D.7.5 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 Adress: Institut fuer Informatik, TU Muenchen

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 frame 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.
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.7.5.1 EGCD		gcd over an algebraic extension field of Q
D.7.5.2 Factor		factorization of f over an algebraic extension field
D.7.5.3 Quotient		quotient q of f w.r.t. g (in f = q*g + remainder)
D.7.5.4 Remainder		remainder of the division of f by g
D.7.5.5 Roots		computes all roots of f in an extension field of Q
D.7.5.6 SQFRNorm		norm of f (f must be squarefree)
D.7.5.7 ZeroSet		zero-set of the 0-dim. ideal I

Auxiliary procedures:

D.7.5.8 EGCDMain		gcd over an algebraic extension field of Q
D.7.5.9 FactorMain		factorization of f over an algebraic extension field
D.7.5.10 InvertNumberMain		inverts an element of an algebraic extension field
D.7.5.11 QuotientMain		quotient of f w.r.t. g
D.7.5.12 RemainderMain		remainder of the division of f by g
D.7.5.13 RootsMain		computes all roots of f, might extend the ground field
D.7.5.14 SQFRNormMain		norm of f (f must be squarefree)
D.7.5.15 ContainedQ		f in data ?
D.7.5.16 SameQ		a == b (list a,b)