Trinomials, singular moduli and Riffaut's conjecture

Separating singular moduli and the primitive element problem

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 $α\ne0,\pm1$ and show that the field $\mathbb Q(x,y)$ is generated by $x+αy$, with a few exceptions occurring when $x$ and $y$ generate the same quadratic field o…

Linear independence of powers of singular moduli of degree 3

We show that two distinct singular moduli $j(τ),j(τ')$, such that for some positive integers $m, n$ the numbers $1,j(τ)^m$ and $j(τ')^n$ are linearly dependent over $\mathbb{Q}$ generate the same number field of degree at most $2$. This completes a result of Riffaut, who proved the above theorem except for two explicit pair of exceptions consisting of numbers of degree $3$. The purpose of this article is to treat these two remaining cases.

Equations in three singular moduli: the equal exponent case

Let $a \in \mathbb{Z}_{>0}$ and $ε_1, ε_2, ε_3 \in \{\pm 1\}$. We classify explicitly all singular moduli $x_1, x_2, x_3$ satisfying either $ε_1 x_1^a + ε_2 x_2^a + ε_3 x_3^a \in \mathbb{Q}$ or $(x_1^{ε_1} x_2^{ε_2} x_3^{ε_3})^{a} \in \mathbb{Q}^{\times}$. In particular, we show that all the solutions in singular moduli $x_1, x_2, x_3$ to the Fermat equations $x_1^a + x_2^a + x_3^a= 0$ and $x_1^a + x_2^a - x_3^a= 0$ satisfy $x_1 x_2 x_3 = 0$. Our proofs use a generalisation of a result of Faye and Riffaut on the fields generated by sums and products of two singular moduli, which we also establ…

二元分解三项式以实现快速模运算

Dyadically resolving trinomials for fast modular arithmetic

研究形式为 $2^n - 2^k + 1$ 的三项式模数，基于多项式结式建立互素性充分条件，结合图论模型与最大团查找算法构造固定比特长度的两两互素模数大集合，并利用图着色与分圆多项式理论证明集合大小上界。

3个奇异模量的有效乘法独立性

Effective multiplicative independence of 3 singular moduli

本文通过有效方法证明了三个不同的非零奇异模量的乘法依赖关系仅在判别式不超过10^10时成立，推广了Riffaut对两个奇异模量的结果。