GroupWithMorphisms compute_cokernel(const std::size_t p, const MatrixQ& f,
const AbelianGroup& Y,
const MatrixQRefList& to_Y_ref,
const MatrixQRefList& from_Y_ref)
{
MatrixQ f_rel_Y(f.height(), f.width() + Y.tor_rank());
f_rel_Y(0, 0, f.height(), f.width()) = f;
f_rel_Y(0, f.width(), Y.tor_rank(), Y.tor_rank()) = Y.torsion_matrix(p);
MatrixQList to_Y_copy = deref(to_Y_ref);
MatrixQList from_Y_copy = deref(from_Y_ref);
MatrixQRefList to_X;
MatrixQRefList from_X;
MatrixQRefList to_Y_copy_ref = ref(to_Y_copy);
MatrixQRefList from_Y_copy_ref = ref(from_Y_copy);
smith_reduce_p(p, f_rel_Y, to_X, from_X, to_Y_copy_ref, from_Y_copy_ref);
std::size_t rank_diff = 0;
std::size_t torsion_rank = 0;
for (std::size_t i = 0; i < std::min(f_rel_Y.height(), f_rel_Y.width());
++i) {
if (f_rel_Y(i, i) == 1)
++rank_diff;
else if (f_rel_Y(i, i) != 0)
++torsion_rank;
else
break;
}
GroupWithMorphisms C(f_rel_Y.height() - rank_diff - torsion_rank,
torsion_rank);
for (std::size_t i = rank_diff; i < rank_diff + torsion_rank; ++i)
C.group(i - rank_diff) =
static_cast<std::size_t>(p_val_q(p, f_rel_Y(i, i)));
for (MatrixQ& g_to_Y : to_Y_copy)
C.maps_to.emplace_back(
g_to_Y(rank_diff, 0, g_to_Y.height() - rank_diff, g_to_Y.width()));
for (MatrixQ& g_from_Y : from_Y_copy)
C.maps_from.emplace_back(g_from_Y(0, rank_diff, g_from_Y.height(),
g_from_Y.width() - rank_diff));
return C;
}
GroupWithMorphisms compute_image(const mod_t p, const MatrixQ& f,
const AbelianGroup& X, const AbelianGroup& Y)
{
MatrixQRefList to_X_dummy;
MatrixQList from_X = {MatrixQ::identity(f.width())};
GroupWithMorphisms K = compute_kernel(p, f, X, Y, to_X_dummy, ref(from_X));
MatrixQList to_X_2 = {MatrixQ::identity(X.rank())};
MatrixQList from_X_2 = {f,MatrixQ::identity(X.rank())};//hacky, since id:X->X doesn't vanish on K.
//but due to the implementation, the columns of this will contain
//representatives in X for the generators of img.
GroupWithMorphisms img =
compute_cokernel(p, K.maps_from[0], X, ref(to_X_2), ref(from_X_2));
return img;
}
AbelianEquationsSolver::AbelianEquationsSolver( const AbelianGroup& a ,
const VectorOf<Word>& v ,
int numOfVar )
: rawA( a ),
A( FPGroup() ),
rawSystem( v ),
system( v.length() ),
b( v.length() ),
x( numOfVar ),
torsion( numOfVar ),
params( numOfVar ),
numberOfVariables( numOfVar ),
sysRank( 0 ),
haveSol( -1 )
{
FPGroup G( a.getFPGroup() );
VectorOf<Chars> q( G.numberOfGenerators() - numberOfVariables );
for( int i = numberOfVariables ; i < G.numberOfGenerators() ; i++ )
q[ i - numberOfVariables ] = (G.namesOfGenerators())[i];
if( G.getRelators().cardinality() )
{
SetOf<Word> s = G.getRelators();
SetIterator<Word> I(s);
SetOf<Word> news;
while( !I.done() )
{
Word w = I.value();
for( int j = 0 ; j < w.length() ; j++ )
{
int p = Generator( w[j] ).hash();
if( p > 0 )
w[j] = Generator( p - numberOfVariables );
else
w[j] = Generator( p + numberOfVariables );
}
news.adjoinElement( w );
I.next();
}
FPGroup G1( q , news );
A = AbelianGroup( G1 );
}
else
A = AbelianGroup( FPGroup(q) );
}
bool morphism_equal(std::size_t p, const MatrixQ& f, const MatrixQ& g,
const AbelianGroup& Y)
{
for (std::size_t i = 0; i < f.height(); i++) {
for (std::size_t j = 0; j < f.width(); j++) {
if (f(i, j) != g(i, j)) {
if (i >= Y.tor_rank() ||
static_cast<std::size_t>(p_val_q(p, f(i, j) - g(i, j))) < Y(i)) {
return false;
}
}
}
}
return true;
}
// lifts a map from f:F -> Y over the map map: X -> Y. We only need relations
// for Y.
// Remark 1: does NOT catch if such a lift doesn't exist!
// Remark 2: If the map X -> Y is injective, then for a map A -> Y the lift F_A
// -> X of the map
// F_A -> Y induces a well-defined map A -> X.
// In general, the problem whether there IS such a lift, and how to
// compute it,
// will involve additional work.
