#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Alexander Hulpke.
##
## Copyright of GAP belongs to its developers, whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-2.0-or-later
##
## This file contains attributes, properties and operations for univariate
## polynomials over the rationals
##
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##
#F APolyProd(<a>,<b>,<p>) . . . . . . . . . . polynomial product a*b mod p
##
## <ManSection>
## <Func Name="APolyProd" Arg='a,b,p'/>
##
## <Description>
## return a</E>b mod p;
## </Description>
## </ManSection>
##
DeclareGlobalFunction("APolyProd");
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##
#F BPolyProd(<a>,<b>,<m>,<p>) . . . . . . polynomial product a*b mod m mod p
##
## <ManSection>
## <Func Name="BPolyProd" Arg='a,b,m,p'/>
##
## <Description>
## return EuclideanRemainder(PolynomialRing(Rationals),a</E>b mod p,m) mod p;
## </Description>
## </ManSection>
##
DeclareGlobalFunction("BPolyProd");
#############################################################################
##
#F PrimitivePolynomial( <f> )
##
## <#GAPDoc Label="PrimitivePolynomial">
## <ManSection>
## <Oper Name="PrimitivePolynomial" Arg='f'/>
##
## <Description>
## takes a polynomial <A>f</A> with rational coefficients and computes a new
## polynomial with integral coefficients, obtained by multiplying with the
## Lcm of the denominators of the coefficients and casting out the content
## (the Gcd of the coefficients). The operation returns a list
## [<A>newpol</A>,<A>coeff</A>] with rational <A>coeff</A> such that
## <C><A>coeff</A>*<A>newpol</A>=<A>f</A></C>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation("PrimitivePolynomial",[IsPolynomial]);
#############################################################################
##
#F BombieriNorm(<pol>)
##
## <#GAPDoc Label="BombieriNorm">
## <ManSection>
## <Func Name="BombieriNorm" Arg='pol'/>
##
## <Description>
## computes weighted Norm [<A>pol</A>]<M>_2</M> of <A>pol</A> which is a
## good measure for factor coefficients (see <Cite Key="BTW93"/>).
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("BombieriNorm");
#############################################################################
##
#A MinimizedBombieriNorm( <f> ) . . . Tschirnhaus transf'd polynomial
##
## <#GAPDoc Label="MinimizedBombieriNorm">
## <ManSection>
## <Attr Name="MinimizedBombieriNorm" Arg='f'/>
##
## <Description>
## This function applies linear Tschirnhaus transformations
## (<M>x \mapsto x + i</M>) to the
## polynomial <A>f</A>, trying to get the Bombieri norm of <A>f</A> small. It returns a
## list <C>[<A>new_polynomial</A>, <A>i_of_transformation</A>]</C>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute("MinimizedBombieriNorm",
IsPolynomial and IsRationalFunctionsFamilyElement);
#############################################################################
##
#F RootBound(<f>)
##
## <ManSection>
## <Func Name="RootBound" Arg='f'/>
##
## <Description>
## returns the bound for the norm of (complex) roots of the rational
## univariate polynomial <A>f</A>.
## </Description>
## </ManSection>
##
DeclareGlobalFunction("RootBound");
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##
#F OneFactorBound(<pol>)
##
## <#GAPDoc Label="OneFactorBound">
## <ManSection>
## <Func Name="OneFactorBound" Arg='pol'/>
##
## <Description>
## returns the coefficient bound for a single factor of the rational
## polynomial <A>pol</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("OneFactorBound");
#############################################################################
##
#F HenselBound(<pol>,[<minpol>,<den>]) . . . Bounds for Factor coefficients
##
## <#GAPDoc Label="HenselBound">
## <ManSection>
## <Func Name="HenselBound" Arg='pol,[minpol,den]'/>
##
## <Description>
## returns the Hensel bound of the polynomial <A>pol</A>.
## If the computation takes place over an algebraic extension, then
## the minimal polynomial <A>minpol</A> and denominator <A>den</A> must be given.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("HenselBound");
#############################################################################
##
#F TrialQuotientRPF(<f>,<g>,<b>)
##
## <ManSection>
## <Func Name="TrialQuotientRPF" Arg='f,g,b'/>
##
## <Description>
## returns <M><A>f</A>/<A>g</A></M> if coefficient bounds are given by list <A>b</A>.
## </Description>
## </ManSection>
##
DeclareGlobalFunction("TrialQuotientRPF");
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##
#F TryCombinations(<f>,...)
##
## <ManSection>
## <Func Name="TryCombinations" Arg='f,...'/>
##
## <Description>
## trial divisions after Hensel factoring.
## </Description>
## </ManSection>
##
DeclareGlobalFunction("TryCombinations");
DeclareGlobalFunction("HeuGcdIntPolsExtRep"); # to permit recursive call
DeclareGlobalFunction("HeuGcdIntPolsCoeffs"); # univariate version
#############################################################################
##
#F PolynomialModP(<pol>,<p>)
##
## <#GAPDoc Label="PolynomialModP">
## <ManSection>
## <Func Name="PolynomialModP" Arg='pol,p'/>
##
## <Description>
## for a rational polynomial <A>pol</A> this function returns a polynomial over
## the field with <A>p</A> elements, obtained by reducing the coefficients modulo
## <A>p</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("PolynomialModP");
