Linear Algebra for Zero-Dimensional Ideals
| Autor: | Anna Maria Bigatti |
|---|---|
| Rok vydání: | 2019 |
| Předmět: |
border bases
cocoa cocoalib ideals of points minimal and characteristic polynomial primary decomposition radical ideals solving polynomial systems zero-dimensional ideals zero-dimensional ideals solving polynomial systems Mathematics::Commutative Algebra primary decomposition business.industry Polynomial ring Univariate radical ideals ideals of points Modular design border bases minimal and characteristic polynomial Algebra Primary decomposition Gröbner basis Minimal polynomial (field theory) cocoa ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION Linear algebra cocoalib business Finite set Mathematics |
| Zdroj: | ISSAC |
| DOI: | 10.1145/3326229.3326279 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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