Is the p-adic Lindemann-Weierstrass Conjecture still open?
up vote
11
down vote
favorite
The p-adic Lindemann-Weierstrass Conjecture: Let $alpha_{1},ldots,alpha_{N}inoverline{mathbb{Q}_{p}}$
be distinct $p$-adic algebraic numbers satisfying $left|alpha_{n}right|_{p}<p^{-frac{1}{p-1}}$
(so that $exp_{p}left(alpha_{n}right)inmathbb{C}_{p}$) 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 Dec 12 at 21:10
MCS
1705
add a comment |
up vote
11
down vote
favorite
The p-adic Lindemann-Weierstrass Conjecture: Let $alpha_{1},ldots,alpha_{N}inoverline{mathbb{Q}_{p}}$
be distinct $p$-adic algebraic numbers satisfying $left|alpha_{n}right|_{p}<p^{-frac{1}{p-1}}$
(so that $exp_{p}left(alpha_{n}right)inmathbb{C}_{p}$) 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 Dec 12 at 21:10
MCS
1705
12
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
Dec 12 at 21:55
I suippose that $alpha_1,dots,alpha_N$ are meant to be distinct. Or maybe linearly independent over the rationals.
– Gerry Myerson
Dec 13 at 19:13
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:
– MCS
Dec 13 at 20:54
(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.
– MCS
Dec 13 at 20:54
add a comment |
up vote
11
down vote
favorite
up vote
11
down vote
favorite
The p-adic Lindemann-Weierstrass Conjecture: Let $alpha_{1},ldots,alpha_{N}inoverline{mathbb{Q}_{p}}$
be distinct $p$-adic algebraic numbers satisfying $left|alpha_{n}right|_{p}<p^{-frac{1}{p-1}}$
(so that $exp_{p}left(alpha_{n}right)inmathbb{C}_{p}$) 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 Dec 12 at 21:10
MCS
1705
The p-adic Lindemann-Weierstrass Conjecture: Let $alpha_{1},ldots,alpha_{N}inoverline{mathbb{Q}_{p}}$
be distinct $p$-adic algebraic numbers satisfying $left|alpha_{n}right|_{p}<p^{-frac{1}{p-1}}$
(so that $exp_{p}left(alpha_{n}right)inmathbb{C}_{p}$) 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 Dec 12 at 21:10
MCS
1705
asked Dec 12 at 21:10
MCS
1705
edited Dec 13 at 20:42
asked Dec 12 at 21:10
MCS
1705
asked Dec 12 at 21:10
MCS
1705
asked Dec 12 at 21:10
MCS
1705
1705
12
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
Dec 12 at 21:55
I suippose that $alpha_1,dots,alpha_N$ are meant to be distinct. Or maybe linearly independent over the rationals.
– Gerry Myerson
Dec 13 at 19:13
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:
– MCS
Dec 13 at 20:54
(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.
– MCS
Dec 13 at 20:54
add a comment |
12
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
Dec 12 at 21:55
I suippose that $alpha_1,dots,alpha_N$ are meant to be distinct. Or maybe linearly independent over the rationals.
– Gerry Myerson
Dec 13 at 19:13
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:
– MCS
Dec 13 at 20:54
(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.
– MCS
Dec 13 at 20:54
12
12
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
Dec 12 at 21:55
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
Dec 12 at 21:55
I suippose that $alpha_1,dots,alpha_N$ are meant to be distinct. Or maybe linearly independent over the rationals.
– Gerry Myerson
Dec 13 at 19:13
I suippose that $alpha_1,dots,alpha_N$ are meant to be distinct. Or maybe linearly independent over the rationals.
– Gerry Myerson
Dec 13 at 19:13
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:
– MCS
Dec 13 at 20:54
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:
– MCS
Dec 13 at 20:54
(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.
– MCS
Dec 13 at 20:54
(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.
– MCS
Dec 13 at 20:54
add a comment |
1 Answer
1
active
oldest
votes
up vote
13
down vote
accepted
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 Dec 12 at 21:47
Carlo Beenakker
72.4k9162271
What is the citation information for that one-page flyer by Nesterenko that you linked? I'd like to use it. :)
– MCS
Dec 13 at 20:56
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
– Carlo Beenakker
Dec 13 at 21:25
Thank you very much. :)
– MCS
Dec 13 at 21:30
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "504"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f317550%2fis-the-p-adic-lindemann-weierstrass-conjecture-still-open%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
13
down vote
accepted
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 Dec 12 at 21:47
Carlo Beenakker
72.4k9162271
What is the citation information for that one-page flyer by Nesterenko that you linked? I'd like to use it. :)
– MCS
Dec 13 at 20:56
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
– Carlo Beenakker
Dec 13 at 21:25
Thank you very much. :)
– MCS
Dec 13 at 21:30
add a comment |
up vote
13
down vote
accepted
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 Dec 12 at 21:47
Carlo Beenakker
72.4k9162271
What is the citation information for that one-page flyer by Nesterenko that you linked? I'd like to use it. :)
– MCS
Dec 13 at 20:56
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
– Carlo Beenakker
Dec 13 at 21:25
Thank you very much. :)
– MCS
Dec 13 at 21:30
add a comment |
up vote
13
down vote
accepted
up vote
13
down vote
accepted
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 Dec 12 at 21:47
Carlo Beenakker
72.4k9162271
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 Dec 12 at 21:47
Carlo Beenakker
72.4k9162271
edited Dec 12 at 22:16
answered Dec 12 at 21:47
Carlo Beenakker
72.4k9162271
answered Dec 12 at 21:47
Carlo Beenakker
72.4k9162271
answered Dec 12 at 21:47
Carlo Beenakker
72.4k9162271
72.4k9162271
What is the citation information for that one-page flyer by Nesterenko that you linked? I'd like to use it. :)
– MCS
Dec 13 at 20:56
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
– Carlo Beenakker
Dec 13 at 21:25
Thank you very much. :)
– MCS
Dec 13 at 21:30
add a comment |
What is the citation information for that one-page flyer by Nesterenko that you linked? I'd like to use it. :)
– MCS
Dec 13 at 20:56
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
– Carlo Beenakker
Dec 13 at 21:25
Thank you very much. :)
– MCS
Dec 13 at 21:30
What is the citation information for that one-page flyer by Nesterenko that you linked? I'd like to use it. :)
– MCS
Dec 13 at 20:56
What is the citation information for that one-page flyer by Nesterenko that you linked? I'd like to use it. :)
– MCS
Dec 13 at 20:56
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
– Carlo Beenakker
Dec 13 at 21:25
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
– Carlo Beenakker
Dec 13 at 21:25
Thank you very much. :)
– MCS
Dec 13 at 21:30
Thank you very much. :)
– MCS
Dec 13 at 21:30
add a comment |
Thanks for contributing an answer to MathOverflow!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f317550%2fis-the-p-adic-lindemann-weierstrass-conjecture-still-open%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
12
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
Dec 12 at 21:55
I suippose that $alpha_1,dots,alpha_N$ are meant to be distinct. Or maybe linearly independent over the rationals.
– Gerry Myerson
Dec 13 at 19:13
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:
– MCS
Dec 13 at 20:54
(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.
– MCS
Dec 13 at 20:54