Timeline for Proving that $(\omega(c_1 x_1^n+\cdots+c_kx_k^n))_{n\ge 1}$ is bounded only if $|x_1|=\cdots=|x_k|$ by the Subspace Theorem
Current License: CC BY-SA 3.0
11 events
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| Nov 15, 2017 at 22:13 | vote | accept | Salvo Tringali | ||
| Nov 15, 2017 at 22:13 | answer | added | Salvo Tringali | timeline score: 2 | |
| Dec 9, 2016 at 19:33 | comment | added | Salvo Tringali | So I had at least in part misinterpreted your words. Based on your comments, I'm going to edit the OP to make the question a little broader in scope (in hindsight, insisting on the p-adic Subspace Theorem was not a good idea), maybe someone else will then give it a try. | |
| Dec 9, 2016 at 19:26 | history | edited | Salvo Tringali | CC BY-SA 3.0 |
Removed some words
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| Dec 9, 2016 at 18:53 | comment | added | Vesselin Dimitrov | Ah, I see now that I had misread your question. I was assuming precisely the thing you wrote at the end of your upvoted comment above, but it was something different: I was misreading the $\omega(\cdot)$ notation, for the number of distrinct prime factors. Well, OK, then of course you need additional work. You may not appeal to the Subspace theorem. (Actually, I wasn't implying this about the sledgehammer against elementary approaches, and I don't agree with this point of view. The Subspace theorem is just one of many approaches. It just doesn't apply to your $\omega$l question.) | |
| Dec 9, 2016 at 13:17 | comment | added | Salvo Tringali | (...) work, as far as I can say, and is precisely what I'd like to avoid, since then the conclusion would be more or less straightforward (with or without the subspace theorem). But maybe I'm missing something, as you write of a direct consequence of Evertse, Schlickewei and Schmidt's theorem on the k-variable unit equation (specialized to the rational field), or at least this is how I interpret your words. So thank you in advance for any further comment. | |
| Dec 9, 2016 at 10:05 | comment | added | Salvo Tringali | I agree that appealing to the subspace theorem (and related results) is a bit like using a sledgehammer to crack a nut, and the elementary approach I was alluding to in the OP is along the lines of what you suggest in your 1st comment. On the other hand, I don't see how to use Theorem 7.4.1 in [BG], which is, I think, what you refer to in your 2nd comment, to prove the claim without first showing that, if the contrary were true, there would exist a finite set $P$ of primes s.t. every prime divisor of $c_1x_1^{2n}+\cdots+c_kx_k^{2n}$ is in $P$ for infinitely many $n$, which needs a bit of (...) | |
| Dec 7, 2016 at 23:34 | comment | added | Vesselin Dimitrov | (I just noticed you required the $x_i$ to be rational integers. In that case, Skolem-Mahler-Lech is not needed, and your statement is directly a consequence of the theorem stating that all but finitely many solutions to the $k$-variable $S$-unit equation have a proper vanishing subsum. This in turn is an easy consequence of the Subspace theorem, explained in chapter 7 of [BG].) | |
| Dec 7, 2016 at 22:56 | comment | added | Vesselin Dimitrov | It is a trivial consequence of Skolem-Mahler-Lech together with the finiteness theorem on the $k$-variable $S$-unit equation, itself a special case of the Subspace theorem. You will find the argument in chapter 7 of Bombieri and Gubler's book. It is much easier to prove directly that the power sums are piecewise monomials on a suitable arithmetic progression. | |
| Dec 7, 2016 at 19:15 | history | edited | Salvo Tringali | CC BY-SA 3.0 |
added 4 characters in body
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| Dec 7, 2016 at 19:10 | history | asked | Salvo Tringali | CC BY-SA 3.0 |