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Given a family of cyclic covers of $\mathbb{P}^1$ and a prime $p$ of good reduction, by [12] the generic Newton polygon (resp. Ekedahl--Oort type) in the family ($\mu$-ordinary) is known. In this paper, we investigate the existence of non-$\mu$-ordinary smooth curves in the family. In particular, under some auxiliary conditions, we show that when $p$ is sufficiently large the complement of the $\mu$-ordinary locus is always non empty, and for $1$-dimensional families with condition on signature type, we obtain a lower bound for the number of non-$\mu$-ordinary smooth curves. In specific examples, for small $m$, the above general statement can be improved, and we establish the non emptiness of all codimension 1 non-$\mu$-ordinary Newton/Ekedahl--Oort strata ({\em almost} $\mu$-ordinary). Our method relies on further study of the extended Hasse-Witt matrix initiated in [12].

Consider a finite field $\mathbb{F}_q$ and positive integers $d,m,r$ with $1\leq r\leq \binom{m+d}{d}$. Let $S_d(m)$ be the $\mathbb{F}_q$ vector space of all homogeneous polynomials of degree $d$ in $X_0,\dots,X_m$. Let $e_r(d,m)$ be the maximum number of $\mathbb{F}_q$-rational points in the vanishing set of $W$ as $W$ varies through all subspaces of $S_d(m)$ of dimension $r$. Ghorpade, Datta and Beelen had conjectured an exact formula of $e_r(d,m)$ when $q\geq d+1$. We prove that their conjectured formula is true when $q$ is sufficiently large in terms of $m,d,r$. The problem of determining $e_r(d,m)$ is equivalent to the problem of computing the $r^{th}$ generalized hamming weights of projective the Reed Muller code $PRM_q(d,m)$. It is also equivalent to the problem of determining the maximum number of points on sections of Veronese varieties by linear subvarieties of codimension $r$.

Given a family of abelian covers of $\mathbb{P}^1$ and a prime $p$ of good reduction, by considering the associated Deligne--Mostow Shimura variety, we obtain lower bounds for the Ekedahl-Oort type, and the Newton polygon, at $p$ of the curves in the family. In this paper, we investigate whether such lower bounds are sharp. In particular, we prove sharpness when the number of branching points is at most five and $p$ sufficiently large. Our result is a generalization under stricter assumptions of [2, Theorem 6.1] by Bouw, which proves the analogous statement for the $p$-rank, and it relies on the notion of Hasse-Witt triple introduced by Moonen in [9].

We fix a number field $K$ and study statistical properties of the ring $\mathcal{O}_K[\gamma]\cap K$ as $\gamma$ varies over algebraic numbers of a fixed degree $n\geq 2$. Given $k\geq 1$, we explicitly compute the density of $\gamma$ for which $\mathcal{O}_K[\gamma]\cap K =\mathcal{O}_K[1/k]$ and show that this does not depend on the number field $K$. In particular, we show that the density of $\gamma$ for which $\mathcal{O}_K[\gamma]\cap K=\mathcal{O}_K$ is $\frac{\zeta(n+1)}{\zeta(n)}$. In a recent paper the authors defined $X(K,\gamma)$ to be a certain finite subset of $\text{Spec}(\mathcal{O}_K)$ and showed that $X(K,\gamma)$ determines the ring $\mathcal{O}_K[\gamma]\cap K$. We show that if $\mathfrak{p}_1,\mathfrak{p}_2\in \text{Spec}(\mathcal{O}_K)$ satisfy $\mathfrak{p}_1\cap \mathbb{Z}\neq\mathfrak{p}_2\cap \mathbb{Z}$, then the events $\mathfrak{p}_1\in X(K,\gamma)$ and $\mathfrak{p}_2\in X(K,\gamma)$ are independent. As $t\to\infty$, we study the asymptotics of the density of $\gamma$ for which $|X(K,\gamma)|=t$.

A numerical set $T$ is a subset of $\mathbb N_0$ that contains $0$ and has finite complement. The atom monoid of $T$ is the set of $x \in \mathbb N_0$ such that $x+T \subseteq T$. Marzuola and Miller introduced the anti-atom problem: how many numerical sets have a given atom monoid? This is equivalent to asking for the number of integer partitions with a given set of hook lengths. We introduce the void poset of a numerical semigroup $S$ and show that numerical sets with atom monoid $S$ are in bijection with certain order ideals of this poset. We use this characterization to answer the anti-atom problem when $S$ has small type.

