`helpers`¶

Miscellaneous mathematical helper functions.

exception sage_acsv.helpers.ACSVException(message, retry=False)[source]¶

property retry¶: Initialize self. See help(type(self)) for accurate signature.

class sage_acsv.helpers.Term(coefficient: Expression | AlgebraicNumber, pi_factor: Expression, base: Expression | AlgebraicNumber, power: Expression | AlgebraicNumber)[source]¶

A dataclass for storing the decomposed terms of an asymptotic expression.

INPUT:

coefficient – The coefficient of the term
pi_factor – The factor of pi in the term
base – The base of the term
power – The power of the term

OUTPUT:

A dataclass with the given attributes.

EXAMPLES:

sage: from sage_acsv.helpers import Term
sage: Term(1, 1/sqrt(pi), 4, -1/2)
Term(coefficient=1, pi_factor=1/sqrt(pi), base=4, power=-1/2)

base: Expression | AlgebraicNumber¶

coefficient: Expression | AlgebraicNumber¶

pi_factor: Expression¶

power: Expression | AlgebraicNumber¶

sage_acsv.helpers.collapse_zero_part(algebraic_number)[source]¶

sage_acsv.helpers.compute_hessian(H, variables, r, critical_point=None)[source]¶

Computes the Hessian of an implicitly defined function.

The computed matrix is the Hessian of the map

\[(t_1,...t_{d-1}) \mapsto \log(g(z_1t_1,...,z_{d-1}t_{d-1}))/g(z_1,...,z_{d-1}) + I\cdot (r_1t_1+...+r_{d-1}t_{d-1})/r_d \]

at a critical point where the partial derivative of \(H\) with respect to \(z_d\) is non-zero, and \(g\) determined implicitly by

\[H(z_1,...,z_{d-1}, g(z_1,...,z_{d-1})) = 0.\]

INPUT:

H – A polynomial; the denominator of the rational generating function \(F = G/H\).
vs – A list of variables z_1, ..., z_d
r – The direction. A vector of length \(d\) with positive algebraic numbers (usually integers) as coordinates.
critical_point – (Optional) A critical point of the map at which to evaluate the Hessian. If not specified, the symbolic Hessian is returned.

OUTPUT:

A matrix representing the specified Hessian.

sage_acsv.helpers.compute_implicit_hessian(Hs, vs, r, subs)[source]¶

Compute the Hessian of an implicitly defined function.

Given a transverse intersection point \(w\) in \(H_1(w),\dots,H_s(w)=0\), we can parametrize \(V(H_1,\dots,H_s)\) near \(w\) by writing \(z_{d-s+j} = g_j(z_1,\dots,z_{d-s})\).

Let \(h(\theta_1,\dots,\theta_{d-s}) = \sum_{j=1}^s r_{d-s+j}\log g_j({w_1 \exp(i\theta_1) \dots w_{d-s} \exp(i\theta_{d-s})})\). This function returns the Hessian of \(h\).

INPUT:

Hs – A list of polynomials \(H\)
vs – A list of variables in the equation
r – A direction vector
subs – a dictionary {v_i: w_i} defining the point w

OUTPUT:

The Hessian of the implicitly defined function \(h\) defined above.

EXAMPLES:

sage: from sage_acsv.helpers import compute_implicit_hessian
sage: R.<x,y,z,w> = PolynomialRing(QQ,4)
sage: Hs = [
....:     z^2+z*w+x*y-4,
....:     w^3+z*x-y
....: ]
sage: compute_implicit_hessian(Hs, [x,y,z,w], [1,1,1,1], {x:1,y:1,z:1,w:1})
[21/32     0]
[    0   7/8]

sage_acsv.helpers.compute_newton_series(phi, variables, series_precision)[source]¶

Computes the series expansion of an implicitly defined function.

The function \(g(x)\) for which a series expansion is computed is a simple root of the expression

\[\Phi(x, g(x)) = 0\]

INPUT:

phi – A polynomial; the equation defining the function that is expanded.
variables – A list of variables in the equation. The last variable in this list is the variable corresponding to \(g(x)\).
series_precision – A positive integer, the precision of the series expansion.

OUTPUT:

A series expansion of the function \(g(x)\).

EXAMPLES:

sage: from sage_acsv.helpers import compute_newton_series
sage: R.<x, T> = QQ[]
sage: compute_newton_series(x*T^2 - T + 1, [x, T], 7)
132*x^6 + 42*x^5 + 14*x^4 + 5*x^3 + 2*x^2 + x + 1

sage_acsv.helpers.generate_linear_form(system, vsT, u_, linear_form=None)[source]¶

Generate a linear form for the input system.

This is an integer linear combination of the variables that, with high probability, takes unique values on the solutions of the system.

INPUT:

system – A polynomial system of equations
vsT – A list of variables in the system
u_ – A variable not in the system
linear_form – (Optional) A precomputed linear form in the variables of the system. If passed, the returned form is based on the given linear form and not randomly generated.

OUTPUT:

A linear form in u_ and the variables of vsT.

sage_acsv.helpers.get_expansion_terms(expr)[source]¶

Determines coefficients for each n^k that appears in the asymptotic expressions returned by diagonal_asymptotics_combinatorial().

