multiplicative_multivariate_generating_series

Multiplicative rational multivariate functions

This module implements the algebra generated by rational functions of the form

\[\frac{P}{\prod (1 - m_i)^{d_i}}\]

where the m_i are monomials and d_i are positive integers.

REFERENCES:

[LHTY] De Loera, R. Hemmecke, J. Tauzer, R. Yoshida

“Effective lattice point counting in rational convex polytopes” (LattE)

[Ba] Barvinok

[St] Stanley “Enumerative Combinatorics” Volume I Cambridge Studies in Advanced Mathematics 49 (1997)

class surface_dynamics.misc.multiplicative_multivariate_generating_series.MultiplicativeMultivariateGeneratingSeries(parent, data, allow_multiple=False, check=True)

as_symbolic()

delta()

Take a derivative (up to lower order terms) with respect to each of the variables.

derivative(var)

EXAMPLES:

sage: from surface_dynamics.misc.multiplicative_multivariate_generating_series import MultiplicativeMultivariateGeneratingSeriesRing

sage: M = MultiplicativeMultivariateGeneratingSeriesRing('x', 2)
sage: R = M.polynomial_ring()
sage: x0, x1 = R.gens()

sage: f = M.term(1, [((1,0),1)])
sage: f.derivative(0)
(1)/((1 - x0)^2)
sage: f.derivative(0).derivative(0)
(2)/((1 - x0)^3)

sage: xx0, xx1 = R.polynomial_ring().gens()
sage: f.taylor(10).derivative(xx0) == f.derivative(0).taylor(9)
True
sage: f.derivative(1)
0

sage: f = M.term(x0 * x1, [((1,1),2)])
sage: f.derivative(0)
(x1)/((1 - x0*x1)^2) + (2*x0*x1^2)/((1 - x0*x1)^3)
sage: f.derivative(0).taylor(10) - f.taylor(10).derivative(xx0)
30*x0^5*x1^6

sage: f = M.term(1, [((1,2),2), ((0,1),2), ((1,0),2), ((1,1),2)])
sage: min((f.taylor(10).derivative(xx0) - f.derivative(0).taylor(10)).degrees())
9
sage: min((f.taylor(10).derivative(xx1) - f.derivative(1).taylor(10)).degrees())
10

sage: f = M.term(1, [((1,0),2), ((1,1),1)])
sage: f.taylor(10).derivative(xx0).derivative(xx1) - f.derivative(0).derivative(1).taylor(9)
-18*x0^9*x1 - 42*x0^8*x1^2 - ... - 72*x0^5*x1^3 - 25*x0^4*x1^4

sage: f = M.zero() + M.term(1/1*x0^2 * x1, [((1,0),1), ((1,1),1)])
sage: f.derivative(0)
(2*x0*x1)/((1 - x0)*(1 - x0*x1)) + (x0^2*x1^2)/((1 - x0)*(1 - x0*x1)^2) + (x0^2*x1)/((1 - x0)^2*(1 - x0*x1))

You can indistinctly use integers, strings or polynomial variables for var:

sage: f = M.term(1, [((1,1),1)])
sage: f.derivative('x0')
(x1)/((1 - x0*x1)^2)
sage: f.derivative(0)
(x1)/((1 - x0*x1)^2)
sage: f.derivative(R.gen(0))
(x1)/((1 - x0*x1)^2)

Checking errors in the input:

sage: f.derivative(-1)
Traceback (most recent call last):
...
ValueError: generator not defined
sage: f.derivative('q')
Traceback (most recent call last):
...
ValueError: 'q' not valid as a variable

derivative_up_to_lower_order_terms(var)

Each term u/v is replaced by -u v'/v^2.

This corresponds to one half of the derivative, the other half being u' / v. This second part can be ignored when asymptotics question are considered.

EXAMPLES:

sage: from surface_dynamics.misc.multiplicative_multivariate_generating_series import MultiplicativeMultivariateGeneratingSeriesRing

sage: M = MultiplicativeMultivariateGeneratingSeriesRing('x', 2)
sage: f = M.term(1, [((1,0),2)])
sage: f
(1)/((1 - x0)^2)
sage: f.derivative_up_to_lower_order_terms(0)
(2)/((1 - x0)^3)

sage: f = M.term(1, [((1,0),1),((0,1),1),((1,1),1)])

factor()

Group all the partial fractions into a unique fraction.

