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/* Copyright (c) 1997-2020
Ewgenij Gawrilow, Michael Joswig, and the polymake team
Technische Universität Berlin, Germany
https://polymake.org
This program 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, or (at your option) any
later version: http://www.gnu.org/licenses/gpl.txt.
This program 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.
--------------------------------------------------------------------------------
*/
#include "polymake/client.h"
#include "polymake/Matrix.h"
#include "polymake/Array.h"
#include "polymake/SparseMatrix.h"
#include "polymake/Integer.h"
#include "polymake/Rational.h"
#include "polymake/common/lattice_tools.h"
#include "polymake/Smith_normal_form.h"
#include "polymake/integer_linalg.h"
namespace polymake { namespace fulton {
// rational_divisor_class_group provides a pair of matrices:
// a projection map to the class group, and a lifting map to a deivisor on the fan
//
// let A be the matrix with rows the primitive generators of the fan
// the projection equals Gale transform of A,
// i.e. a matrix G such that the cols g of G span the kernel of A in the form g^tA=0
// this implies that if divisors x,y
// (i.e. right hand sides for the polytope defined by the normal fan spanned by A)
// are mapped to the same element:
// if x^tG=y^tG, then there is p such that x-y=Ap, i.e. x and y differ by a linear map
// (or, seen as right hand side of a polytope, they differ by a translation)
//
// for the converse direction we write G as D=LGR for unimodular matrices L,R (Smith normal form)
// by construction, D is diagonal (as far as possible), and there are only ones on the diagonal
// we can then lift with R*L.minor(range(0,rk(D)-1),All)
std::pair<Matrix<Integer>, Matrix<Integer>> rational_divisor_class_group(BigObject fan)
{
Matrix<Rational> rational_rays = fan.give("RAYS");
Matrix<Integer> projection;
Matrix<Integer> t_int_rays = T(common::primitive(rational_rays));
// here we call the perl function "integer_kernel" defined in the
// lll-extension (http://polymake.org/polytopes/paffenholz/data/polymake/extensions/lll).
// this should be replaced by a c++-call
// once we can detect the ntl extension (see ticket #504), or ntl is a core extension
//
// Rem: pm now has integer_linalg.
// CallPolymakeFunction("integer_kernel",t_int_rays,false) >> projection;
projection = null_space_integer(t_int_rays);
SmithNormalForm<Integer> SNF = smith_normal_form(T(projection));
// the matrix D has only +1,-1,0 on the diagonal, but we need to know the positions of the -1's
// hence, we can't do the following, but really need to multiply (or find them otherwise)
//Matrix<Integer> lifting(R*L.minor(range(0,r-1),All));
projection = T(T(projection) * inv(SNF.right_companion));
Matrix<Integer> lifting(T(SNF.form) * inv(SNF.left_companion));
std::pair<Matrix<Integer>, Matrix<Integer> > class_group_data;
class_group_data.first = T(projection);
class_group_data.second = lifting;
return class_group_data;
}
Function4perl(&rational_divisor_class_group, "rational_divisor_class_group($)");
} }
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