On the Shapes of Surfaces and the Solutions to DEs

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Minor Update: [tex]\varphi_{i} \in \mathbb{Q} \cup[/tex] {0}

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Remark: Oops! Zero is a rational number.

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[tex]\varphi_{i} \in \mathbb{Q}[/tex]

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We let [tex]\Omega[/tex] represent the set of all integral solutions for our Diophantine equation, and we assume card([tex]\Omega[/tex]) [tex]\ge 1[/tex].

We know X is an integral solution, but we want to find more solutions. How do we proceed?

Solve T([tex]\varphi_{1}x_{1 }[/tex], [tex]\varphi_{2}x_{2 }[/tex], [tex]\varphi_{3}x_{3 }[/tex], ..., [tex]\varphi_{n}x_{n }[/tex]) = k

for some [tex]\varphi_{i} \in \mathbb{Q}[/tex]...

Remark: This is a difficult problem. And we are clueless!

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Remark: On the shape of space as defined by T, we may want to consider the 'generic' hypercube initially since we must establish limits or bounds for all possible solutions of T on a multidimensional surface...

https://en.m.wikipedia.org/wiki/Hypercube

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Remark: On the shape of space as defined by T, we may want to consider the 'generic' hypercube initially since we must establish limits or bounds for all possible solutions of T on a multidimensional surface...

https://en.m.wikipedia.org/wiki/Hypercube

FYI: 'Counting Rational Points on Algebraic Varieties' by Prof. D.R. Heath-Brown,

https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.155.5182&rep=rep1&type=pdf.

Enjoy!

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Question: Does this stuff (previous posts) makes sense?

We hope so! Please share your comments, corrections, solutions, etc.

Thank you!

Go Blue!

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FYI: 'DIOPHANTINE PROBLEMS IN MANY VARIABLES: THE ROLE OF ADDITIVE NUMBER THEORY' by Prof. T. D. Wooley,

https://www.math.purdue.edu/~twooley/publ/1999%20dpv.pdf.

Enjoy!

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Remark: We believe we have sufficient math resources (excellent published papers on DEs or related topics, ideas, methods, etc.) to solve Hilbert's Tenth Problem either positively or negatively soon (a year or less). And we hope the problem has a positive answer (a DE algorithm).

Good Luck!

Go Blue!

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Final Remark: We have decided to write and publish a manuscript on our solution to David Hilbert's Tenth Problem. We hope to complete our work by January 6, 2023.

Manuscript Reference Link: https://theory-of-energy.org/2021/11/22/david-hilberts-tenth-problem-and-the-art-and-science-of-problem-solving/.

Dave.

Go Blue!

P.S. The initial work will be made public...

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Final Remark: We have decided to write and publish a manuscript on our solution to David Hilbert's Tenth Problem. We hope to complete our work by January 6, 2023.

Manuscript Reference Link: https://theory-of-energy.org/2021/11/22/david-hilberts-tenth-problem-and-the-art-and-science-of-problem-solving/.

Dave.

Go Blue!

P.S. The initial work will be made public...

Oops! I cannot complete the difficult work, but I expect the definitive work will be completed or has been completed by the top experts in algebraic geometry/number theory, etc. Good luck!

Dave.

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FYI: The Work of Prof. June Huh Go Blue!

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FYI: He, June Huh, Dropped Out to Become a Poet. Now He’s Won a Fields Medal. Go Blue!

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FYI: The Number of Curves of Genus Two with Elliptic Differentials;

Helpful Article: Mathematical theorem used to crack US government encryption algorithm.

Enjoy!

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FYI: Prof. Bhargav Bhatt;

Some Published Papers by Prof. Bhargav Bhatt.

Enjoy!

Go Blue!

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FYI: RATIONAL LINEAR SUBSPACES OF HYPERSURFACES OVER FINITE FIELDS.

Enjoy!

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FYI: Prof. Mircea Mustaţă

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