MathOverflow

Let $n$ be a positive integer and let $\mathbb Z_n=\mathbb Z/n \mathbb Z$. Consider the ring of Laurent polynomials $R=\mathbb Z_n[x_1^\pm,\dots,x_D^\pm]$. $R$-modules of the form $M=M_0 \otimes_{\mathbb Z_n} R$, where $M_0$ is a $\mathbb Z_n$-module...

Is there an efficient algorithm to calculate the inclusion probabilities (the probability that an item will be included in a sample) in the Yates-Grundy draw-by-draw sampling?
Sampling description: We have $n$ items each with probabilities $p_1, \dots, p_n$ of being selected in the first draw. We then sample without replacement the items one by one with probabilities proportional to the original $p_i$....

According to Cox (Primes of the form $x^2+ny^2$, Theorem $9.2$): Given a positive integer $n$, there exists a polynomial $f_n(x)$ of degree $h(-4n)$ which splits completely modulo $p$ if and only if $p$ can be expressed as $x^2+ny^2$ for some integers...

Here are the relevant papers:
Searching for modular companions, by Shashank Kanade (on arXiv, 2019)
The dilogarithm function, by Don Zagier link at Zagier's page (page 46)