def newton_data(H, exceptional=False):
r"""Determines the "newton data" associated with each side of the polygon.
For each side :math:`\Delta` of the Newton polygon of `H` we
associate the data :math:`(q,m,l,`phi)` where
.. math::
\Delta: qj + mi = l \\
\phi_{\Delta}(t) = \sum_{(i,j) \in \Delta} a_{ij} t^{(i-i_0)/q}
Here, :math:`a_ij x^j y_i` is a term in the polynomial :math:`H` and
:math:`i_0` is the smallest value of :math:`i` belonging to the
polygon side :math:`\Delta`.
Parameters
----------
H : sympy.Poly
Polynomial in `x` and `y`.
Returns
-------
list
A list of the tuples :math:`(q,m,l,\phi)`.
"""
R = H.parent()
x, y = R.gens()
if exceptional:
newton = newton_polygon_exceptional(H)
else:
newton = newton_polygon(H)
# special case when the newton polygon is a single point
if len(newton[0]) == 1:
return []
# for each side dtermine the corresponding newton data: side slope
# information and corresponding side characteristic polynomial, phi
result = []
for side in newton:
i0, j0 = side[0]
i1, j1 = side[1]
slope = QQ(j1 - j0) / QQ(i1 - i0)
q = slope.denom()
m = -slope.numer()
l = min(q * j0 + m * i0, q * j1 + m * i1)
phi = sum(
H.coefficient({
y: i,
x: j
}) * x**((i - i0) / q) for i, j in side)
phi = phi.univariate_polynomial()
result.append((q, m, l, phi))
return result
def newton_data(H, exceptional=False):
r"""Determines the "newton data" associated with each side of the polygon.
For each side :math:`\Delta` of the Newton polygon of `H` we
associate the data :math:`(q,m,l,`phi)` where
.. math::
\Delta: qj + mi = l \\
\phi_{\Delta}(t) = \sum_{(i,j) \in \Delta} a_{ij} t^{(i-i_0)/q}
Here, :math:`a_ij x^j y_i` is a term in the polynomial :math:`H` and
:math:`i_0` is the smallest value of :math:`i` belonging to the
polygon side :math:`\Delta`.
Parameters
----------
H : sympy.Poly
Polynomial in `x` and `y`.
Returns
-------
list
A list of the tuples :math:`(q,m,l,\phi)`.
"""
R = H.parent()
x,y = R.gens()
if exceptional:
newton = newton_polygon_exceptional(H)
else:
newton = newton_polygon(H)
# special case when the newton polygon is a single point
if len(newton[0]) == 1:
return []
# for each side dtermine the corresponding newton data: side slope
# information and corresponding side characteristic polynomial, phi
result = []
for side in newton:
i0,j0 = side[0]
i1,j1 = side[1]
slope = QQ(j1-j0)/QQ(i1-i0)
q = slope.denom()
m = -slope.numer()
l = min(q*j0 + m*i0, q*j1 + m*i1)
phi = sum(H.coefficient({y:i,x:j})*x**((i-i0)/q) for i,j in side)
phi = phi.univariate_polynomial()
result.append((q,m,l,phi))
return result