EXAMPLES:

sage: from surface_dynamics.misc.multiplicative_multivariate_generating_series import MultiplicativeMultivariateGeneratingSeriesRing
sage: M = MultiplicativeMultivariateGeneratingSeriesRing('x', 3)

sage: f = M.term(1, [([1,3,0],1),([1,0,-1],1)])
sage: g = M.term(1, [([1,1,0],1),([1,0,-1],2)])
sage: (f + g).factor()
(-x0*x1^3 + x0^2*x1*x2^-1 - x0*x1 - x0*x2^-1 + 2)/((1 - x0*x2^-1)^2*(1 - x0*x1)*(1 - x0*x1^3))

sage: x0, x1, x2 = M.polynomial_ring().gens()
sage: f = M.term(x0 + x1, [([1,3,0],1),([1,0,1],1)])
sage: g = M.term(x2 - 1, [([1,1,0],1),([1,0,1],2)])
sage: h = f + g
sage: t1 = h.taylor(8)
sage: t2 = h.factor().taylor(8)
sage: assert all(sum(e) >= 8 for e in (t1 - t2).exponents())

Simplification:

sage: f1 = M.term(x0, [([1,0,0],1)])
sage: f2 = M.term(x1, [([0,1,0],1)])
sage: f3 = M.term(x0*x1 - 1, [([1,0,0],1), ([0,1,0],1)])
sage: (f1 + f2 + f3).factor()
(-1)

sage: f1 + f2 + 1 == -f3  # indirect doctest
True

is_one()

is_zero()

numerator()

Return the numerator if there is at most one summand.

EXAMPLES:

sage: from surface_dynamics.misc.multiplicative_multivariate_generating_series import MultiplicativeMultivariateGeneratingSeriesRing
sage: M = MultiplicativeMultivariateGeneratingSeriesRing('x', 3)
sage: x0, x1, x2 = M.polynomial_ring().gens()

sage: f = M.term(x0 + x1^2*x2, [([1,3,0],1),([1,0,-1],1)])
sage: f.numerator()
x1^2*x2 + x0
sage: g = M.term(1 - x2, [([1,1,0],1),([1,0,-1],2)])
sage: (f + g).numerator()
Traceback (most recent call last):
...
ValueError: not a simple fraction
sage: (f + g).factor().numerator()
x0^2*x1^3 - x0*x1^3 + x0^3*x1*x2^-1 - x0^2*x1 - x0*x1^2 + x1^2*x2 - x0^2*x2^-1 + x0 - x2 + 1

numerator_subs(*args, **kwds)

residue()

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})])

subs_numerator(*args, **kwds)

TESTS:

sage: from surface_dynamics.misc.multiplicative_multivariate_generating_series import MultiplicativeMultivariateGeneratingSeriesRing
sage: M = MultiplicativeMultivariateGeneratingSeriesRing('x', 2)
sage: R = M.polynomial_ring()
sage: x0, x1 = R.gens()
sage: f = M.term(-x0 + 2*x0*x1^3, [((1,1), 1), ((1,2),2)])
sage: f.subs_numerator(x0=1, x1=1)
(1)/((1 - x0*x1)*(1 - x0*x1^2)^2)

summands()

Return the list of elementary summands

EXAMPLES:

sage: from surface_dynamics.misc.multiplicative_multivariate_generating_series import *
sage: M = MultiplicativeMultivariateGeneratingSeriesRing(2, 'x')
sage: m1 = M.term(1, [((1,-1),1), ((0,1),1)])
sage: m2 = M.term(1, [((-1,1),1), ((1,0),1)])
sage: f = (m1 + m2)**2
sage: for s in f.summands(): print(s)
(2)/((1 - x0^-1*x1)*(1 - x1)*(1 - x0*x1^-1)*(1 - x0))
(1)/((1 - x0^-1*x1)^2*(1 - x0)^2)
(1)/((1 - x1)^2*(1 - x0*x1^-1)^2)

taylor(prec, R=None)

