Basi bordo: confronti e approccio algebrico alla stabilità numerica

This thesis is in the framework of Computational and Commutative Algebra and it concerns the theory of border bases. We can identify three main macro areas in this thesis: properties and characterizations of border bases with connections to Groebner theory, study of their numerical behaviour, connections to marked bases theory. In this abstract we focus on the last two parts, that contain some original contributions. We start giving an intuitive idea of border bases’ utility. Consider a field K, a polynomial ring P over K and a zero-dimensional ideal I in P. Our aim is to describe I by an “internal”and “external”point of view: the focus is not only on a generating set of I but also on a basis of the quotient P/I as K-vector space. Groebner theory is fully satisfactory for the internal point of view but not enough for the external one, for instance, Groebner bases haven’t a good numerical behaviour. Consider a system of generators of an ideal I and a perturbation of this system, so that the geometrical configuration of the polynomial equations associated is “very similar”to the initial configuration. Nevertheless, the Groebner basis of the starting ideal I is significantly different from the Groebner basis of the perturbed ideal, though the geometrical configuration is very similar. Border bases are more “stable”than Groebner bases in numerical behaviour, and that is one on the main reasons of their success. One of the aims of this thesis is deepening the idea of “numerical stability”, finding a formal definition of stability of border bases from an algebraic point of view. First, we study several examples in literature. Kreuzer and Robbiano talk about border bases’ “preservation of continuity”, pointing out the continuity between small perturbations of the starting ideal and of its corresponding border basis. More precisely, the order ideal on which the border basis is defined is the same before and after the perturbation. Thanks to one of the border bases criterions proved in the first part of this thesis, we can show explicitly that the announced preservation of continuity is valid for a bigger class of Kreuzer and Robbiano’s examples, considered by Mourrain. After this first partial result, we can formulate a formal definition of stable border bases in the current context and investigate some perturbations which are stable in this sense. In the end, we study the paper by Abbott, Fassino and Torrente. Their context is different from Kreuzer and Robbiano’s one because they focus on vanishing ideals of sets of points. Given a set of different points, their aim is finding a border basis of the corresponding vanishing ideal “stable”under perturbations in the coordinates of points, so they introduce new notions of stable order ideals and stable border bases in this particular context. Studying several examples we can connect their theory to Kreuzer and Robbiano’s one and formulate a conjecture about the link between admissible perturbations of vanishing ideals and corresponding Hilbert schemes. In the final part of the thesis, we start from the recent work by Bertone and Cioffi, where they study the close relation between border bases and Pommaret bases. We continue this work in the thesis and in particular we highlight that the existing algorithms to compute a border basis for a given zero-dimensional ideal can be easily adapted to the computation of a Pommaret marked basis of the same ideal.