Given a zero-dimensional ideal I in a polynomial ring, many algorithms start by finding univariate polynomials in~I, or by computing a lex-Groebner basis of~I. These are related to considering the minimal polynomial of an element in P/I, which may be computed using Linear Algebra from a Groebner Basis (for any term-ordering). In this tutorial we'll see algorithms for computing minimal polynomials, applications of modular methods, and then some applications, namely algorithms for computing radicals and primary decompositions of zero-dimensional ideals, and also for testing radicality and maximality. We'll also address a kind of opposite problem: given a "geometrical description'', such as a finite set of points, find the ideal of polynomials which vanish at it. We start from the original Buchberger-Moeller algorithm, and we show some developments. All this will be done with a special eye on the practical implementations, and with demostrations in CoCoA.

### Linear Algebra for Zero-Dimensional Ideals

#### Abstract

Given a zero-dimensional ideal I in a polynomial ring, many algorithms start by finding univariate polynomials in~I, or by computing a lex-Groebner basis of~I. These are related to considering the minimal polynomial of an element in P/I, which may be computed using Linear Algebra from a Groebner Basis (for any term-ordering). In this tutorial we'll see algorithms for computing minimal polynomials, applications of modular methods, and then some applications, namely algorithms for computing radicals and primary decompositions of zero-dimensional ideals, and also for testing radicality and maximality. We'll also address a kind of opposite problem: given a "geometrical description'', such as a finite set of points, find the ideal of polynomials which vanish at it. We start from the original Buchberger-Moeller algorithm, and we show some developments. All this will be done with a special eye on the practical implementations, and with demostrations in CoCoA.
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2019
9781450360845
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Descrizione: Articolo principale
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11567/965914`