def compute_series_truncations(f,x,y,alpha,T):
"""
Computes the Puiseux series expansions at the `x`-point `x=a` with
the necessary number of terms in order to compute the integral
basis of the algebraic functions field corresponding to `f`. The
Puiseux series is returned in parametric form for computational
efficiency. (Sympy doesn't do as well with fractional exponents.)
"""
# compute the first terms of the Puiseux series expansions
p = puiseux(f,x,y,alpha,nterms=1,parametric=False)
# compute the expansion bounds
N = compute_expansion_bounds(p,x,alpha)
Nmax = max(N)
# compute Puiseux series and truncate using the expansion bounds.
r = puiseux(f,x,y,alpha,degree_bound=Nmax)
n = len(r)
for i in xrange(n):
ri = r[i].subs(x,x+alpha).expand(log=False,power_base=False,
power_exp=False,multinomial=False,
basic=False,force=True)
terms = ri.collect(x,evaluate=False)
ri_trunc = sum( coeff*term for term,coeff in terms.iteritems()
if term.as_coeff_exponent(x)[1] < N[i] )
r[i] = ri_trunc
return r
def _delta_invariant(f,x,y,singular_pt):
"""
Returns the delta invariant corresponding to the singular point
`singular_pt` = [alpha, beta, gamma] on the plane algebraic curve
f(x,y) = 0.
"""
# compute the Puiseux series at the projective point [alpha,
# beta, gamma]. If on the line at infinity, make the
# appropriate variable transformation.
g,u,v,u0,v0 = _transform(f,x,y,singular_pt)
# compute Puiseux expansions at u=u0 and filter out
# only those with v(t=0) == v0
P = puiseux(g,u,v,u0,nterms=1,parametric=_t)
P_v0 = [(X,Y) for X,Y in P if Y.subs(_t,0) == v0]
P_v0_x = []
for X,Y in P_v0:
solns = sympy.solve(u-X,_t)
P_v0_x.append(Y.subs(_t,solns[0]).simplify().collect(u-u0))
P_x = puiseux(g,u,v,u0,nterms=1,parametric=False)
# for each place compute its contribution to the delta invariant
delta = sympy.Rational(0,1)
for i in range(len(P_v0_x)):
yhat = P_v0_x[i]
j = P_x.index(yhat)
IntPj = Int(j,P_x,u,u0)
rj = (P[i][0]-u0).expand().leadterm(_t)[1]
delta += sympy.Rational(rj * IntPj - rj + 1, 2)
return int(delta)
def _multiplicity(f,x,y,singular_pt):
"""
Returns the multiplicity of the place (alpha : beta : 1) from the
Puiseux series P at the place.
For each (parametric) Puiseux series
Pj = { x = x(t)
{ y = y(t)
at (alpha : beta : 1) the contribution from Pj to the multiplicity
is min( deg x(t), deg y(t) ).
"""
# compute the Puiseux series at the projective point [alpha,
# beta, gamma]. If on the line at infinity, make the
# appropriate variable transformation.
g,u,v,u0,v0 = _transform(f,x,y,singular_pt)
# compute Puiseux expansions at u=u0 and filter out
# only those with v(t=0) == v0
P = puiseux(g,u,v,u0,nterms=1,parametric=_t)
m = 0
for X,Y in P:
X = X - u0 # Shift so no constant
Y = Y - v0 # term remains.
ri = abs( X.leadterm(_t)[1] ) # Get order of lead term
si = abs( Y.leadterm(_t)[1] )
m += min(ri,si)
return int(m)
def _branching_number(f,x,y,singular_pt):
"""
Returns the branching number of the place [alpha : beta : 1]
from the Puiseux series P at the place.
The braching number is simply the number of distinct branches
(i.e. non-interacting branches) at the place. In parametric form,
this is simply the number of Puiseux series at the place.
"""
# compute the Puiseux series at the projective point [alpha,
# beta, gamma]. If on the line at infinity, make the
# appropriate variable transformation.
g,u,v,u0,v0 = _transform(f,x,y,singular_pt)
# compute Puiseux expansions at u=u0 and filter out
# only those with v(t=0) == v0
P = puiseux(g,u,v,u0,nterms=1,parametric=_t)
P_v0 = [(X,Y) for X,Y in P if Y.subs(_t,0) == v0]
return len(P_v0)