def solve_for_one_nonzero_variable(conditions,vars):
conds=[]
for var in vars:
conds.append(conditions+[var != 0])
#print conds
solutions=SE.solve_ext(conds,*vars)
return solutions
def max_index_cond_k(cond_tuples,k,var_vec,in_pos_eigenspace_conds):
cond_k = cond_tuples[k]
rest_conds = map(lambda cts:map(lambda ct: ct[1], cts),
cond_tuples[0:k] + cond_tuples[k+1:])
rest_conditions_combinations = map(lambda css: list(itertools.chain(*css)),
SE.any_combination_of(rest_conds))
vars=in_pos_eigenspace_conds[1]+in_pos_eigenspace_conds[2]
# print "cond_k",cond_k
# print "rest_conds",rest_conds
# print "combinations",rest_conditions_combinations
for idx,conds in cond_k:
cond_disjunct=map(lambda cs:cs+in_pos_eigenspace_conds[0]+conds,rest_conditions_combinations)
# print idx, cond_disjunct
# print idx, in_pos_eigenspace_conds[0]
# print idx, conds
# print idx, in_pos_eigenspace_conds[1]+in_pos_eigenspace_conds[2]
solutions=SE.solve_ext(cond_disjunct,*vars)
# print "solutions:", solutions
if len(solutions) > 0:
return (idx,solutions)
return (None,[])
def find_Q_min(S_min,space_exts):
def solve_for_one_nonzero_variable(conditions,vars):
conds=[]
for var in vars:
conds.append(conditions+[var != 0])
#print conds
solutions=SE.solve_ext(conds,*vars)
return solutions
def gen_vars(col_count,exts):
vardict=dict()
vars=[]
for c in range(col_count):
vardict[c]=dict()
for e_id in range(len(exts)):
v=var("v_c"+str(c)+"_e"+str(e_id))
vardict[c][exts[e_id]]=v
vars.append(v)
assume(v,'rational')
return vardict,vars
def solution_to_coefficients(solution,vardict,vars):
idict=dict()
for eq in solution:
for v in vars:
if eq.operator() == operator.eq:
if eq.lhs() == v:
idict[v]=eq.rhs()
if eq.rhs() == v:
idict[v]=eq.lhs()
if eq.operator() == operator.ne:
if (eq.lhs() == v and eq.rhs() == 0) or (eq.rhs() == v and eq.lhs() == 0):
idict[v] = SR(1)
idict = dict(map(lambda x:(x[0],x[1].substitute(idict)),idict.items())) # maybe mor than one subtitution step is needed
coeffs=[]
for c in vardict.keys():
coeff=0
for ext in vardict[c].keys():
v = vardict[c][ext]
if not idict.has_key(v):
idict[v]=0
coeff = coeff + idict[v]*ext.radical_expression()
coeffs.append(coeff)
# print "idict",idict
return coeffs
S_min_matrix = S_min.matrix().transpose()
dims = S_min_matrix.dimensions()
vecs = S_min_matrix.columns()
exts = map(abs,[AA(1)]+space_exts) # it does not suffice to take the space extension of S_min
# print exts
conditions=[]
vardict,vars=gen_vars(dims[1],exts)
for r in range(dims[0]):
#print "row", S_min_matrix.row(r)
d=dict()
for c in range(dims[1]):
for e_id in range(len(exts)):
num_part=QQbar(S_min_matrix[r,c])*exts[e_id]
nf_gens=QQbar(num_part).as_number_field_element()[2].im_gens()
# print "np:",vardict[c][exts[e_id]],num_part,nf_gens
signf=1
if QQbar(num_part) < 0:
signf = -1
for gen in nf_gens:
#v=var("v_c"+str(c)+"_e"+str(e_id))
v=vardict[c][exts[e_id]]
k=abs(gen) # only use positive extensions as key
if d.has_key(k):
d[k].append(signf*v)
else:
d[k] = [signf*v]
# print "d:",d
for ext in d.keys():
#print ext,ext != 1
if ext != 1:
conditions.append(sum(d[ext]) == 0)
# print "basic conditions", conditions
# print vardict
# print vars
solutions= solve_for_one_nonzero_variable(conditions,vars) # find as many possible solutions to get all vectors of Q_min
# print solutions
# reduce to real solutions not containing 0 != 0
solutions= SE.solve_ext(solutions,vars)
# print solutions
forget() # forget that the variables are rational
coeffs=map(lambda s: solution_to_coefficients(s,vardict,vars),solutions)
# print "coeffs",coeffs
vecs=map(lambda v:MX.linear_combination_of(vector(v),S_min_matrix),coeffs)
# print vecs
V=VectorSpace(QQ,S_min.degree())
Q_min=V.span(vecs)
return Q_min