Comments on: Open question: effective Skolem-Mahler-Lech theorem
https://terrytao.wordpress.com/2007/05/25/open-question-effective-skolem-mahler-lech-theorem/
Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence TaoMon, 26 Sep 2016 16:56:02 +0000
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By: The subspace theorem approach to Siegel’s theorem on integral points on curves via nonstandard analysis | What's new
https://terrytao.wordpress.com/2007/05/25/open-question-effective-skolem-mahler-lech-theorem/#comment-388660
Mon, 07 Jul 2014 10:20:42 +0000http://terrytao.wordpress.com/2007/05/25/open-question-effective-skolem-mahler-lech-theorem/#comment-388660[…] may be bounded effectively; cf. the situation with the Skolem-Mahler-Lech theorem, discussed in this previous blog post.) Once again, the lower bound here is basically sharp except for the factor and the implied […]
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By: From letter exchange to blog exchange | To All You Zombies
https://terrytao.wordpress.com/2007/05/25/open-question-effective-skolem-mahler-lech-theorem/#comment-223058
Sun, 07 Apr 2013 11:54:32 +0000http://terrytao.wordpress.com/2007/05/25/open-question-effective-skolem-mahler-lech-theorem/#comment-223058[…] https://terrytao.wordpress.com/2007/05/25/open-question-effective-skolem-mahler-lech-theorem/, retrieved 7.4.2013 […]
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By: Peter Komjath
https://terrytao.wordpress.com/2007/05/25/open-question-effective-skolem-mahler-lech-theorem/#comment-49337
Sun, 02 Jan 2011 18:40:32 +0000http://terrytao.wordpress.com/2007/05/25/open-question-effective-skolem-mahler-lech-theorem/#comment-49337This paper may be of interest (sorry if already mentioned):
Bruce E. Litow: A Decision Method for the Rational Sequence Problem Electronic Colloquium on Computational Complexity (ECCC) 4(55): (1997)
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By: Richard Stanley
https://terrytao.wordpress.com/2007/05/25/open-question-effective-skolem-mahler-lech-theorem/#comment-46975
Tue, 14 Sep 2010 02:40:36 +0000http://terrytao.wordpress.com/2007/05/25/open-question-effective-skolem-mahler-lech-theorem/#comment-46975I am curious about generalizations of Mahler-Skolem-Lech. I believe it is still
open for algebraic power series or even D-finite power series. Is this correct?
In other words, if is algebraic (satisfies a polynomial equation whose coefficients are polynomials in ) or more generally D-finite (satisfies a linear differential equation whose coefficients are polynomials in ), then is the set of ‘s for which eventually a union of arithmetic progressions? This is known to be false for algebraically differential series , i.e., satisfies an equation that is a polynomial in .
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By: Vagabond
https://terrytao.wordpress.com/2007/05/25/open-question-effective-skolem-mahler-lech-theorem/#comment-46954
Sun, 12 Sep 2010 04:56:41 +0000http://terrytao.wordpress.com/2007/05/25/open-question-effective-skolem-mahler-lech-theorem/#comment-46954Incidentally a lemma due to Turan comes very close to proving Skolem Mahler Lech theorem. Its says if one knows the values of an exponential polynomial along an n length AP then one can find an estimate of it at the n+1 th point. Applied to the the specific case under consideration ( i.e. an exponential polynomial of order n) it says if the exponential polynomial vanishes at an arithmetic progression of length n then it vanishes on the complete arithmetic progression. So Szemeredi’s theorem tells us that the structure of the zero set has to be union of complete AP plus a set of dnsity zero set ( having no AP of length n). Though it does not prove Skolem Mahler Lech theorem it’s tantalizingly close and provides the extra information about the geometry of the zero sets …. that the exceptional set do not have an n length AP and using estimates from Ramsey theory one can find a bound on how many zeros can be there in an interval ?
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By: The Skolem-Mahler-Lech theorem « Cam's Blog
https://terrytao.wordpress.com/2007/05/25/open-question-effective-skolem-mahler-lech-theorem/#comment-43917
Fri, 19 Mar 2010 01:22:00 +0000http://terrytao.wordpress.com/2007/05/25/open-question-effective-skolem-mahler-lech-theorem/#comment-43917[…] proof of the SML theorem that I’ll give was essentially lifted from Terrence Tao’s blog. He in turn essentially lifted it from this paper of […]
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By: Mathematical Embarrassments « Gödel’s Lost Letter and P=NP
https://terrytao.wordpress.com/2007/05/25/open-question-effective-skolem-mahler-lech-theorem/#comment-42913
Sat, 26 Dec 2009 14:14:00 +0000http://terrytao.wordpress.com/2007/05/25/open-question-effective-skolem-mahler-lech-theorem/#comment-42913[…] linear recursion zero problem. An ME due to Tao is a problem about linear recurrences. Given a linear recurrence over the […]
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By: The Orbit Problem « Gödel’s Lost Letter and P=NP
https://terrytao.wordpress.com/2007/05/25/open-question-effective-skolem-mahler-lech-theorem/#comment-41284
Fri, 04 Sep 2009 12:30:27 +0000http://terrytao.wordpress.com/2007/05/25/open-question-effective-skolem-mahler-lech-theorem/#comment-41284[…] this a theorem? Terry Tao raises a related question in his famous blog on recurrence sequences. He gives it as an example of a question that we should know the answer to, […]
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By: davidspeyer
https://terrytao.wordpress.com/2007/05/25/open-question-effective-skolem-mahler-lech-theorem/#comment-36270
Mon, 02 Mar 2009 22:41:13 +0000http://terrytao.wordpress.com/2007/05/25/open-question-effective-skolem-mahler-lech-theorem/#comment-36270That’s interesting, but it isn’t hard to give a bound in the coin case — if I recall correctly, the product of the coin values works. The NP-hard problem is determining the actual position of that last zero. Whereas in Skolem-Mahler-Lech, we seem to actually have a computability issue — there is no known bound at all.
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By: Qiaochu Yuan
https://terrytao.wordpress.com/2007/05/25/open-question-effective-skolem-mahler-lech-theorem/#comment-36268
Mon, 02 Mar 2009 21:54:02 +0000http://terrytao.wordpress.com/2007/05/25/open-question-effective-skolem-mahler-lech-theorem/#comment-36268It occurred to me today that it’s possible to embed the Frobenius coin problem into this problem, as follows: the number of ways to represent a non-negative integer as a non-negative sum of a fixed set of non-negative integers has a rational generating function, equivalently, is a sequence that satisfies an integer linear recurrence. The coin problem asks for the largest number such that no such representations exist, i.e. the largest zero of such a sequence. And the coin problem is known to be NP-hard (in general) as well (although the existence of zeroes is trivial here). Perhaps this would be an instructive special case?
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