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Here is a short proof that, for an infinite field $k$, and all non-zero polynomials $F \in k[x_1,\ldots,x_n]$ in $n$ variables, there exists an $n$-tuple $a_1,\ldots,a_n \in k$ such that $$ F(a_1,\ldo …

The number of rational solutions to your equation is finite. In short: your equation defines a genus $3$ curve, as follows from a straightforward computation and an application of Riemann--Hurwitz; fi …

This came up in a discussion with a colleague of mine, who studies PDEs. He was asking for a function $f \colon \mathbb{N} \rightarrow \mathbb{N}$ such that, for all but finitely many $n$, the equatio …

By a variety over a field $k$, I mean a scheme that is separated and of finite type over $k$. I indicate changes of the base ring by subscripts. Does there exist a smooth and projective variety $V$ …