def residue(self):
r"""
denominator: each (1 - mon)^k in denom is replaced with -> mon^k
numerator: evaluate at (1,1,...,1)
OUTPUT: a pair '(degree, value)`
EXAMPLES::
sage: from surface_dynamics.misc.multiplicative_multivariate_generating_series import MultiplicativeMultivariateGeneratingSeriesRing
sage: M = MultiplicativeMultivariateGeneratingSeriesRing(2, 'x')
sage: R = M.polynomial_ring()
sage: x0,x1 = R.gens()
sage: f = M.term(x0, [((1,1),2)])
sage: f.residue()
(2, [(1, {(1, 1): 2})])
sage: f = M.term(x0, [((1,1),2)]) + M.term(1, [((1,0),1),((0,1),1),((1,1),1)])
sage: r = f.residue()
sage: r # random
(3, [(1, {(1, 0): 1, (0, 1): 1, (1, 1): 1})])
sage: f = M.term(x0, [((1,1),2)]) + M.term(1, [((1,0),1),((1,1),1)])
sage: r = f.residue()
sage: r # random
(2, [(1, {(1, 1): 2}), (1, {(1, 0): 1, (1, 1): 1})])
"""
R = self.parent().polynomial_ring()
one = QQ.one()
values = {g: one for g in R.gens()}
ans = []
d = -1
for den, num in self._data.items():
if den.degree() >= d:
if den.degree() > d:
ans = []
d = den.degree()
num = QQ(num.subs(values))
if num.is_zero():
raise NotImplementedError('zero numerator')
ans.append((num, den))
return d, ans
def residue(self):
r"""
denominator: each (1 - mon)^k in denom is replaced with -> mon^k
numerator: evaluate at (1,1,...,1)
OUTPUT: a pair '(degree, value)`
EXAMPLES::
sage: from surface_dynamics.misc.multivariate_generating_series import MultivariateGeneratingSeriesRing
sage: M = MultivariateGeneratingSeriesRing(2, 'x')
sage: R = M.laurent_polynomial_ring()
sage: x0,x1 = R.gens()
sage: f = M.term(x0, [((1,1),2)])
sage: f.residue()
(2, [(1, {(1, 1): 2})])
sage: f = M.term(x0, [((1,1),2)]) + M.term(1, [((1,0),1),((0,1),1),((1,1),1)])
sage: f.residue()
(3, [(1, {(0, 1): 1, (1, 0): 1, (1, 1): 1})])
sage: f = M.term(x0, [((1,1),2)]) + M.term(1, [((1,0),1),((1,1),1)])
sage: f.residue()
(2, [(1, {(1, 0): 1, (1, 1): 1}), (1, {(1, 1): 2})])
"""
R = self.parent().laurent_polynomial_ring()
one = QQ.one()
values = {g: one for g in R.gens()}
ans = []
d = -1
for den, num in self._data.items():
if den.degree() >= d:
if den.degree() > d:
ans = []
d = den.degree()
num = QQ(num.subs(values))
if num.is_zero():
raise NotImplementedError('zero numerator')
ans.append((num, den))
return d, ans