Denote the set of algebraic numbers as $\overline{\mathbb{Q}}$ and the set of algebraic integers as $\overline{\mathbb{Z}}$. For $\gamma\in\overline{\mathbb{Q}}$, consider its irreducible polynomial in $\mathbb{Z}[x]$, $F_{\gamma}(x)=a_nx^n+\dots+a_0$. Denote $e(\gamma)=\gcd(a_{n},a_{n-1},\dots,a_1)$. Drungilas, Dubickas and Jankauskas show in a recent paper that $\mathbb{Z}[\gamma]\cap \mathbb{Q}=\{\alpha\in\mathbb{Q}\mid \{p\mid v_p(\alpha)<0\}\subseteq \{p\mid p|e(\gamma)\}\}$. Given a number field $K$ and $\gamma\in\overline{\mathbb{Q}}$, we show that there is a subset $X(K,\gamma)\subseteq \text{Spec}(\mathcal{O}_K)$, for which $\mathcal{O}_K[\gamma]\cap K=\{\alpha\in K\mid \{\mathfrak{p}\mid v_{\mathfrak{p}}(\alpha)<0\}\subseteq X(K,\gamma)\}$. We prove that $\mathcal{O}_K[\gamma]\cap K$ is a principal ideal domain if and only if the primes in $X(K,\gamma)$ generate the class group of $\mathcal{O}_K$. We show that given $\gamma\in \overline{\mathbb{Q}}$, we can find a finite set $S\subseteq \overline{\mathbb{Z}}$, such that for every number field $K$, we have $X(K,\gamma)=\{\mathfrak{p}\in\text{Spec}(\mathcal{O}_K)\mid \mathfrak{p}\cap S\neq \emptyset\}$. We study how this set $S$ relates to the ring $\overline{\mathbb{Z}}[\gamma]$ and the ideal $\mathfrak{D}_{\gamma}=\{a\in\overline{\mathbb{Z}}\mid a\gamma\in\overline{\mathbb{Z}}\}$ of $\overline{\mathbb{Z}}$. We also show that $\gamma_1,\gamma_2\in \overline{\mathbb{Q}}$ satisfy $\mathfrak{D}_{\gamma_1}=\mathfrak{D}_{\gamma_2}$ if and only if $X(K,\gamma_1)=X(K,\gamma_2)$ for all number fields $K$.

We study statistical properties of numerical semigroups of genus $g$ as $g$ goes to infinity. More specifically, we answer a question of Eliahou by showing that as $g$ goes to infinity, the proportion of numerical semigroups of genus $g$ with embedding dimension close to $g/\sqrt{5}$ approaches $1$. We prove similar results for the type and weight of a numerical semigroup of genus $g$., Comment: 35 pages

A generalised numerical semigroup (GNS) is a submonoid $S$ of $\mathbb{N}^d$ for which the complement $\mathbb{N}^d\setminus S$ is finite. The points in the complement $\mathbb{N}^d\setminus S$ are called gaps. A gap $F$ is considered Frobenius allowable if there is some relaxed monomial ordering on $\mathbb{N}^d$ with respect to which $F$ is the largest gap. We characterise the Frobenius allowable gaps of a GNS. A GNS that has only one Frobenius allowable gap is called a Frobenius GNS. We estimate the number of Frobenius GNS with a given Frobenius gap $F=(F^{(1)},\dots,F^{(d)})\in\mathbb{N}^d$ and show that it is close to $\sqrt{3}^{(F^{(1)}+1)\cdots (F^{(d)}+1)}$ for large $d$. We define notions of quasi-irreducibility and quasi-symmetry for GNS. While in the case of $d=1$ these notions coincide with irreducibility and symmetry of GNS, they are distinct in higher dimensions., Comment: 20 pages

A numerical set $S$ is a cofinite subset of $\mathbb{N}$ which contains $0$. We use the natural bijection between numerical sets and Young diagrams to define a numerical set $\widetilde{S}$, such that their Young diagrams are complements. We determine various properties of $\widetilde{S}$, particularly with an eye to closure under addition (for both $S$ and $\widetilde{S}$), which promotes a numerical set to become a numerical semigroup., Comment: 12 pages

A numerical semigroup is a sub-semigroup of the natural numbers that has a finite complement. Some of the key properties of a numerical semigroup are its Frobenius number F, genus g and type t. It is known that for any numerical semigroup $\frac{g}{F+1-g}\leq t\leq 2g-F$. Numerical semigroups with $t=2g-F$ are called almost symmetric, we introduce a new property that characterises them. We give an explicit characterisation of numerical semigroups with $t=\frac{g}{F+1-g}$. We show that for a fixed $\alpha$ the number of numerical semigroups with Frobenius number $F$ and type $F-\alpha$ is eventually constant for large $F$. Also the number of numerical semigroups with genus $g$ and type $g-\alpha$ is also eventually constant for large $g$., Comment: 20 pages

A numerical semigroup is a sub-semigroup of the natural numbers that has a finite complement. The size of its complement is called the genus and the largest number in the complement is called its Frobenius number. We consider the set of numerical semigroups with a fixed Frobenius number $f$ and analyse their genus. We find the asymptotic distribution of genus in this set of numerical semigroups and show that it is a product of a Gaussian and a power series. We show that almost all numerical semigroups with Frobenius number $f$ have genus close to $\frac{3f}{4}$. We denote the number of numerical semigroups of Frobenius number $f$ by $N(f)$. While $N(f)$ is not monotonic we prove that $N(f)

A numerical set is a co-finite subset of the natural numbers that contains zero. Its Frobenius number is the largest number in its complement. Each numerical set has an associated semigroup $A(T)=\{t\mid t+T\subseteq T\}$, which has the same Frobenius number as $T$. For a fixed Frobenius number $f$ there are $2^{f-1}$ numerical sets. It is known that there is a number $\gamma$ close to $0.484$ such that the ratio of these numerical sets that are mapped to $N_f=\{0\}\cup(f,\infty)$ is asymptotically $\gamma$. We identify a collection of families $N(D,f)$ of numerical semigroups such that for a fixed $D$ the ratio of the $2^{f-1}$ numerical sets that are mapped to $N(D,f)$ converges to a positive limit as $f$ goes to infinity. We denote the limit as $\gamma_D$, these constants sum up to $1$ meaning that they asymptotically account for almost all numerical sets., Comment: 13 pages