INPUT:

expr – An asymptotic expression, symbolic expression, ACSV tuple, or list of ACSV tuples

OUTPUT:

A list of Term objects (with attributes coefficient, pi_factor, base and power), each representing a summand in the fully expanded expression.

EXAMPLES:

sage: from sage_acsv import diagonal_asymptotics_combinatorial, get_expansion_terms
sage: var('x y z')
(x, y, z)
sage: res = diagonal_asymptotics_combinatorial(1/(1 - x - y), r=[1,1], expansion_precision=2)
sage: coefs = sorted(get_expansion_terms(res), reverse=True)
sage: coefs
[Term(coefficient=1, pi_factor=1/sqrt(pi), base=4, power=-1/2),
 Term(coefficient=-1/8, pi_factor=1/sqrt(pi), base=4, power=-3/2)]
sage: res = diagonal_asymptotics_combinatorial(1/(1 - x - y), r=[1,1], expansion_precision=2, output_format="tuple")
sage: sorted(get_expansion_terms(res)) == sorted(coefs)
True
sage: res = diagonal_asymptotics_combinatorial(1/(1 - x - y), r=[1,1], expansion_precision=2, output_format="symbolic")
sage: sorted(get_expansion_terms(res)) == sorted(coefs)
True

sage: res = diagonal_asymptotics_combinatorial(1/(1 - x^7))
sage: get_expansion_terms(res)
[Term(coefficient=1/7, pi_factor=1, base=0.6234898018587335? + 0.7818314824680299?*I, power=0),
 Term(coefficient=1/7, pi_factor=1, base=0.6234898018587335? - 0.7818314824680299?*I, power=0),
 Term(coefficient=1/7, pi_factor=1, base=-0.2225209339563144? + 0.9749279121818236?*I, power=0),
 Term(coefficient=1/7, pi_factor=1, base=-0.2225209339563144? - 0.9749279121818236?*I, power=0),
 Term(coefficient=1/7, pi_factor=1, base=-0.9009688679024191? + 0.4338837391175582?*I, power=0),
 Term(coefficient=1/7, pi_factor=1, base=-0.9009688679024191? - 0.4338837391175582?*I, power=0),
 Term(coefficient=1/7, pi_factor=1, base=1, power=0)]

sage: res = diagonal_asymptotics_combinatorial(1/(1 - x - y^2))
sage: coefs = get_expansion_terms(res); coefs
[Term(coefficient=0.6123724356957945?, pi_factor=1/sqrt(pi), base=-2.598076211353316?, power=-1/2),
 Term(coefficient=0.6123724356957945?, pi_factor=1/sqrt(pi), base=2.598076211353316?, power=-1/2)]
sage: coefs[0].coefficient.parent()
Algebraic Field
sage: coefs[0].coefficient.radical_expression()
1/2*sqrt(3/2)

sage: F2 = (1+x)*(1+y)/(1-z*x*y*(x+y+1/x+1/y))
sage: res = diagonal_asymptotics_combinatorial(F2, expansion_precision=3)
sage: coefs = get_expansion_terms(res); coefs
[Term(coefficient=4, pi_factor=1/pi, base=4, power=-1),
 Term(coefficient=1, pi_factor=1/pi, base=-4, power=-3),
 Term(coefficient=-6, pi_factor=1/pi, base=4, power=-2),
 Term(coefficient=19/2, pi_factor=1/pi, base=4, power=-3)]

sage: res = diagonal_asymptotics_combinatorial(3/(1 - x))
sage: get_expansion_terms(res)
[Term(coefficient=3, pi_factor=1, base=1, power=0)]

sage: res = diagonal_asymptotics_combinatorial((x - y)/(1 - x - y))
sage: get_expansion_terms(res)
[]

sage_acsv.helpers.is_contributing(vs, pt, r, factors, c)[source]¶

Determines if a minimal critical point pt such that the singular variety has transverse square-free factorization is contributing; that is, whether \(r\) is in the interior of the scaled log-normal cone of factors at pt

INPUT:

vs – A list of variables
pt – A point
r – A direction vector
factors – A list of factors of \(H\) for which \(pt\) vanishes
c – The co-dimension of the intersection of factors

OUTPUT:

True or False verifying if vs is contributing

EXAMPLES:

sage: from sage_acsv.helpers import is_contributing
sage: R.<x,y> = PolynomialRing(QQ, 2)
sage: is_contributing([x,y], [1,1], [17/24, 7/24], [1-(2*x+y)/3,1-(3*x+y)/4], 2)
True
sage: is_contributing([x,y], [1,1], [1, 1], [1-(2*x+y)/3,1-(3*x+y)/4], 2)
False

sage_acsv.helpers.rational_function_reduce(G, H)[source]¶

Reduction of the rational function \(G/H\) by dividing \(G\) and \(H\) by their GCD.

INPUT:

G, H – polynomials

OUTPUT:

A tuple (G/g, H/g), where g is the GCD of G and H.

helpers¶

`helpers`¶