Singular

D.8.6 zeroset_lib

Library:

zeroset.lib

Purpose:

Procedures for roots and factorization

Author:

Thomas Bayer, email: tbayer@mathematik.uni-kl.de,
http://wwwmayr.informatik.tu-muenchen.de/personen/bayert/
Current address: Hochschule Ravensburg-Weingarten

Overview:

Algorithms for finding the zero-set of a zero-dim. ideal in Q(a)[x_1,..,x_n], roots and factorization of univariate polynomials over Q(a)[t] where a is an algebraic number. Written in the scope of the diploma thesis (advisor: Prof. Gert-Martin Greuel) 'Computations of moduli spaces of semiquasihomogeneous singularities and an implementation in Singular'. This library is meant as a preliminary extension of the functionality of @sc{Singular} for univariate factorization of polynomials over simple algebraic extensions in characteristic 0.

NOTE:
Subprocedures with postfix 'Main' require that the ring contains a variable 'a' and no parameters, and the ideal 'mpoly', where 'minpoly' from the basering is stored.

Procedures:

D.8.6.1 Quotient		quotient q of f w.r.t. g (in f = q*g + remainder)
D.8.6.2 remainder		remainder of the division of f by g
D.8.6.3 roots		computes all roots of f in an extension field of Q
D.8.6.4 sqfrNorm		norm of f (f must be squarefree)
D.8.6.5 zeroSet		zero-set of the 0-dim. ideal I
D.8.6.6 egcdMain		gcd over an algebraic extension field of Q
D.8.6.7 factorMain		factorization of f over an algebraic extension field
D.8.6.8 invertNumberMain		inverts an element of an algebraic extension field
D.8.6.9 quotientMain		quotient of f w.r.t. g
D.8.6.10 remainderMain		remainder of the division of f by g
D.8.6.11 rootsMain		computes all roots of f, might extend the ground field
D.8.6.12 sqfrNormMain		norm of f (f must be squarefree)
D.8.6.13 containedQ		f in data ?
D.8.6.14 sameQ		a == b (list a,b)