EXAMPLES:

sage: from surface_dynamics.misc.multiplicative_multivariate_generating_series import MultiplicativeMultivariateGeneratingSeriesRing

sage: M = MultiplicativeMultivariateGeneratingSeriesRing('t,u')
sage: t,u = M.polynomial_ring().gens()
sage: f = M.term(t + u^3, [((1,1), 1), ((1,2),2)])
sage: f
(u^3 + t)/((1 - t*u)*(1 - t*u^2)^2)
sage: f.taylor(10)
2*t^4*u^8 + 4*t^3*u^9 + t^4*u^7 + 3*t^3*u^8 + ... + t*u^4 + 2*t^2*u^2 + t^2*u + u^3 + t

TODO: this is only working term by term but for example containing negative powers as:

sage: M = MultiplicativeMultivariateGeneratingSeriesRing('x', 2)   # not tested
sage: m1 = M.term(1, [((1,-1),1), ((0,1),1)])   # not tested
sage: m2 = M.term(1, [((-1,1),1), ((1,0),1)])   # not tested
sage: f = m1 + m2                               # not tested
sage: f.taylor(10)                              # not tested

TESTS:

sage: M = MultiplicativeMultivariateGeneratingSeriesRing('x', 2)
sage: R = M.polynomial_ring()
sage: x0, x1 = R.gens()

sage: import sage.rings.polynomial.multi_polynomial_libsingular
sage: isinstance(M.term(1, [((1,0),1)]).taylor(10), sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular)
True

class surface_dynamics.misc.multiplicative_multivariate_generating_series.MultiplicativeMultivariateGeneratingSeriesRing(poly_ring)

EXAMPLES:

sage: from surface_dynamics.misc.multiplicative_multivariate_generating_series import MultiplicativeMultivariateGeneratingSeriesRing

sage: MultiplicativeMultivariateGeneratingSeriesRing('x', 3)
Multiplicative multivariate generating series on x0, x1, x2

sage: MultiplicativeMultivariateGeneratingSeriesRing(['x', 'y', 'z'])
Multiplicative multivariate generating series on x, y, z

sage: MultiplicativeMultivariateGeneratingSeriesRing('a,b,c')
Multiplicative multivariate generating series on a, b, c

sage: MultiplicativeMultivariateGeneratingSeriesRing('y', 2)
Multiplicative multivariate generating series on y0, y1

sage: M = MultiplicativeMultivariateGeneratingSeriesRing('x', 3)
sage: M.zero()
0
sage: M.one()
(1)
sage: m1 = M.term(1, [((1,1,0),1)])
sage: m1
(1)/((1 - x0*x1))
sage: m2 = M.term(-2, [((1,0,0),1),((0,1,0),2)])
sage: m2
(-2)/((1 - x1)^2*(1 - x0))

Element

alias of MultiplicativeMultivariateGeneratingSeries

residue_ring()

Return the residue ring (the additive version of self).

EXAMPLES:

sage: from surface_dynamics.misc.multiplicative_multivariate_generating_series import MultiplicativeMultivariateGeneratingSeriesRing
sage: M = MultiplicativeMultivariateGeneratingSeriesRing('x', 3)
sage: M.residue_ring()
Additive multivariate generating series on x0, x1, x2

surface_dynamics.misc.multiplicative_multivariate_generating_series.latte_generating_series(L, M=None)

EXAMPLES:

sage: from surface_dynamics.misc.multiplicative_multivariate_generating_series import latte_generating_series
sage: ieqs = [[0, 1, 0, 0, 0, 0, 0],
....:         [0, 0, 1, -1, 1, 0, 0],
....:         [0, 0, 0, 0, 1, 0, 0],
....:         [0, 0, 0, 1, 0, 0, 0],
....:         [0, 0, 1, 0, 0, 0, 0]]
sage: eqns = [[0, 0, 1, -1, 1, 0, -1], [0, 0, 1, -1, 1, -1, 0]]
sage: L = Polyhedron(ieqs=ieqs, eqns=eqns)
sage: latte_generating_series(L)   # optional: latte_int
(1)/((1 - x2^-1*x4*x5)*(1 - x2*x3)*(1 - x1*x2)*(1 - x0)) + (1)/((1 - x3*x4*x5)*(1 - x2*x4^-1*x5^-1)*(1 - x1*x4*x5)*(1 - x0))

surface_dynamics.misc.multiplicative_multivariate_generating_series.parse_latte_generating_series(M, s)

INPUT:

• M - a multivariate generating series ring

• s - a string as given by LattE

OUTPUT: multivariate short rational function