# Questions tagged [diophantine-equations]

Diophantine equations are polynomial equations $F=0$, or systems of polynomial equations $F_1=\ldots=F_k=0$, where $F,F_1,\ldots,F_k$ are polynomials in either $\mathbb{Z}[X_1,\ldots,X_n]$ of $\mathbb{Q}[X_1,\ldots,X_n]$ of which it is asked to find solutions over $\mathbb{Z}$ or $\mathbb{Q}$. Topics: Pell equations, quadratic forms, elliptic curves, abelian varieties, hyperelliptic curves, Thue equations, normic forms, K3 surfaces ...

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### Persistence of KAM tori as a function of dimension

I have tried posting this question in MSE, but I think it might be too technical so I'm trying again here. In KAM theory one tries to describe the persistence of quasi-periodic motion when an ...
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### Does this conic have a rational point?

Consider the conic $$C = \{X^2+uY^2+vZ^2=0\}\subset\mathbb{P}^2_{\mathbb{Q}(u,v)}$$ over the function field $\mathbb{Q}(u,v)$. Does $C$ have a $\mathbb{Q}(u,v)$-rational point?
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### Rational points on genus 3 curves defined by short equations

(a) Find all pairs of rational numbers $(x,y)$ such that $$y^3-y=x^4-x.$$ (b) Find all pairs of rational numbers $(x,y)$ such that $$y^3+y=x^4+x.$$ If not a complete answer, I would be happy to ...
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### Is every even number greater than $44$ not divisible by $8$ of the form $x^2+y^2+z^4+t^4$?

Related to this question, where Bogdan Grechuk suggested this question. Q1 Is every even number greater than $44$ not divisible by $8$ of the form $x^2+y^2+z^4+t^4$...
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### Complexity of finding solutions of trapdoored polynomial?

Related to this question Cryptography signature scheme based on hardness of finding points on varieties. Working over $K=\mathbb{Q}[x_1,...,x_n,y_1,...y_m]$. By abuse of notation, for polynomial $f$, ...