additive_multivariate_generating_series

Additive rational multivariate functions

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

\[\frac{P}{\prod L_i^{d_i}}\]

where the L_i are linear forms (ie homogeneous degree one) and d_i are positive integers.

class surface_dynamics.misc.additive_multivariate_generating_series.AdditiveMultivariateGeneratingSeries(parent, data, allow_multiple=False, check=True)

EXAMPLES:

sage: from surface_dynamics.misc.additive_multivariate_generating_series import AdditiveMultivariateGeneratingSeriesRing

sage: A = AdditiveMultivariateGeneratingSeriesRing('x', 3)

factor()

Group all the partial fractions into a unique fraction.

EXAMPLES:

sage: from surface_dynamics.misc.additive_multivariate_generating_series import AdditiveMultivariateGeneratingSeriesRing

sage: A = AdditiveMultivariateGeneratingSeriesRing('x', 3)

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

Simplification:

sage: x0, x1, x2 = A.polynomial_ring().gens()
sage: f1 = A.term(1, [([1,0,0],1)])
sage: f2 = A.term(1, [([0,1,0],1)])
sage: f3 = A.term(x0 + x1, [([1,0,0],1), ([0,1,0],1)])
sage: (f1 + f2 - f3).factor()
0

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

integral_sum_as_mzv()

Make the sum over all possible positive integers and express the result as a linear combination of multiple zeta values.

EXAMPLES:

sage: from surface_dynamics.misc.additive_multivariate_generating_series import AdditiveMultivariateGeneratingSeriesRing

sage: A = AdditiveMultivariateGeneratingSeriesRing('x', 3)
sage: f = A.term(1, [((1,0,0),2), ((0,1,0),2), ((0,0,1),2)])
sage: f.integral_sum_as_mzv()
36*ζ(1,1,4) + 24*ζ(1,2,3) + 12*ζ(1,3,2) + 12*ζ(2,1,3) + 6*ζ(2,2,2)

subs(*args, **kwds)

Variable substitution by linear forms.

EXAMPLES:

sage: from surface_dynamics.misc.additive_multivariate_generating_series import AdditiveMultivariateGeneratingSeriesRing

sage: A = AdditiveMultivariateGeneratingSeriesRing('x', 2)
sage: R = A.polynomial_ring()
sage: x0, x1 = R.gens()

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

sage: f.subs(x0=x0+x1)
(x0)/((x0 + x1)*(x0 + 2*x1)^2) + (2)/((x0 + x1)^2*(x0 + 3*x1)^3)

sage: f.subs(x1=x0)
(2)/((x0)^2*(3*x0)^3)

sage: f.subs(x0=1)
Traceback (most recent call last):
...
ValueError: only homogeneous degree 1 substitution are allowed
sage: f.subs(x0=x0*x1)
Traceback (most recent call last):
...
ValueError: only homogeneous degree 1 substitution are allowed

to_rational_function(R=None)

Return the multivariate rational function corresponding to this element.

EXAMPLES:

sage: from surface_dynamics.misc.additive_multivariate_generating_series import AdditiveMultivariateGeneratingSeriesRing
sage: A = AdditiveMultivariateGeneratingSeriesRing('x', 3)
sage: x0, x1, x2 = A.polynomial_ring().gens()
sage: f1 = A.term(1, [((1,0,0),2), ((0,1,0),2), ((0,0,1),2)])
sage: f2 = A.term(x0*x1 - 1, [((1,1,1),3)])
sage: g = (f1 + f2).to_rational_function()
sage: g
(x0^3*x1^3*x2^2 - x0^2*x1^2*x2^2 + x0^3 + 3*x0^2*x1 + 3*x0*x1^2 + x1^3 + 3*x0^2*x2 + 6*x0*x1*x2 + 3*x1^2*x2 + 3*x0*x2^2 + 3*x1*x2^2 + x2^3)/(x0^5*x1^2*x2^2 + 3*x0^4*x1^3*x2^2 + 3*x0^3*x1^4*x2^2 + x0^2*x1^5*x2^2 + 3*x0^4*x1^2*x2^3 + 6*x0^3*x1^3*x2^3 + 3*x0^2*x1^4*x2^3 + 3*x0^3*x1^2*x2^4 + 3*x0^2*x1^3*x2^4 + x0^2*x1^2*x2^5)
sage: f1.to_rational_function() + f2.to_rational_function() == g
True

class surface_dynamics.misc.additive_multivariate_generating_series.AdditiveMultivariateGeneratingSeriesRing(poly_ring)

Element

alias of AdditiveMultivariateGeneratingSeries