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Post Reopened by R.P., Yemon Choi, Alexey Ustinov, David C, Alex M.
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I am wondering if anyone has ever seen or used methods fromWhat are some examples of sheaf theory to solve a Diophantine equation or to say whether solutions exist. If we just have an element of a polynomial ring in several variables over the integers, can a statement about sheaves be used to say whether or not that Diophantine equation haseither provide solutions. That is, given $f\in \mathbb{Z}[x_1, \dots, x_n]$, can sheaf theory be used (or, can it ever play a part in an answer) to say whether $V(f)$ contains any $\mathbb{Q}-$pointsDiophantine equations, or points in $\mathbb{Z}^n$.

Thanks,

Reggieto state that no such solutions exist?

I am wondering if anyone has ever seen or used methods from sheaf theory to solve a Diophantine equation or to say whether solutions exist. If we just have an element of a polynomial ring in several variables over the integers, can a statement about sheaves be used to say whether or not that Diophantine equation has solutions. That is, given $f\in \mathbb{Z}[x_1, \dots, x_n]$, can sheaf theory be used (or, can it ever play a part in an answer) to say whether $V(f)$ contains any $\mathbb{Q}-$points, or points in $\mathbb{Z}^n$.

Thanks,

Reggie

What are some examples of sheaf theory used to either provide solutions to Diophantine equations, or to state that no such solutions exist?

Post Closed as "Needs more focus" by abx, David Hansen, ARG, LeechLattice, Alex M.
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Methods of Sheaf Theorysheaf theory for Solvingsolving Diophantine Equaitonsequations

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locally trivial
locally trivial
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Methods of Sheaf Theory for Solving Diophantine Equaitons

I am wondering if anyone has ever seen or used methods from sheaf theory to solve a Diophantine equation or to say whether solutions exist. If we just have an element of a polynomial ring in several variables over the integers, can a statement about sheaves be used to say whether or not that Diophantine equation has solutions. That is, given $f\in \mathbb{Z}[x_1, \dots, x_n]$, can sheaf theory be used (or, can it ever play a part in an answer) to say whether $V(f)$ contains any $\mathbb{Q}-$points, or points in $\mathbb{Z}^n$.

Thanks,

Reggie