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Is there a general method to solve the equation $P(x_1,x_2,...,x_n)=0$ with $P$ is a polynomial in $n$ variables with integer coefficients and $x_k=\cos(q_k\pi)$ with $q_k$ is a rational number?

This type of equation appears in the problem of classifying all tetrahedra whose dihedral angles are multiples of $\pi$ and I want to know how to solve this kind of problem generally.

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    $\begingroup$ If you know a lower bound for the denominator of all $q_k$, then it can be written as a diophantine equation in a cyclotomic field. There isn't a "general" method, but many methods that can help depending on the shape of the equation. $\endgroup$
    – Chris Wuthrich
    Sep 27, 2022 at 8:43

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This is really a question about solving multivariable polynomials in roots of unity, which in turn can be deduced from finding all linear combinations of roots of unity with certain coefficients that equal 0. For this look at the old paper of Conway and Jones. I worked something like this out in https://www.impan.pl/pl/wydawnictwa/czasopisma-i-serie-wydawnicze/acta-arithmetica/all/60/3/107463/finding-integers-k-for-which-a-given-diophantine-equation-has-no-solution-in-kth-powers-of-integers

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    $\begingroup$ In 1975, Conway suggested I use the methods of Conway-Jones to work on the rational tetrahedra problem to which Veronica refers. I didn't make much progress. $\endgroup$
    – Gerry Myerson
    Sep 27, 2022 at 23:19

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