Let ${𝒳}_g$ be a genus $g\ge 2$ superelliptic curve, $F$ its field of moduli, and $K$ the minimal field of definition. In this short note we construct an equation of the curve ${𝒳}_g$ over its minimal field of definition $K$ when ${𝒳}_g$ has extra automorphisms. We make use of the dihedral invariants of superelliptic curves as defined by Shaska in [6] and results on the automorphism groups of superelliptic curves as in [10].

Let $X\subset {\mathrm{\mathbb{P}}}^n$ be an integral and non-degenerate variety. Fix $P\in {\mathrm{\mathbb{P}}}^n$. In this paper we discuss the minimal integer $\sum _{i=1}^k\#\left(S_i\right)$ such that $S_i\subset X$ and $\left\{P\right\}={\cap }_{i=1}^k\langle S_{\mathit{i}}\rangle$, where $\langle \rangle$ denote the linear span (in positive characteristic sometimes this integer is $+\infty$). We use tools introduced for the study of the $X$-rank of $P$. Our main results are when $X$ is a Veronese embedding of ${\mathrm{\mathbb{P}}}^m$ (it is related to the symmetric tensor rank of $P$) or when $X$ is a curve.

This paper is the first version of a project of classifying all superelliptic curves of genus $g\le 48$ according to their automorphism group. We determine the parametric equations in each family, the corresponding signature of the group, the dimension of the family, and the inclussion among such families. At a later stage it will be determined the decomposition of the Jacobians and each locus in the moduli spaces of curves.

In this short paper we present symmetric encryption algorithm based on family of bipartite graphs $\widetilde{D(n,q)}$. It is a fast stream cipher with computation speed $O\left(n\right)$ for the key of fixed length. If the key is linear function from $n$ the efficiency is $O\left(n^2\right)$. Encryption map has a multivariate nature, so the security level can be evaluated via degrees and other parameters of corresponding multivariate polynomials. We show that the degree of graph based encryption maps are growing with the growth of the dimension of the plain space. Therefore this algorithm is resistant to linearization attacks.

Given a real finite field extension $K/\mathrm{\mathbb{Q}}$ of degree $d$ and class number $h_K$ and a positive integer $a$, we show that there is a set of rational prime numbers of relative density at least $1/\left(2d{h}_K\right)$ that have a principal prime factor $\pi {𝒪}_K\subset {𝒪}_K$ of degree one such that the equation $a{\pi }^2=x^4-y^2$ has no nontrivial solutions in ${𝒪}_K$.

We use Gröbner basis techniques to study the balanced canonical ribbon in each odd genus $g\ge 5$. We obtain equations and syzygies of the ribbon, give a Gröbner interpretation of part of Alper, Fedorchuk, and Smyth’s proof of finite Hilbert stability for canonical curves, and discuss the obstacles in using ribbons to give a new proof of Generic Green’s Conjecture (Voisin’s Theorem).