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 .

We present an algorithm for computing the Berkovich skeleton of a superelliptic curve y n = 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.

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.

In this paper we show that for any prime number p not equal to 11 or 19, the Tribonacci number T p−1 is divisible by p if and only if p is of the form x 2 + 11y 2 . We first use class field theory on the Galois closure of the number field corresponding to the polynomial x 3 − x 2 − x − 1 to give the splitting behavior of primes in this number field. After that, we apply these results to the explicit exponential formula for T p−1 . We also give a connection between the Tribonacci numbers and the Fourier coefficients of the unique newform of weight 2 and level 11.

We give an elementary proof of the fact that any elliptic curve E over an algebraically closed nonarchimedean field K with residue characteristic = 2, 3 and with v(j(E)) < 0 can be faithfully tropicalized. We first define an adapted form of minimal models over non-discrete valuation rings and we recover several well-known theorems from the discrete case. Using these, we create an explicit family of marked elliptic curves (E, P ), where E has multiplicative reduction and P is a three-torsion point that reduces to the singular point on the reduction of E. We then follow the strategy for proving faithfulness as in [BPR16, Theorem 6.2] and construct an embedding such that its tropicalization contains a cycle of length −v(j(E)). A key difference between this approach and the approach in [BPR16] is that the proof in this text does not use any of the analytic theory on Berkovich spaces such as the Poincaré-Lelong formula or [BPR16, Theorem 5.25] for checking the faithfulness of tropicalizations.