Stable Complete Intersections

Abstract

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.

An Algebraic Approach to Hough Transforms

“…Proof. We know that the morphism Φ |dσ(a) is flat and that G(a, x) specializes to the reduced σ-Gröbner basis of I(α, x) for every α ∈ U σ (see for instance the proof of Proposition 2.3 in [8]). By assumption, the term ordering σ is degree-compatible, hence all the fibers of Φ |dσ(a) share the same affine Hilbert function (see Chapter 5 of [7]), hence they share the same dimension.…”