58 research outputs found
On Border Basis and Groebner Basis Schemes
Hilbert schemes of zero-dimensional ideals in a polynomial ring can be
covered with suitable affine open subschemes whose construction is achieved
using border bases. Moreover, border bases have proved to be an excellent tool
for describing zero-dimensional ideals when the coefficients are inexact. And
in this situation they show a clear advantage with respect to Groebner bases
which, nevertheless, can also be used in the study of Hilbert schemes, since
they provide tools for constructing suitable stratifications.
In this paper we compare Groebner basis schemes with border basis schemes. It
is shown that Groebner basis schemes and their associated universal families
can be viewed as weighted projective schemes. A first consequence of our
approach is the proof that all the ideals which define a Groebner basis scheme
and are obtained using Buchberger's Algorithm, are equal. Another result is
that if the origin (i.e. the point corresponding to the unique monomial ideal)
in the Groebner basis scheme is smooth, then the scheme itself is isomorphic to
an affine space. This fact represents a remarkable difference between border
basis and Groebner basis schemes. Since it is natural to look for situations
where a Groebner basis scheme and the corresponding border basis scheme are
equal, we address the issue, provide an answer, and exhibit some consequences.
Open problems are discussed at the end of the paper.Comment: Some typos fixed, some small corrections done. The final version of
the paper will be published on "Collectanea Mathematica
The Geometry of Border Bases
The main topic of the paper is the construction of various explicit flat
families of border bases. To begin with, we cover the punctual Hilbert scheme
Hilb^\mu(A^n) by border basis schemes and work out the base changes. This
enables us to control flat families obtained by linear changes of coordinates.
Next we provide an explicit construction of the principal component of the
border basis scheme, and we use it to find flat families of maximal dimension
at each radical point. Finally, we connect radical points to each other and to
the monomial point via explicit flat families on the principal component
Deformations of Border Bases
Here we study the problem of generalizing one of the main tools of Groebner
basis theory, namely the flat deformation to the leading term ideal, to the
border basis setting. After showing that the straightforward approach based on
the deformation to the degree form ideal works only under additional
hypotheses, we introduce border basis schemes and universal border basis
families. With their help the problem can be rephrased as the search for a
certain rational curve on a border basis scheme. We construct the system of
generators of the vanishing ideal of the border basis scheme in different ways
and study the question of how to minimalize it. For homogeneous ideals, we also
introduce a homogeneous border basis scheme and prove that it is an affine
space in certain cases. In these cases it is then easy to write down the
desired deformations explicitly.Comment: 21 page
Stable Complete Intersections
A complete intersection of n polynomials in n indeterminates has only a
finite number of zeros. In this paper we address the following question: how do
the zeros change when the coefficients of the polynomials are perturbed? In the
first part we show how to construct semi-algebraic sets in the parameter space
over which all the complete intersection ideals share the same number of
isolated real zeros. In the second part we show how to modify the complete
intersection and get a new one which generates the same ideal but whose real
zeros are more stable with respect to perturbations of the coefficients.Comment: 1 figur
An Algebraic Approach to Hough Transforms
The main purpose of this paper is to lay the foundations of a general theory
which encompasses the features of the classical Hough transform and extend them
to general algebraic objects such as affine schemes. The main motivation comes
from problems of detection of special shapes in medical and astronomical
images. The classical Hough transform has been used mainly to detect simple
curves such as lines and circles. We generalize this notion using reduced
Groebner bases of flat families of affine schemes. To this end we introduce and
develop the theory of Hough regularity. The theory is highly effective and we
give some examples computed with CoCoA
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