Inversion and fraction fields for power series rings

[simon-king-jena]

This ticket is about at least three bugs related with inversion of elements of power series rings.

Here is the first:

sage: R.<x> = ZZ[[]]
sage: (1/x).parent()
Laurent Series Ring in x over Integer Ring
sage: (x/x).parent()
Power Series Ring in x over Integer Ring

Both parents are wrong. Usually, the parent of a/b is the fraction field of the parent of a,b, even if a==b. And neither above parent is a field.

Next bug:

sage: (1/(2*x)).parent()
ERROR: An unexpected error occurred while tokenizing input
The following traceback may be corrupted or invalid
The error message is: ('EOF in multi-line statement', (919, 0))

---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
... very long traceback
TypeError: no conversion of this rational to integer

And the third:

sage: F = FractionField(R)
sage: 1/x in F
False

With the proposed patch, we solve all bugs except

sage: (x/x).parent()
Power Series Ring in x over Integer Ring

---

Apply (old):

• attachment: trac-8972_fraction_of_power_series_combined.patch

CC: @tscrim @mkoeppe @nbruin @jhpalmieri @dkrenn @bgrenet @sagetrac-tmonteil @videlec @DaveWitteMorris

Component: algebra

Keywords: power series ring, fraction field

Work Issues: doctest failures

Author: Simon King, Michael Jung

Branch/Commit: 5aca068

Reviewer: Robert Bradshaw, Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/8972

[simon-king-jena]

Work Issues: segfault of div of Laurent series

[simon-king-jena]

comment:1

OK, here is a patch.

Idea:

• Introduce a fraction_field method to power series rings that returns the Laurent series ring over the fraction field of the base ring. I hope that the base_extend method for Laurent polynomials works stably enough, otherwise it should be rewritten.
• In the division method of power serise, construct the fraction field of the parent, move numerator and denominator into the fraction field, and divide there. The original form of the method is preserved as a fail safe.

With the patch, I get:

sage: P.<t> = ZZ[]
sage: R.<x> = P[[]]
sage: FractionField(R)
Laurent Series Ring in x over Fraction Field of Univariate Polynomial Ring in t over Integer Ring

O dear. I just realise that there will be more work. There is a segmentation fault, as follows:

sage: R1.<x> = ZZ[[]]
sage: F = FractionField(R1)
sage: F
Laurent Series Ring in x over Rational Field
sage: F(x)
x
sage: ~F(x)
/home/king/SAGE/sage-4.3.1/local/bin/sage-sage: line 206:  4437 Segmentation fault      sage-ipython "$@" -i

So, the division of elements of a Laurent series ring fails with a segmentation fault. By consequence, division of power series segfaults as well, with the patch. "Needs work", I presume.

[simon-king-jena]

comment:2

Strangely, without the patch, the construction works:

sage: R.<x> = ZZ[[]]
sage: F = LaurentSeriesRing(FractionField(R.base()),R.variable_names())
sage: 1/F(x)
x^-1

The patch does this construction as well -- but segfaults. There seems to be a nasty side effect.

[simon-king-jena]

comment:3

The segfault problem seems to come from the fact that the div method for Laurent series relies on the div method for power series -- and with my patch, the div method for power series uses the div method for Laurent series. Not good. But that seems solvable.

[simon-king-jena]

Attachment: 8972_power_series_inverses.patch.gz

Bugfixes for fraction field and inverses of power series over non-fields

[simon-king-jena]

Changed work issues from segfault of div of Laurent series to none

[simon-king-jena]

comment:4

OK, I replaced the old patch, and now it seems to work!

For example:

sage: P.<t> = ZZ[]
sage: R.<x> = P[[]]
sage: 1/(t*x)
1/t*x^-1
sage: (1/x).parent() is FractionField(R)
True
sage: Frac(R)
Laurent Series Ring in x over Fraction Field of Univariate Polynomial Ring in t over Integer Ring

The doc tests for the two modified files pass. So, ready for review!

[simon-king-jena]

Author: Simon King