The p-adic Lindemann-Weierstrass Conjecture: Let $\alpha_{1},\ldots,\alpha_{N}\in\overline{\mathbb{Q}_{p}}$ be distinct $p$-adic algebraic numbers satisfying $\left|\alpha_{n}\right|_{p}<p^{-\frac{1}{p-1}}$ (so that $\exp_{p}\left(\alpha_{n}\right)\in\mathbb{C}_{p}$) for all $n$. Then, $\exp_{p}\left(\alpha_{1}\right),\ldots,\exp_{p}\left(\alpha_{N}\right)$ are algebraically independent over $\mathbb{Q}$.

I'm a graduate student who is considering taking on this problem for my doctoral dissertation

This article from 2008 by M. Waldschmidt says that the conjecture is still open (it lists it as conjecture 5.16).

I was wondering if that was still the case.

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    $\begingroup$ The question does not ask for advise, but I wonder whether it is advisable to choose for a Ph.D. project a problem that for a decade has resisted solution by experts. $\endgroup$ – Carlo Beenakker Dec 12 '18 at 21:55
  • $\begingroup$ I suippose that $\alpha_1,\dots,\alpha_N$ are meant to be distinct. Or maybe linearly independent over the rationals. $\endgroup$ – Gerry Myerson Dec 13 '18 at 19:13
  • $\begingroup$ Yes, they are distinct. I've fixed that. :) @CarloBeenakker: Call me crazy (I probably am), but I think I might have made a breakthrough on the problem. I've gone through my proof line-by-line several times over already, and nothing is out of place. To give a hint of what I'm doing, the argument hinges on two propositions: $\endgroup$ – MCS Dec 13 '18 at 20:54
  • $\begingroup$ (continued) 1) (Already known): certain zeroes of power series over a complete non-archimedean field are algebraic over said field. 2) (I had to prove it): a non-trivial linear combination of algebraic translates of the Iwasawa logarithm is never analytic at the point at infinity of the complex p-adic numbers. $\endgroup$ – MCS Dec 13 '18 at 20:54

Here is a 2018 paper, A Note on One-dimensional Varieties Over the Complex p-adic Field, that still lists the "full" statement as a conjecture; "half" of the statement, meaning that at least $\lfloor N/2\rfloor$ of the exponents are independent, has been proven by Nesterenko.

  • $\begingroup$ What is the citation information for that one-page flyer by Nesterenko that you linked? I'd like to use it. :) $\endgroup$ – MCS Dec 13 '18 at 20:56
  • $\begingroup$ the full article is: Yu.V. Nesterenko, Algebraic independence of $p$-adic numbers, Izv. Math. 72, 565-579 (2008); doi.org/10.1070/IM2008v072n03ABEH002411 ; "half of the Lindemann–Weierstrass theorem" is corollary 2 in that paper $\endgroup$ – Carlo Beenakker Dec 13 '18 at 21:25

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