On the Shapes of Surfaces and the Solutions to DEs

On the Shapes of Surfaces and the Solutions to DEs

Postby Guest » Sun Oct 10, 2021 2:39 am

Some Food for Thought:

Geometry meets number theory.

Please ponder curved surfaces, flat surfaces, and boundaries of surfaces when seeking solutions to DEs (Diophantine Equations).

And please think big (beyond 3 dimensions).

Good luck!

Go Blue!
Guest
 

Re: On the Shapes of Surfaces and the Solutions to DEs

Postby Guest » Sun Oct 10, 2021 4:02 am

Remark: There may be a deep connection (theorem) between the shape of space (surfaces including boundaries, curvature, other metrics) and the solvability of DEs.
Guest
 

Re: On the Shapes of Surfaces and the Solutions to DEs

Postby Guest » Mon Oct 11, 2021 1:15 am

A Point of Clarity: The shape of space is broadly determined by the function associated with a Diophantine equation.
Guest
 

Re: On the Shapes of Surfaces and the Solutions to DEs

Postby Guest » Mon Oct 11, 2021 2:31 am

Remark: There are three possibilities.

1. DEs with integral solutions;

2. DEs without integral solutions;

3. DEs that cannot be determined either way (1 or 2).

Let a DE be described by T(X) = k. The shape of space is determined by T(X).

And how does the shape of space determine 1, 2, and 3?
Guest
 

Re: On the Shapes of Surfaces and the Solutions to DEs

Postby Guest » Mon Oct 11, 2021 2:51 am

Remark: We are clueless! :(
Guest
 

Re: On the Shapes of Surfaces and the Solutions to DEs

Postby Guest » Mon Oct 11, 2021 2:35 pm

Guest wrote:Remark: We are clueless! :(


Hah! We can do better, and we will do better! :)

"We must know. We will know!" -- David Hilbert.

FYI: 'Hilbert metric',

https://en.m.wikipedia.org/wiki/Hilbert_metric.
Guest
 

Re: On the Shapes of Surfaces and the Solutions to DEs

Postby Guest » Mon Oct 11, 2021 8:16 pm

"Everything should be made as simple as possible, but no simpler." -- Albert Einstein.

Remark: Analysis is vital...

And we are in a period of learning and pondering.
Guest
 

Re: On the Shapes of Surfaces and the Solutions to DEs

Postby Guest » Tue Oct 12, 2021 2:42 am

Remark: Yes! Depending on k, T(X) filters most or all relevant X [tex]\in \mathbb{Z^{n}}[/tex]. And we should be able to determine when it does.

Moreover, we are close to claiming that Hilbert's Tenth Problem has an affirmative answer, but we have not done the necessary computations yet. So we are not sure about our tentative claim.

We will find answer soon.

Thank Lord GOD! Amen!

Go Blue! Thank you! :D
Guest
 

Re: On the Shapes of Surfaces and the Solutions to DEs

Postby Guest » Tue Oct 12, 2021 3:00 am

Remark: Hilbert's Tenth Problem is a difficult problem!
And it is also a great problem.

Good Luck!
Guest
 

Re: On the Shapes of Surfaces and the Solutions to DEs

Postby Guest » Tue Oct 12, 2021 3:19 am

Remark: We expect an answer to our problem before Hilbert's 160th birthday on January 23, 2022.
Guest
 

Re: On the Shapes of Surfaces and the Solutions to DEs

Postby Guest » Tue Oct 12, 2021 3:31 am

Why should T(X) filter most or all relevant X [tex]\in \mathbb{Z^{n}}[/tex]?
Guest
 

Re: On the Shapes of Surfaces and the Solutions to DEs

Postby Guest » Tue Oct 12, 2021 3:53 am

Guest wrote:Why should T(X) filter most or all relevant X [tex]\in \mathbb{Z^{n}}[/tex]?



Are you asking why T(X) allows a few or no integral solutions to T(X) = k?
Guest
 

Re: On the Shapes of Surfaces and the Solutions to DEs

Postby Guest » Tue Oct 12, 2021 3:54 am

Yes!
Guest
 

Re: On the Shapes of Surfaces and the Solutions to DEs

Postby Guest » Tue Oct 12, 2021 4:11 am

Guest wrote:
Guest wrote:Why should T(X) filter most or all relevant X [tex]\in \mathbb{Z^{n}}[/tex]?


Easy Answer: [tex]card(\mathbb{Z^{n}}) << card(\mathbb{R^{n}})[/tex].
Guest
 

Re: On the Shapes of Surfaces and the Solutions to DEs

Postby Guest » Tue Oct 12, 2021 4:37 am

... Better Answer: [tex]\frac{|\mathbb{Z^{n}}|}{| \mathbb{R^{n}} |} = 0[/tex]
Guest
 

Re: On the Shapes of Surfaces and the Solutions to DEs

Postby Guest » Tue Oct 12, 2021 4:47 am

Remark: We believe Hilbert metric is quite useful here.

But it is not clear to us on how we should proceed. We have much to learn and to consider.
Guest
 

Re: On the Shapes of Surfaces and the Solutions to DEs

Postby Guest » Wed Oct 13, 2021 11:10 pm

An Observation:

T(X) = 0 may have no solutions or a finite number of solutions.

T(X) = 1 may have no solutions or a finite number of solutions.

T(X) = 2 may have no solutions or a finite number of solutions.

...

Why?
Guest
 

Re: On the Shapes of Surfaces and the Solutions to DEs

Postby Guest » Wed Oct 13, 2021 11:14 pm

Update:

Guest wrote:An Observation:

T(X) = 0 may have no integral solutions or a finite number of integral solutions.

T(X) = 1 may have no integral solutions or a finite number of integral solutions.

T(X) = 2 may have no integral solutions or a finite number of integral solutions.

...

Why?
Guest
 

Re: On the Shape of Space and the Solutions to DEs

Postby Guest » Thu Oct 14, 2021 12:03 pm

Hey, don't forget to consider the negative integers, k, when searching for integral solutions to the Diophantine equation, T(X) = k.
Guest
 

Re: On the Shapes of Surfaces and the Solutions to DEs

Postby Guest » Mon Oct 18, 2021 1:13 pm

There may be solutions galore, [tex]X \in \mathbb{R^{n}}[/tex], for the Diophantine equation, T(X) = k. But there could also be no solutions, [tex]X \in \mathbb{Z^{n}}[/tex], such that T(X) = k for n > 3.

Go figure!

Exhaustive Search:

We assume T(X) = k and [tex]X \in \mathbb{R^{n}}[/tex] such [tex]X \notin \mathbb{Z^{n}}[/tex] for n > 3.

Let [tex]X + \triangle X \ne X[/tex] such that T([tex]X + \triangle X[/tex]) = k where [tex]\triangle X \in \mathbb{R^{n}}[/tex].

Does the sum, [tex]X + \triangle X \in \mathbb{Z^{n}}[/tex]?

It's time to do some computing. Good Luck!

While there are some or all [tex]x_{i } \in X[/tex] that are not integers, we try to adjust [tex]\triangle x_{i } \in \triangle X \in \mathbb{R^{n}}[/tex] so that [tex]x_{i } + \triangle x_{i } \in \mathbb{Z^{n}}[/tex].

And me must solve T([tex]X + \triangle X[/tex]) = k.

We repeat the process until [tex]X + \triangle X \in \mathbb{Z^{n}}[/tex].

Does the process determines a solution or no solution?
Guest
 

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