MatrixQ lift_from_free(const mod_t p, const MatrixQ& f, const MatrixQ& map,
const AbelianGroup& Y)
{
MatrixQ rel_y_map(map.height(), Y.tor_rank() + map.width());
rel_y_map(Y.free_rank(), 0, Y.tor_rank(), Y.tor_rank()) = Y.torsion_matrix(p);
rel_y_map(0, Y.tor_rank(), map.height(), map.width()) = map;
MatrixQ proj(map.width(), Y.tor_rank() + map.width());
proj(0, Y.tor_rank(), map.width(), map.width()) =
MatrixQ::identity(map.width());
MatrixQ f_copy(f);
MatrixQRefList to_X_dummy;
MatrixQRefList from_Y_dummy;
MatrixQRefList from_X_ref;
MatrixQRefList to_Y_ref;
from_X_ref.emplace_back(proj);
to_Y_ref.emplace_back(f_copy);
smith_reduce_p(p, rel_y_map, to_X_dummy, from_X_ref, to_Y_ref, from_Y_dummy);
MatrixQ lift(rel_y_map.width(), f.width());
dim_t d_max = 0;
while (d_max < rel_y_map.height() && d_max < rel_y_map.width() &&
rel_y_map(d_max, d_max) != 0) {
d_max++;
}
for (dim_t i = 0; i < d_max; i++) {
for (dim_t j = 0; j < f.width(); j++) {
lift(i, j) = f_copy(i, j) / rel_y_map(i, i);
}
}
return proj * lift;
}
GroupWithMorphisms compute_kernel(const std::size_t p, const MatrixQ& f,
const AbelianGroup& X, const AbelianGroup& Y,
const MatrixQRefList& to_X_ref,
const MatrixQRefList& from_X_ref)
{
MatrixQ f_rel_Y(f.height(), f.width() + Y.tor_rank());
f_rel_Y(0, 0, f.height(), f.width()) = f;
f_rel_Y(0, f.width(), Y.tor_rank(), Y.tor_rank()) = Y.torsion_matrix(p);
MatrixQ rel_x_lift(f.width() + Y.tor_rank(), X.tor_rank());
rel_x_lift(0, 0, X.tor_rank(), X.tor_rank()) = X.torsion_matrix(p);
// rel_x_lift(f.width(), 0, Y.tor_rank(), X.tor_rank()) = -lift of f\circ
// rel_x over rel_Y.
// Can be computed by multiplying the columns of f with the orders of X,
// and dividing the rows by the orders of Y.
for (std::size_t i = 0; i < Y.tor_rank(); ++i) {
for (std::size_t j = 0; j < X.tor_rank(); ++j) {
long order_diff = static_cast<long>(X(j) - Y(i));
rel_x_lift(f.width() + i, j) = -f(i, j) * p_pow_q(p, order_diff);
}
}
// build to_X_rel_Y, from_X_rel_Y.
MatrixQList to_X_rel_Y;
for (MatrixQ& g_to_X : to_X_ref) {
to_X_rel_Y.emplace_back(f.width() + Y.tor_rank(), g_to_X.width());
to_X_rel_Y.back()(0, 0, f.width(), g_to_X.width()) = g_to_X;
MatrixQ fg = f * g_to_X;
for (std::size_t i = 0; i < Y.tor_rank(); ++i) {
for (std::size_t j = 0; j < g_to_X.width(); ++j) {
to_X_rel_Y.back()(f.width() + i, j) = -fg(i, j) / p_pow_z(p, Y(i));
}
}
}
MatrixQList from_X_rel_Y;
for (MatrixQ& g_from_X : from_X_ref) {
from_X_rel_Y.emplace_back(g_from_X.height(), f.width() + Y.tor_rank());
from_X_rel_Y.back()(0, 0, g_from_X.height(), f.width()) = g_from_X;
}
MatrixQRefList to_X_rel_Y_ref = ref(to_X_rel_Y);
MatrixQRefList from_X_rel_Y_ref = ref(from_X_rel_Y);
to_X_rel_Y_ref.emplace_back(rel_x_lift);
MatrixQRefList to_Y;
MatrixQRefList from_Y;
smith_reduce_p(p, f_rel_Y, to_X_rel_Y_ref, from_X_rel_Y_ref, to_Y, from_Y);
std::size_t rank_diff;
for (rank_diff = 0; rank_diff < std::min(f_rel_Y.height(), f_rel_Y.width());
++rank_diff) {
if (f_rel_Y(rank_diff, rank_diff) == 0) break;
}
// next, restrict attention to the entries corresponding to zero columns of
// f_rel_Y:
// for rel_x_lift as well as the entries of to_X_rel_Y, take the submatrices
// formed
// by the corresponding rows. For from_X_rel_Y, take the submatrices formed
// by
// the corresponding columns.
MatrixQ rel_K = rel_x_lift(rank_diff, 0, rel_x_lift.height() - rank_diff,
rel_x_lift.width());
MatrixQList to_free_K;
for (MatrixQ& g_to_X_rel_Y : to_X_rel_Y) {
to_free_K.emplace_back(g_to_X_rel_Y(
rank_diff, 0, g_to_X_rel_Y.height() - rank_diff, g_to_X_rel_Y.width()));
}
MatrixQList from_free_K;
for (MatrixQ& g_from_X_rel_Y : from_X_rel_Y) {
from_free_K.emplace_back(
g_from_X_rel_Y(0, rank_diff, g_from_X_rel_Y.height(),
g_from_X_rel_Y.width() - rank_diff));
}
MatrixQRefList to_free_K_ref = ref(to_free_K);
MatrixQRefList from_free_K_ref = ref(from_free_K);
AbelianGroup free_K(rel_K.height(), 0);
// then, compute the cokernel of the new rel_x_lift with the respective
// to_Y, from_Y.
return compute_cokernel(p, rel_K, free_K, to_free_K_ref, from_free_K_ref);
}