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// polys.cc : implements interface to NTL polynomials
//////////////////////////////////////////////////////////////////////////
//
// Copyright 1990-2023 John Cremona
//
// This file is part of the eclib package.
//
// eclib is free software; you can redistribute it and/or modify it
// under the terms of the GNU General Public License as published by the
// Free Software Foundation; either version 2 of the License, or (at your
// option) any later version.
//
// eclib is distributed in the hope that it will be useful, but WITHOUT
// ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
// FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
// for more details.
//
// You should have received a copy of the GNU General Public License
// along with eclib; if not, write to the Free Software Foundation,
// Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA
//
//////////////////////////////////////////////////////////////////////////
#include <eclib/polys.h>
FqPoly reduce(const ZPoly& f, const galois_field& Fq)
{
NewFqPoly(Fq,fmodq);
SetDegree(fmodq,Degree(f));
for(int i=0; i<=Degree(f); i++)
SetCoeff(fmodq,i,ZtoGF(Fq,PolyCoeff(f,i)));
return fmodq;
}
vector<gf_element> roots(const FqPoly& f)
{
// make f monic:
FqPoly f1=f;
MakeMonic(f1);
// reduce to distinct roots case:
ZZ_pX X; SetX(X);
ZZ_pX g = PowerXMod(ZZ_p::modulus(),f1)-X;
vec_ZZ_p r; FindRoots(r,GCD(f1,g));
vector<gf_element>ans;
for(int i=0; i<r.length(); i++) ans.push_back(r[i]);
return ans;
}
vector<bigint> rootsmod(const vector<bigint>& coeffs, bigint q)
{
galois_field Fq(q);
NewFqPoly(Fq,f);
unsigned long i, deg = coeffs.size()-1;
SetDegree(f,deg);
for (i=0; i<=deg; i++) SetCoeff(f,i,ZtoGF(Fq,coeffs[i]));
vector<gf_element> r = roots(f);
vector<bigint>ans;
for(i=0; i<r.size(); i++) ans.push_back(LiftGF(r[i]));
sort(ans.begin(),ans.end());
return ans;
}
//#define TRACE_ROOTS
vector<bigrational> roots(const ZPoly& f)
{
#ifdef TRACE_ROOTS
cout<<"Finding rational roots of polynomial f = "<<f<<endl;
#endif
vector<bigrational> ans;
int i;
ZPoly g;
bigrational root;
ZZ c;
vec_pair_ZZX_long factors;
factor(c,factors,f);
#ifdef TRACE_ROOTS
cout<<"f has " << factors.length() << " factors" << endl;
#endif
for(i=0; i<factors.length(); i++)
{
g = factors[i].a;
#ifdef TRACE_ROOTS
cout<<"factor "<<g<<" has degree "<<deg(g)<<endl;
#endif
if(deg(g)==1)
{
root = bigrational(-coeff(g,0),coeff(g,1));
#ifdef TRACE_ROOTS
cout<<"root "<<root<<endl;
#endif
ans.push_back(root);
}
}
sort(ans.begin(), ans.end());
return ans;
}
// Return the list of *integral* roots of an integral polynomial.
// Intended for monic polys, but that is not a requirement
vector<bigint> introots(const ZPoly& f)
{
#ifdef TRACE_ROOTS
cout<<"Finding integer roots of polynomial f = "<<f<<endl;
#endif
vector<bigrational> ratroots = roots(f);
vector<bigint> ans;
if (ratroots.size()==0)
return ans;
for( const auto& r : ratroots)
if (den(r)==1)
ans.push_back(num(r));
sort(ans.begin(), ans.end());
return ans;
}
vector<bigrational> roots(const vector<bigint>& coeffs)
{
#ifdef TRACE_ROOTS
cout<<"Finding rational roots of polynomial f with coefficients "<<coeffs<<endl;
#endif
ZZX f;
vector<bigrational> ans;
int i, d = coeffs.size()-1; // degree
if(d<1)
return ans;
for(i=0; i<=d; i++)
SetCoeff(f,d-i,coeffs[i]);
#ifdef TRACE_ROOTS
cout<<"f = "<<f<<endl;
#endif
ans = roots(f);
#ifdef TRACE_ROOTS
cout<<"roots of f: "<< ans << endl;
#endif
return ans;
}
// root-finding functions for monic integer cubics and quartics
//
// With NTL we factor the polynomial in Z[X] and pick out degree 1 factors
vector<bigint> Introotscubic(const bigint& a, const bigint& b, const bigint& c)
{
ZZX f;
SetCoeff(f,3); // sets it to 1
SetCoeff(f,2,a);
SetCoeff(f,1,b);
SetCoeff(f,0,c);
return introots(f);
}
vector<bigint> Introotsquartic(const bigint& a, const bigint& b, const bigint& c, const bigint& d)
{
ZZX f; vec_pair_ZZX_long factors; bigint cont;
SetCoeff(f,4); // sets it to 1
SetCoeff(f,3,a);
SetCoeff(f,2,b);
SetCoeff(f,1,c);
SetCoeff(f,0,d);
return introots(f);
}
// find the number of roots of X^3 + bX^2 + cX + d = 0 (mod p)
int nrootscubic(const bigint& b, const bigint& c, const bigint& d, const bigint& p)
{
vector<bigint> coeffs;
coeffs.push_back(d);
coeffs.push_back(c);
coeffs.push_back(b);
coeffs.push_back(bigint(1));
return rootsmod(coeffs,p).size();
}
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