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I was interested in counting (and more generally having somehow an interesting expression) the numbers of solution of cubic equations modulo a prime $p$. So here are my thoughts. Let take a cubic po …

Considering the Jacobi theta: $\theta_3(z) = \sum_{n\in\mathbb{Z}} q^{n^2}$, we can invert $\theta_3-1$ in a small enough neighbourhood of 0. Routine computation with Lagrange-Burmann inversion gives …

Let $n$ be a positive integer, and $\zeta_n$ a primitive $n$-root of unity. I consider the polynomial $P(X) = \sum_{k=0}^{\phi(n)-1} \left[ \sum_{l \in \mathbb{Z}_n^*}^n \zeta_n^{kl} \right]X^k = \su …

Let $k$ a CM-field of conjugation $\bar \cdot$ and maximal totally real subfield $k^+$ and $k^{++}$ its positive part $k^{++}$. Given a totally positive element $x$ in the ring of integers of $k^+$, w …

Let $n,H$ two fixed positive integers. Let $P\in\mathbb{Z}[X]$ a monic integral polynomial of height $H$ and degree $n$ taken uniformly at random (i.e. each of the $n$ free coefficients of $P$ is sam …