implement multivariate truncated power series arithmetic

[williamstein]

Multivariate truncated power series arithmetic has been requested a lot.

Apply:

1. attachment: trac_1956_combined_6.patch

This is a single patch combining all of the partial patches now attached to this ticket.

CC: @mwhansen @simon-king-jena @malb @unzvfu

Component: commutative algebra

Keywords: multivariate power series

Author: Niles Johnson

Reviewer: Martin Albrecht, Simon King

Merged: sage-4.7.1.alpha2

Issue created by migration from https://trac.sagemath.org/ticket/1956

[sagetrac-mabshoff]

comment:2

Hi Mike,

hasn't something happened in that direction already or am I confused?

Cheers,

Michael

[nilesjohnson]

Author: niles

[nilesjohnson]

comment:3

I just attached a first version of multivariate power series. In order to pass all tests, one also needs the patches at #9457 and #9443. Without these, everything still works though.

The implementation is based on an idea mentioned in this sage-devel thread:
use univariate power series over multivariate polynomial ring to do arithmetic and track total-degree precision, but display as multivariate power series.

Although there are some limitations of this approach, I think there's a useful amount of functionality here and I'm happy with how this first version works. Having said that, here are some of the issues that are still unresolved. I'd be happy to hear recommendations about how to prioritize them, or others :)

• sparse: I tried to include sparse=True in the right places, but I haven't really tested to see whether it's actually better.

• inherited functions: some of them (e.g. 'variable') don't make sense; is there a way to block inheritance of certain attributes? (del or delattr?) For now, they just return "NotImplementedError"s

• inherited functions 2: for multi_power_series_ring_element, there is a pile of functions that could be implemented but aren't (e.g. sqrt, derivative, generating functions, etc). They're noted in the documentation and lumped together in the code.

• (rich) comparison: comparisons seem to work, but someone who understands the past and future of sage/python comparisons should take a look at it. (The classes I'm inheriting from seem to use an older system.)

• _l_action_ for multivariate power series: the function works, but doesn't get called when it should; I don't understand how to fix that. (This only affects cases where the scalar isn't already in the base ring, like when you want to multiply a power series over ZZ by a rational and have the result automatically move to a power series over QQ.)

• division of multivariate power series: works fine when the denominator is a unit, but not in other cases (even when the result would not be a Laurent series).

• un/pickling: I don't understand this so I haven't done anything with it.

• cython where appropriate: I don't know C or python, so I've ignored this.

• unifying multivariate and univariate power series, as has been done for polynomials: seems like a project for another time.

[nilesjohnson]

comment:4

Uploaded a revised patch which now includes doctests for all* functions. And yes, I did find a fix a number of bugs while doing that. So thanks, whoever decided to require 100% doctest coverage :)

• note: multi_power_series_ring.py has 100% doctest coverage,
  multi_power_series_ring_element.py has 61% doctest coverage, but the functions with missing doctests are all inherited functions that (as described above) raise "NotImplementedError"s.

further note: I commented out the function _im_gens_ which was supposed to be based on the same function for multivariate polynomial rings. I couldn't get it to work even for the polynomial case, so I let it go (after little effort, I admit).

[malb]

comment:5

Hi, I took a quick look, here are some examples:

• You don't need #random for random output, the seed of the pseudo random number generator is reset for each doctest plot
• It would be nice if you could add line breaks around 80 for the doc strings
• the arithmetic functions doctests do not test correctness of the output

I know very little about multivariate power series, thus I'm clueless about whether your implementation strategy is (a) standard and/or (b) clever. Is it something that is usually done?