### D.8.10 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 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.8.10.1 Quotient |  | quotient q of f w.r.t. g (in f = q*g + remainder)
> D.8.10.2 remainder |  | remainder of the division of f by g
> D.8.10.3 roots |  | computes all roots of f in an extension field of Q
> D.8.10.4 sqfrNorm |  | norm of f (f must be squarefree)
> D.8.10.5 zeroSet |  | zero-set of the 0-dim. ideal I
> D.8.10.6 egcdMain |  | gcd over an algebraic extension field of Q
> D.8.10.7 factorMain |  | factorization of f over an algebraic extension field
> D.8.10.8 invertNumberMain |  | inverts an element of an algebraic extension field
> D.8.10.9 quotientMain |  | quotient of f w.r.t. g
> D.8.10.10 remainderMain |  | remainder of the division of f by g
> D.8.10.11 rootsMain |  | computes all roots of f, might extend the ground field
> D.8.10.12 sqfrNormMain |  | norm of f (f must be squarefree)
> D.8.10.13 containedQ |  | f in data ?
> D.8.10.14 sameQ |  | a == b (list a,b)

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