Umemura Siegel Elliptic Functions for Roots of a Polynomial
94Hi, again.
Another applied problem arose today which requires me to compute all the roots of a polynomial
with real coefficients. The degree needs to be left arbitrary.
I only see vague references on the internet to the very abstract, symbolic "formula"
(if one wishes to call it that) for all the roots of a polynomial in terms of siegel
(siegal? seagel?) elliptic modular functions. No one ever seems to try to use it.
The only person I know who ever wrote out the formula was Hiroshi Umemura, professor
from Nagoya University in Japan. I actually contacted him once, back in 1994. But
we could not say much to each other over the internet. I saw his article in which
he expressed this formula in the back of a book called "Tata Lectures" Volume 2.
I'd like to at least see how far I can get with it before giving up.
I like the features of both Mathematica and Maple that, rather than "give up", they at least
express some formula in some sort of symbolic fashion, rather than "FAIL". I still find
it helpful, say, when I am computing partial fractions.
I am currently using Maple for college courses in microbiology and biotechnology.
I wrote my own little heuristic algorithm of how to choose functions (I chose
certain very specific rational functions) to model probability distribution functions.
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Umemura Siegel Elliptic Functions for Roots of a Polynomial
I guess you're referring to this? Of course I don't know how this could be implemented in Maple.
THAT's the book! Thanks!
Thank you, Robert Israel! Yes, the Mumford book "Tata Lectures on Theta II".
In the appendix is Umemura's formula - as much as you wish to
call it a formula. I've seen it nowhere else.
Many of the polynomials with which I deal ARE highly structured.
Or, if the polynomials themselves are not highly structured,
then the specific things I do WITH them have a lot of structure -
for example, computing differential resolvents of such polynomials.
Ideally, at the end of the day (i.e. at the end of my lifetime)
I would like some sort of "advanced category-theory type
description" of all this computational work I've done with polynomials
and differential equations.
Closed-forms not always "useful"
There are times when closed-forms are really quite useless - and this is one of them. Even for polynomials of low degree (3,4 in terms of radicals, 5,6 in terms of hypergeometric, 7,8 in terms of Lauricella functions), the closed-forms are awful for any further computations. Maple's RootOf is a much better answer, since it can be manipulated (symbolically) quite easily.
Now, it may turn out that your polynomials are highly structured, so that closed-forms are useful. In that case, I would suggest turning your polynomials into systems of differential equations (for the roots, in terms of the parameters). In some situations, dsolve is a fair bit more powerful than solve!
Thank you, JacquesC!
I've found a use, if one can call it that, for these so-called "useless"
closed-form solutions. Specifically, I am examining the structure of
differential resolvents of polynomials. In April 1999
I found a way to factor certain terms of my powersum formula for a
differential resolvent. I have wrestled with how to factor the remaining
terms.
By factor, in my case, I mean to give a matrix determinantal formula,
e.g. my formula for the k-th term of a differential resolvent is given
as det(A(k)) of a certain matrix A(k). Since 1999 I have tried to
factor the A(k)'s, i.e determine the entries of matrices B(k)
such that
A(k) = M*B(k) where M is the matrix which is known to factor out of
A(k) for SOME of the k (but I can't figure out how it
factors out for the others).
I started a paper this year, and am still writing it, for joint
differential resolvents of polynomials.
In all cases, I run up to computational difficulty. Namely,
enormous intermediate blowup problems. For example, just last night,
I calculated that my powersum formula for a 6-th order
joint differential resolvent of two polynomials would consist
of computing the determinant of roughly a 684 x 684 matrix with
polynomial entries.
These closed-form solutions of polynomials may give me insights,
even if I never use them for direct computations.
Thank you for informing me of Lauricella functions. I have never
heard of them.
I am just furious and frustrated at always hitting a mathematical
complexity wall no matter which approach I take in any math paper
I start to write.
Sounds interesting
Can you give us a few details? I know of a number of papers on differential resolvants and related works that might be interesting to you, but I would like to know a bit more to steer you towards the right ones.
Maybe you could post the results at order 2 and 3 here? I am sure that a number of people on primes would have good ideas on where to look next. This is definitely an area where symbolic computation should be able to help you. In fact, there are a lot of known algorithms for (fast!) computations of various factorizations of matrices with polynomial entries! [Most of the world experts on that domain are actually loosely associated with Maplesoft]. Now, some of those algorithms are implemented in Maple itself, but they are not always easy to get at, since they require you to set up your problem in very specific ways. Again, there are readers here who can help. There are also algorithms that are not yet integrated in Maple, but they are usually in LinBox, which can be accessed from Maple, AFAIK.
Seeking the Complete Factorization of the Cubic Resolvent
Please let me know if my file NahayCubicResolvent.PDF makes any sense to you. Thanks.
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Thank you, Scott03!
Thank you, Scott03, for pointing JacquesC into the right direction!
A lead
I think a recent paper (see also arXiv version) relates to this.
ISSAC 2007 paper
Yes, I have this paper. It references my doctoral dissertation, my one paper
published in the Journal of Differential Equations, and one of my four papers
published in the International Journal of Mathematics and Mathematical Sciences.
This ISSAC2007 paper is well written and very informative.