What is some good literature to look at for linear forms in logs and $L^3$ B.R.A.
$begingroup$
I am doing some research on solutions to Exponential Diophantine Equations (eDe's), and in a lot of published papers the use of linear forms in logarithms comes up concurrently with lattice basis reduction algorithms. I was wondering what some free to low cost books/papers are out there on this math that would serve as an introduction, and would help me understand how to apply these methods to other eDe's. I found a lot of articles citing examples of equations these methods were used to solve, but no resources on how to apply them to other problems or the mathematics behind them either. I understand the linear forms in logarithms a little more than the reduction algorithms, although more information on both would definitely be more beneficial. For example, I get the part of Baker's Theorem regarding
$$beta_0 + beta_1 log(alpha_1)+ ... + beta_k log(alpha_k) > H^{-C}$$
(Although I'm a little unsure on how to calculate the height H of a an algebraic number). In general, a bookarticle regarding finding the solution set to an eDe would suffice, provided it assumes little to no prior knowledge in the field. If nothing of this sort is realistic, can you refer me to a class or subject that should be studied or looked into before jumping into literature at this level.
Additionally
Here are some of the papers I looked over and found the need for either linear forms in logs andor lattice basis reduction. Linear Forms in Logarithms - a paper detailing many applications of linear forms and lattice basis reduction in specific cases of linear and nonlinear Diophantine equations. Linear Forms in Logarithms and Diophantine Equations - pretty much replicates a lot of the information in the previous document. I also found this paper which proved somewhat more helpful, but only studied particular cases of Diophantine Equations. Again, I am not familiar enough with methods of solutions in order to take the applications to these problems and transfer them into my problems.
Modular Arithmetic is something I have also come across in solving Diophantine Equations, although the ones I am dealing with are considerably too complex to be studied in modular form, so I have ruled this out as a means of determining solutions.
number-theory reference-request diophantine-equations
asked Dec 24 '18 at 1:09
Ryan SheslerRyan Shesler
86
$endgroup$
add a comment |
$begingroup$
I am doing some research on solutions to Exponential Diophantine Equations (eDe's), and in a lot of published papers the use of linear forms in logarithms comes up concurrently with lattice basis reduction algorithms. I was wondering what some free to low cost books/papers are out there on this math that would serve as an introduction, and would help me understand how to apply these methods to other eDe's. I found a lot of articles citing examples of equations these methods were used to solve, but no resources on how to apply them to other problems or the mathematics behind them either. I understand the linear forms in logarithms a little more than the reduction algorithms, although more information on both would definitely be more beneficial. For example, I get the part of Baker's Theorem regarding
$$beta_0 + beta_1 log(alpha_1)+ ... + beta_k log(alpha_k) > H^{-C}$$
(Although I'm a little unsure on how to calculate the height H of a an algebraic number). In general, a bookarticle regarding finding the solution set to an eDe would suffice, provided it assumes little to no prior knowledge in the field. If nothing of this sort is realistic, can you refer me to a class or subject that should be studied or looked into before jumping into literature at this level.
Additionally
Here are some of the papers I looked over and found the need for either linear forms in logs andor lattice basis reduction. Linear Forms in Logarithms - a paper detailing many applications of linear forms and lattice basis reduction in specific cases of linear and nonlinear Diophantine equations. Linear Forms in Logarithms and Diophantine Equations - pretty much replicates a lot of the information in the previous document. I also found this paper which proved somewhat more helpful, but only studied particular cases of Diophantine Equations. Again, I am not familiar enough with methods of solutions in order to take the applications to these problems and transfer them into my problems.
Modular Arithmetic is something I have also come across in solving Diophantine Equations, although the ones I am dealing with are considerably too complex to be studied in modular form, so I have ruled this out as a means of determining solutions.
number-theory reference-request diophantine-equations
asked Dec 24 '18 at 1:09
Ryan SheslerRyan Shesler
86
$endgroup$
add a comment |
$begingroup$
I am doing some research on solutions to Exponential Diophantine Equations (eDe's), and in a lot of published papers the use of linear forms in logarithms comes up concurrently with lattice basis reduction algorithms. I was wondering what some free to low cost books/papers are out there on this math that would serve as an introduction, and would help me understand how to apply these methods to other eDe's. I found a lot of articles citing examples of equations these methods were used to solve, but no resources on how to apply them to other problems or the mathematics behind them either. I understand the linear forms in logarithms a little more than the reduction algorithms, although more information on both would definitely be more beneficial. For example, I get the part of Baker's Theorem regarding
$$beta_0 + beta_1 log(alpha_1)+ ... + beta_k log(alpha_k) > H^{-C}$$
(Although I'm a little unsure on how to calculate the height H of a an algebraic number). In general, a bookarticle regarding finding the solution set to an eDe would suffice, provided it assumes little to no prior knowledge in the field. If nothing of this sort is realistic, can you refer me to a class or subject that should be studied or looked into before jumping into literature at this level.
