Invariants for trees of non-archimedean polynomials and skeleta of superelliptic curves

In this paper we generalize the j-invariant criterion for the semistable reduction type of an elliptic curve to superelliptic curves X given by $$y^{n}=f(x)$$ y n = f ( x ) . We first define a set of tropical invariants for f(x) using symmetrized Plücker coordinates and we show that these invariants determine the tree associated to f(x). This tree then completely determines the reduction type of X for n that are not divisible by the residue characteristic. The conditions on the tropical invariants that distinguish between the different types are given by half-spaces as in the elliptic curve case. These half-spaces arise naturally as the moduli spaces of certain Newton polygon configurations. We give a procedure to write down their equations and we illustrate this by giving the half-spaces for polynomials of degree $$d\le {5}$$ d ≤ 5 .

On effective computations in subsemigroups of affine Cremona semigroup and implentations of new postquantum multivariate cryptosystems

Multivariate cryptography (MC) together with Latice Based, Hash based, Code based and Superelliptic curves based Cryptographies form list of the main directions of Post Quantum Cryptography.Investigations in the framework of tender of National Institute of Standardisation Technology (the USA) indicates that the potential of classical MC working with nonlinear maps of bounded degree and without the usage of compositions of nonlinear transformation is very restricted. Only special case of Rainbow like Unbalanced Oil and Vinegar digital signatures is remaining for further consideration. The remaining public keys for encryption procedure are not of multivariate. nature. The paper presents large semigroups and groups of transformations of finite affine space of dimension n with the multiple composition property. In these semigroups the composition of n transformations is computable in polynomial time. Constructions of such families are given together with effectively computed homomorphisms between members of the family. These algebraic platforms allow us to define protocols for several generators of subsemigroup of affine Cremona semigroups with several outputs. Security of these protocols rests on the complexity of the word decomposition problem, Finally presented algebraic protocols expanded to cryptosystems of El Gamal type which is not a public key system.

Tropical superelliptic curves

We present an algorithm for computing the Berkovich skeleton of a superelliptic curve yn = f(x) over a valued field. After defining superelliptic weighted metric graphs, we show that each one is realizable by an algebraic superelliptic curve when n is prime. Lastly, we study the locus of superelliptic weighted metric graphs inside the moduli space of tropical curves of genus g.

Division by 1–ζ on Superelliptic Curves and Jacobians

Yuri Zarhin gave formulas for “dividing a point on a hyperelliptic curve by 2”. Given a point $P$ on a hyperelliptic curve $\mathcal{C}$ of genus $g$, Zarhin gives the Mumford representation of an effective degree $g$ divisor $D$ satisfying $2(D - g \infty ) \sim P - \infty $. The aim of this paper is to generalize Zarhin’s result to superelliptic curves; instead of dividing by 2, we divide by $1 - \zeta $. There is no Mumford representation for divisors on superelliptic curves, so instead we give formulas for functions that cut out a divisor $D$ satisfying $(1 - \zeta ) D \sim P - \infty $. Additionally, we study the intersection of $(1 - \zeta )^{-1} \mathcal{C}$ and the theta divisor $\Theta $ inside the Jacobian $\mathcal{J}$. We show that the intersection is contained in $\mathcal{J}[1 - \zeta ]$ and compute the intersection multiplicities.