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Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

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Counting $\mathrm{mod}\:p$ solutions of Diophantine equation in two variables taking $O(p^2)...

Are there Diophantine equations in two variables such that counting solutions $\mathrm{mod}\:p$ requires $O(p^2)$ time? Geometrically this means we have to sort through a positive proportion of the af …
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