3 problems

A partial complex Hadamard matrix $H\in M_{M\times N}(\mathbb T)$ is regular when the scalar products between its rows decompose as sums of cycles. A Butson matrix is a complex Had…

Conjecture. For any $k\geq 3$ there exists a number field $K$ with $\ell(K)=k$.

Let $n=2pq$, where $p$ and $q$ are distinct primes, and let $H_n$ denote the set of $0,1$-polynomials divisible by $\Phi_n(x)$ as defined in the paper. Lowest-degree three-term pol…