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
asked 7 hours ago
MCS
1483
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up vote
7
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
asked 7 hours ago
MCS
1483
8
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
6 hours ago
add a comment |
up vote
7
down vote
favorite
up vote
7
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
asked 7 hours ago
MCS
1483
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
asked 7 hours ago
MCS
1483
asked 7 hours ago
MCS
1483
asked 7 hours ago
MCS
1483
asked 7 hours ago
MCS
1483
asked 7 hours ago
MCS
1483
1483
8
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
6 hours ago
add a comment |
8
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
6 hours ago
8
8
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
6 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
6 hours ago
add a comment |
1 Answer
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6
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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.
answered 6 hours ago
Carlo Beenakker
72.2k9161269
add a comment |
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1 Answer
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
6
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.
answered 6 hours ago
Carlo Beenakker
72.2k9161269
add a comment |
up vote
6
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.
answered 6 hours ago
Carlo Beenakker
72.2k9161269
add a comment |
up vote
6
down vote
up vote
6
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.
answered 6 hours ago
Carlo Beenakker
72.2k9161269
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.
answered 6 hours ago
Carlo Beenakker
72.2k9161269
edited 6 hours ago
answered 6 hours ago
Carlo Beenakker
72.2k9161269
answered 6 hours ago
Carlo Beenakker
72.2k9161269
answered 6 hours ago
Carlo Beenakker
72.2k9161269
72.2k9161269
add a comment |
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8
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
6 hours ago