of Rn are called s-equivalent at a common point P if the Hausdorff distance between their intersections with the sphere centered at P of radius r vanishes to order > s when r tends to 0. In this paper we prove that every s-equivalence class of a closed semianalytic set contains a semialgebraic representative of the same dimension. In other words any semianalytic set can be locally approximated of any order s by means of a semialgebraic set and hence, by previous results, also by means of an algebraic one

Massimo Ferrarotti; Elisabetta Fortuna; Les Wilson

Two subanalytic subsets of ℝ n are s-equivalent at a common point, say O, if the Hausdorff distance between their intersections with the sphere centered at O of radius r goes to zero faster than r s . In the present paper we investigate the existence of an algebraic representative in every s-equivalence class of subanalytic sets. First we prove that such a result holds for the zero-set V(f) of an analytic map f when the regular points of f are dense in V(f). Moreover we present some results concerning the algebraic approximation of the image of a real analytic map f under the hypothesis that f ―1 (O) = {O} . .

Elisabetta Fortuna; Patrizia M. Gianni; Barry M. Trager

In this paper we show that the ideal of any algebraic curve in affine 3-space whose Jacobian matrix has rank at least 1 at every singular point of the curve can be generated by three polynomials and we give constructive procedures to compute such generators.

Elisabetta Fortuna; Patrizia M. Gianni; Paola Parenti

We address two basic questions for real algebraic curves. The first one is how to decide whether a real algebraic curve in the n-projective space contains some real point. We present an algorithm that reduces the original question to deciding whether the zero-set of a zero-dimensional ideal contains real points. The second part of the paper is devoted to giving necessary and sufficient conditions for the existence of a real line disjoint from a given real plane algebraic curve. An algorithm for testing whether these conditions are fulfilled is given. are fulfilled is given.

Francesca Acquistapace; Fabrizio Broglia; Elisabetta Fortuna

Let M be a compact non-singular real affine algebraic variety and let A, B be open disjoint semialgebraic subsets of M . Define (where —4 denotes the Zariski closure).

Elisabetta Fortuna; Patrizia M. Gianni; Domenico Luminati; Paola Parenti

The paper deals with the question of recognizing the mutual positions of the connected components of a non-singular real projective surface S in the real projective 3-space. We present an algorithm that answers this question through the computation of the adjacency graph of the surface; it also allows to decide whether each connected component is contractible or not. The algorithm, combined with a previous one returning as an output the topology of the surface, computes a set of data invariant up to ambient-homeomorphism which, though not sufficient to determine the pair , give information about the nature of the surface as an embedded object.