AsymptoticRing.coefficients_of_generating_function: hardcoded exponents in QQ

[cheuberg]

The method AsymptoticRing.coefficients_of_generating_function enforces exponents to be in QQ:

sage: R.<n> = AsymptoticRing('n^QQbar', QQbar)
sage: R.coefficients_of_generating_function(lambda z: (1-z)^(1/2), [1], 1)
-1/2/sqrt(pi)*n^(-3/2) + O(n^(-5/2))
sage: R.coefficients_of_generating_function(lambda z: (1-z)^QQbar(sqrt(2)), [1], 1)
Traceback (most recent call last):
...
ValueError: Cannot include T^(-1.414213562373095?) with parent Exact Term Monoid
T^(Algebraic Field) * log(T)^(Algebraic Field) with coefficients in Symbolic
Ring in Asymptotic Ring <T^QQ * log(T)^QQ> over Symbolic Ring
> *previous* ValueError: T^(-1.414213562373095?) is not in Growth Group T^QQ * log(T)^QQ

The problem is that the singular expansion is always constructed in

A = AsymptoticRing('T^QQ * log(T)^QQ', coefficient_ring=SR,
                   default_prec=precision)

Depends on #21963

CC: @behackl @dkrenn

Component: asymptotic expansions

Keywords: singularity analysis

Work Issues: remove fix for QQbar(1/2)

Author: Clemens Heuberger

Branch/Commit: u/dkrenn/21659/singularity-analysis-qqbar @ 71af378

Issue created by migration from https://trac.sagemath.org/ticket/21659

[cheuberg]

Branch: u/cheuberg/21659/singularity-analysis-qqbar

[cheuberg]

Commit: 382abb8

[cheuberg]

comment:2

Here is a first version of a fix. Unfortunately, some workaround is needed due to the fact that gamma(QQbar(1/2)) does not work, see post on sage-devel.

What I do not understand is that

sage: B.<n> = AsymptoticRing('n^QQbar', SR)
sage: B.coefficients_of_generating_function(
....:     lambda z: (1-z)^QQbar(sqrt(2)), [1], precision=1, exponent_ring=QQbar)
0.384695296551923*n^(-2.414213562373095?) + O(n^(-3.414213562373095?))

works but

sage: B.<n> = AsymptoticRing('QQbar^n * n^QQbar', SR)
sage: B.coefficients_of_generating_function(
....:     lambda z: (1-z)^QQbar(sqrt(2)), [1], precision=1, exponent_ring=QQbar)
Traceback (most recent call last)
...
TypeError: Cannot apply the substitution rules {Z: n} on 0.384695296551923*Z^(-2.414213562373095?) + 0.656715949434358*Z^(-3.414213562373095?) + 0.979573650974364*Z^(-4.414213562373095?) + O(Z^(-5.414213562373095?)) in Asymptotic Ring <Z^(Algebraic Field)> over Symbolic Ring.
> *previous* ValueError: Cannot substitute in 0.384695296551923*Z^(-2.414213562373095?) + 0.656715949434358*Z^(-3.414213562373095?) + 0.979573650974364*Z^(-4.414213562373095?) + O(Z^(-5.414213562373095?)) in Asymptotic Ring <Z^(Algebraic Field)> over Symbolic Ring.
>> *previous* ValueError: Cannot substitute in O(Z^(-5.414213562373095?)) in O-Term Monoid Z^(Algebraic Field) with implicit coefficients in Symbolic Ring.
>>> *previous* ValueError: Cannot substitute in Z^(-5.414213562373095?) in Growth Group Z^(Algebraic Field).
>>>> *previous* ValueError: Cannot take n to the exponent -5.414213562373095?.
>>>>> *previous* TypeError: no canonical coercion from Algebraic Field to Rational Field

does not.

---

New commits:

382abb8

Trac #21659: Singularity Analysis with exponents from QQbar

[cheuberg]

Author: Clemens Heuberger

[cheuberg]

comment:3

Replying to @cheuberg:

What I do not understand is that

sage: B.<n> = AsymptoticRing('n^QQbar', SR)
sage: B.coefficients_of_generating_function(
....:     lambda z: (1-z)^QQbar(sqrt(2)), [1], precision=1, exponent_ring=QQbar)
0.384695296551923*n^(-2.414213562373095?) + O(n^(-3.414213562373095?))

works but

sage: B.<n> = AsymptoticRing('QQbar^n * n^QQbar', SR)
sage: B.coefficients_of_generating_function(
....:     lambda z: (1-z)^QQbar(sqrt(2)), [1], precision=1, exponent_ring=QQbar)
Traceback (most recent call last)
...
TypeError: Cannot apply the substitution rules {Z: n} on 0.384695296551923*Z^(-2.414213562373095?) + 0.656715949434358*Z^(-3.414213562373095?) + 0.979573650974364*Z^(-4.414213562373095?) + O(Z^(-5.414213562373095?)) in Asymptotic Ring <Z^(Algebraic Field)> over Symbolic Ring.
> *previous* ValueError: Cannot substitute in 0.384695296551923*Z^(-2.414213562373095?) + 0.656715949434358*Z^(-3.414213562373095?) + 0.979573650974364*Z^(-4.414213562373095?) + O(Z^(-5.414213562373095?)) in Asymptotic Ring <Z^(Algebraic Field)> over Symbolic Ring.
>> *previous* ValueError: Cannot substitute in O(Z^(-5.414213562373095?)) in O-Term Monoid Z^(Algebraic Field) with implicit coefficients in Symbolic Ring.
>>> *previous* ValueError: Cannot substitute in Z^(-5.414213562373095?) in Growth Group Z^(Algebraic Field).
>>>> *previous* ValueError: Cannot take n to the exponent -5.414213562373095?.
>>>>> *previous* TypeError: no canonical coercion from Algebraic Field to Rational Field

does not.

This is independent of singularity analysis; see #21665.

[cheuberg]

comment:4

The problem with QQbar(1/2) seems to be fixed in pynac/pynac#201 ,
so wait for #21963.

[cheuberg]

Dependencies: #21963

[cheuberg]

Work Issues: remove fix for QQbar(1/2)

[dkrenn]

Changed branch from u/cheuberg/21659/singularity-analysis-qqbar to u/dkrenn/21659/singularity-analysis-qqbar

[dkrenn]

comment:6

Merged in SageMath 8.6

---

New commits:

71af378

Merge tag '8.6' into u/cheuberg/21659/singularity-analysis-qqbar

[dkrenn]

Changed commit from 382abb8 to 71af378