Re: [sage-devel] Question about multivariate power series.
Niles Johnson
I have a proposal for implementing basic multivariate power series in
sage. Looking through the sage-devel history, I can see that this has
come up before, and that a number of people have thought hard about
this. What I have in mind is something of a stop-gap, but there are
two reasons I think it's valuable; I'd like to see if some others of
you agree:
1. I already have working code, and some other code that uses this to
do universal formal group law calculations (relevant for algebraic
topologists)
2. William Stein has noted that multivariate power series usually
don't get off the ground because people find out that they can do what
they want with multivariate polynomials. This happened to me, and the
way I handled it has, I believe, the potential to be useful for
everyone else in this position.
Here's the proposal: for power series in x,y,z over a base ring R,
use a dummy variable t and the ring
R[x,y,z] [[t]]
as a substitute for
R[[x,y,z]]
That is, I work with total-degree power series precision, and replace
\sum a_ijk x^i y^j z^k + O(x,y,z)^n
with
\sum a_ijk (x*t)^i (y*t)^j (z*t)^k+ O(t)^n.
Then most of the operations for multivariable power series can be
reduced to operations for multivariate polynomials or univariate power
series.
The only thing left to do is build functions which translate nicely
between these different representations, so that multivariate power
series can be constructed and printed without the user having to think
about the dummy variable t.
This is probably not a new idea, but I haven't seen it mentioned here
before. I have seen suggestions of using Maxima, Axiom, or something
else to implement multivariate power series . . . I can't deny that
seems like a better way, but it has the disadvantage of not being done
already and that I don't know those languages already. Let me know if
the idea above seems worth finishing.
-Niles
William Stein
I can.
> but it has the disadvantage of not being done
> already and that I don't know those languages already. Let me know if
> the idea above seems worth finishing.
It does to me. It would be way better than nothing, which is what we have now.
William
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--
William Stein
Professor of Mathematics
University of Washington
http://wstein.org