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تسجيل مستخدم جديدTrinomials, singular moduli and Riffauts conjecture
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Riffaut (2019) conjectured that a singular modulus of degree $hge 3$ cannot be a root of a trinomial with rational coefficients. We show that this conjecture follows from the GRH, and obtain partial unconditional results.
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We prove that $|x-y|ge 800X^{-4}$, where $x$ and $y$ are distinct singular moduli of discriminants not exceeding $X$. We apply this result to the primitive element problem for two singular moduli. In a previous article Faye and Riffaut show that the
number field $mathbb Q(x,y)$, generated by two singular moduli $x$ and $y$, is generated by $x-y$ and, with some exceptions, by $x+y$ as well. In this article we fix a rational number $alpha e0,pm1$ and show that the field $mathbb Q(x,y)$ is generated by $x+alpha y$, with a few exceptions occurring when $x$ and $y$ generate the same quadratic field over $mathbb Q$. Together with the above-mentioned result of Faye and Riffaut, this gives a drastic generalization of a theorem due to Allombert et al. (2015) about solution of linear equations in singular moduli.
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