Counting Rotational Sets for Laminations of the Unit Disk from First Principles

By studying laminations of the unit disk, we can gain insight into the structure of Julia sets of polynomials and their dynamics in the complex plane. The polynomials of a given degree, d, have a parameter space. The hyperbolic components of such parameter spaces are in correspondence to rotational polygons, or classes of rotational sets\u27\u27, which we study in this paper. By studying the count of such rotational sets, and therefore the underlying structure of these rotational sets and polygons, we can gain insight into the interrelationship among hyperbolic components of the parameter space of these polynomials. These rotational sets are created by uniting rotational orbits, as we define in this paper. The number of such sets for a given degree d, rotation number p/q, and cardinality k can be determined by analyzing the potential placements of pre-images of zero on the unit circle with respect to the rotational set under the d-tupling map. We obtain a closed-form formula for the count. Though this count is already known based upon some sophisticated results, our count is based upon elementary geometric and combinatorial principles, and provides an intuitive explanation

Finite Posets as Prime Spectra of Commutative Noetherian Rings

We study finite partially ordered sets of prime ideals as found in commutative Noetherian rings. In doing so, we establish that these posets have a bipartite structure and devise a construction for finding ring spectra that are order-isomorphic to many such posets. Specifically, we prove that any finite complete bipartite graph is order-isomorphic to the spectrum of a ring of essentially finite type over the field of rational numbers. Furthermore, we prove that prime spectra of such rings can also depict any finite path or even cycle

The Intricacies of Pairwise Modular Multiplicative Inverse in Lucas Numbers

Let (p,q) be a pair of relatively prime integers greater than 1. The pairwise modular multiplicative inverse (PMMI) of (p,q) is defined as the unique pair of positive integers (p′, q′) such that p p′ ≡ 1 (mod q), p′ \u3c q, qq′ ≡ 1 (mod p), q′ \u3c p. In this paper, we determine all pairs of Lucas numbers such that their PMMIs are pairs of Lucas numbers

Counting Rotational Subsets of the Circle R/Z\mathbb{R}/ \mathbb{Z} under the Angle-Multiplying Map t↦dtt\mapsto dt

A rotational set is a finite subset $A$ of the unit circle $\mathbb{R}/ \mathbb{Z}$ such that the angle-multiplying map $\sigma_{d}:t\mapsto dt$ maps $A$ onto itself by a cyclic permutation of its elements. Each rotational set has a geometric rotation number $p/q$. Lisa Goldberg introduced these sets to study the dynamics of complex polynomial maps. In this paper, we provide a necessary and sufficient condition for a set to be $\sigma_{d}$-rotational with rotation number $p/q$. As applications of our condition, we recover two classical results and enumerate $\sigma_d$-rotational sets with rotation number $p/q$ that consist of a given number of orbits

Proportion of p-adic Polynomials Which Are Irreducible

We attempt to quantify the exact proportion of p-adic polynomials of degree n which are irreducible. We find an exact answer to this when n is prime and p != n, and also when n = 4 and p != 2. Our answers are rational functions in p. This relates to previous work done to find exact proportions of p-adic polynomials of degree n which have k roots

The Frequency of Elliptic Curves Over Q[i]\mathbb{Q}[i] with Fixed Torsion

Mazur\textsc{\char13}s Theorem states that there are precisely 15 possibilities for the torsion subgroup of an elliptic curve defined over the rational numbers. It was previously shown by Harron and Snowden that the number of isomorphism classes of elliptic curves of height up to $X$ that have a specific torsion subgroup $G$ is on the order of $X^{1/{d(G)}}$, for some positive $d(G)$ depending on $G$. We compute $d(G)$ for these groups over \Qi. Furthermore, in a collection of recent papers it was proven that there are 9 more possibilities for the torsion subgroup in the base field \Qi. We compute the value of $d(G)$ for these new groups