Is the p-adic Lindemann-Weierstrass Conjecture still open?











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The p-adic Lindemann-Weierstrass Conjecture: Let $alpha_{1},ldots,alpha_{N}inoverline{mathbb{Q}_{p}}$
be $p$-adic algebraic numbers satisfying $left|alpha_{n}right|_{p}<p^{-frac{1}{p-1}}$
(so that $exp_{p}left(alpha_{n}right)inmathbb{C}_{p}$) 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.






transcendental-number-theory p-adic







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  • 4




    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.
    – Carlo Beenakker
    5 hours ago















up vote
5
down vote

favorite












The p-adic Lindemann-Weierstrass Conjecture: Let $alpha_{1},ldots,alpha_{N}inoverline{mathbb{Q}_{p}}$
be $p$-adic algebraic numbers satisfying $left|alpha_{n}right|_{p}<p^{-frac{1}{p-1}}$
(so that $exp_{p}left(alpha_{n}right)inmathbb{C}_{p}$) 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.






transcendental-number-theory p-adic







share|cite|improve this question


















  • 4




    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.
    – Carlo Beenakker
    5 hours ago













up vote
5
down vote

favorite









up vote
5
down vote

favorite











The p-adic Lindemann-Weierstrass Conjecture: Let $alpha_{1},ldots,alpha_{N}inoverline{mathbb{Q}_{p}}$
be $p$-adic algebraic numbers satisfying $left|alpha_{n}right|_{p}<p^{-frac{1}{p-1}}$
(so that $exp_{p}left(alpha_{n}right)inmathbb{C}_{p}$) 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.






transcendental-number-theory p-adic







share|cite|improve this question













The p-adic Lindemann-Weierstrass Conjecture: Let $alpha_{1},ldots,alpha_{N}inoverline{mathbb{Q}_{p}}$
be $p$-adic algebraic numbers satisfying $left|alpha_{n}right|_{p}<p^{-frac{1}{p-1}}$
(so that $exp_{p}left(alpha_{n}right)inmathbb{C}_{p}$) 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.






transcendental-number-theory p-adic




transcendental-number-theory p-adic






share|cite|improve this question













share|cite|improve this question











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share|cite|improve this question










asked 5 hours ago









MCS

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1383








  • 4




    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.
    – Carlo Beenakker
    5 hours ago














  • 4




    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.
    – Carlo Beenakker
    5 hours ago








4




4




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.
– Carlo Beenakker
5 hours ago




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.
– Carlo Beenakker
5 hours ago










1 Answer
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4
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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 $lfloor N/2rfloor$ of the exponents are independent, has been proven by Nesterenko.






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    1 Answer
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    up vote
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    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 $lfloor N/2rfloor$ of the exponents are independent, has been proven by Nesterenko.






    share|cite|improve this answer



























      up vote
      4
      down vote













      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 $lfloor N/2rfloor$ of the exponents are independent, has been proven by Nesterenko.






      share|cite|improve this answer

























        up vote
        4
        down vote










        up vote
        4
        down vote









        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 $lfloor N/2rfloor$ of the exponents are independent, has been proven by Nesterenko.






        share|cite|improve this answer














        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 $lfloor N/2rfloor$ of the exponents are independent, has been proven by Nesterenko.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited 4 hours ago

























        answered 5 hours ago









        Carlo Beenakker

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        72.1k9161269






























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