Re: S-Unit questions

* [email protected] [2026-03-14 10:38]:
> Below SageMath solver run for S={2,3} took less than 5 minutes.
> I tried with S={2,3,5,7} and aborted after 22.5h on AMD 7950X CPU with boost
> frequency 5.4GHz run.
> https://gist.github.com/Hermann-SW/b538c991c922b1e8fc06abbab554b3ef?permalink_comment_id=6027812#gistcomment-6027812
> Then I tried with S={2,3,5} and aborted after 13.5h.
> 
> Since SageMath S-Unit equation solver is not efficient,
> an efficient PARI/GP solution somehow would be desirable:

Unlikely in the foreseeable future. There's a standard algorithm
to find all exceptional S-units in a given number field (as Bill wrote:
linear forms on logarithms, Wildanger's method and more recent
developments). But painful to implement and intrinsically inefficient as |S|
increases. I believe SageMath implements the state of the art
algorithm, possibly inefficiently but it will be hard to do much better.

In any case, this is a full scale research project. Somewhat analogous
to the ideas in thue / thuinit (= Guillaume Hanrot's PhD thesis).
I started an implementation when Wildanger's preprint appeared, around
1998-2000, and quickly dumped it for more promising projects.

In practice, there will be obvious small solutions, rare medium sized
solutions with hidden structure exhibited by lattice basis reduction
(LLL, etc.) and one must prove there are no big ones. In my 
applications (to algebraic K-Theory: Steinberg symbols, tame
kernels), I required *some* solutions, not all solutions. And |S| was large,
in the thousands and completely out of reach. But a sample of solution
up to bounded height was good enough (until a tame kernel is generated,
say). That can be made efficient, but obviously not rigorous if finding
all solutions is required.

That's the main reason why there are no algorithm in PARI to find
(all) integral points on elliptic curves (which reduces to S-unit equations).

Cheers,

    K.B.
-- 
Pr. Karim Belabas, U. Bordeaux, Vice-président en charge du Numérique
Institut de Mathématiques de Bordeaux UMR 5251 - (+33) 05 40 00 29 77
http://www.math.u-bordeaux.fr/~kbelabas/