Additionally
Here are some of the papers I looked over and found the need for either linear forms in logs andor lattice basis reduction. Linear Forms in Logarithms - a paper detailing many applications of linear forms and lattice basis reduction in specific cases of linear and nonlinear Diophantine equations. Linear Forms in Logarithms and Diophantine Equations - pretty much replicates a lot of the information in the previous document. I also found this paper which proved somewhat more helpful, but only studied particular cases of Diophantine Equations. Again, I am not familiar enough with methods of solutions in order to take the applications to these problems and transfer them into my problems.
Modular Arithmetic is something I have also come across in solving Diophantine Equations, although the ones I am dealing with are considerably too complex to be studied in modular form, so I have ruled this out as a means of determining solutions.
number-theory reference-request diophantine-equations
asked Dec 24 '18 at 1:09
Ryan SheslerRyan Shesler
86
$endgroup$
I am doing some research on solutions to Exponential Diophantine Equations (eDe's), and in a lot of published papers the use of linear forms in logarithms comes up concurrently with lattice basis reduction algorithms. I was wondering what some free to low cost books/papers are out there on this math that would serve as an introduction, and would help me understand how to apply these methods to other eDe's. I found a lot of articles citing examples of equations these methods were used to solve, but no resources on how to apply them to other problems or the mathematics behind them either. I understand the linear forms in logarithms a little more than the reduction algorithms, although more information on both would definitely be more beneficial. For example, I get the part of Baker's Theorem regarding
$$beta_0 + beta_1 log(alpha_1)+ ... + beta_k log(alpha_k) > H^{-C}$$
(Although I'm a little unsure on how to calculate the height H of a an algebraic number). In general, a bookarticle regarding finding the solution set to an eDe would suffice, provided it assumes little to no prior knowledge in the field. If nothing of this sort is realistic, can you refer me to a class or subject that should be studied or looked into before jumping into literature at this level.
Additionally
Here are some of the papers I looked over and found the need for either linear forms in logs andor lattice basis reduction. Linear Forms in Logarithms - a paper detailing many applications of linear forms and lattice basis reduction in specific cases of linear and nonlinear Diophantine equations. Linear Forms in Logarithms and Diophantine Equations - pretty much replicates a lot of the information in the previous document. I also found this paper which proved somewhat more helpful, but only studied particular cases of Diophantine Equations. Again, I am not familiar enough with methods of solutions in order to take the applications to these problems and transfer them into my problems.
Modular Arithmetic is something I have also come across in solving Diophantine Equations, although the ones I am dealing with are considerably too complex to be studied in modular form, so I have ruled this out as a means of determining solutions.
number-theory reference-request diophantine-equations
number-theory reference-request diophantine-equations
asked Dec 24 '18 at 1:09
Ryan SheslerRyan Shesler
86
asked Dec 24 '18 at 1:09
Ryan SheslerRyan Shesler
86
edited Dec 24 '18 at 1:27
Ryan Shesler
asked Dec 24 '18 at 1:09
Ryan SheslerRyan Shesler
86
asked Dec 24 '18 at 1:09
Ryan SheslerRyan Shesler
86
asked Dec 24 '18 at 1:09
Ryan SheslerRyan Shesler
86
86
add a comment |
add a comment |
0
active
oldest
votes
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: "69"
};
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',
autoActivateHeartbeat: false,
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%2fmath.stackexchange.com%2fquestions%2f3050852%2fwhat-is-some-good-literature-to-look-at-for-linear-forms-in-logs-and-l3-b-r-a%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- 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.
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%2fmath.stackexchange.com%2fquestions%2f3050852%2fwhat-is-some-good-literature-to-look-at-for-linear-forms-in-logs-and-l3-b-r-a%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
