Deepesh Singhal
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View article: Largest zero-dimensional intersection of $r$ degree $d$ hypersurfaces
Largest zero-dimensional intersection of $r$ degree $d$ hypersurfaces Open
Suppose we have $r$ hypersurfaces in $\mathbb{P}^m$ of degree $d$, whose defining polynomials are linearly independent, and their intersection has dimension $0$. Then what is the largest possible intersection of the $r$ hypersurfaces? We c…
View article: On the smallest partition associated to a numerical semigroup
On the smallest partition associated to a numerical semigroup Open
The set of hook lengths of an integer partition $λ$ is the complement of some numerical semigroup $S$. There has been recent interest in studying the number of partitions with a given set of hook lengths. Very little is known about the dis…
View article: Non-$μ$-ordinary smooth cyclic covers of $\mathbb{P}^1$
Non-$μ$-ordinary smooth cyclic covers of $\mathbb{P}^1$ Open
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 ($μ$-ordinary) is known. In this paper, we investigate the existence of non-$μ…
View article: On a conjecture of Beelen, Datta and Ghorpade for the number of points of varieties over finite fields
On a conjecture of Beelen, Datta and Ghorpade for the number of points of varieties over finite fields Open
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 …
View article: Enumerating numerical sets associated to a numerical semigroup
Enumerating numerical sets associated to a numerical semigroup Open
A numerical set T is a subset of N0 that contains 0 and has finite complement. The atom monoid of T is the set of x∈N0 such that x+T⊆T. Marzuola and Miller introduced the anti-atom problem: how many numerical sets have a given atom monoid?…
View article: Abelian covers of $\mathbb{P}^1$ of $p$-ordinary Ekedahl-Oort type
Abelian covers of $\mathbb{P}^1$ of $p$-ordinary Ekedahl-Oort type Open
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 cu…
View article: Denisty questions in rings of the form $\mathcal{O}_K[γ]\cap K$
Denisty questions in rings of the form $\mathcal{O}_K[γ]\cap K$ Open
We fix a number field $K$ and study statistical properties of the ring $\mathcal{O}_K[γ]\cap K$ as $γ$ varies over algebraic numbers of a fixed degree $n\geq 2$. Given $k\geq 1$, we explicitly compute the density of $γ$ for which $\mathcal…
View article: The expected embedding dimension, type and weight of a numerical semigroup
The expected embedding dimension, type and weight of a numerical semigroup Open
We study statistical properties of numerical semigroups of genus g as g goes to infinity.More specifically, we answer a question of Delgado and Eliahou by showing that as g goes to infinity, the proportion of numerical semigroups of genus …
View article: Enumerating numerical sets associated to a numerical semigroup
Enumerating numerical sets associated to a numerical semigroup Open
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 m…
View article: Primes in denominators of algebraic numbers
Primes in denominators of algebraic numbers Open
Denote the set of algebraic numbers as $\overline{\mathbb{Q}}$ and the set of algebraic integers as $\overline{\mathbb{Z}}$. For $γ\in\overline{\mathbb{Q}}$, consider its irreducible polynomial in $\mathbb{Z}[x]$, $F_γ(x)=a_nx^n+\dots+a_0$…
View article: The Expected Embedding Dimension, type and weight of a Numerical Semigroup
The Expected Embedding Dimension, type and weight of a Numerical Semigroup Open
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 $g$ w…
View article: Frobenius Allowable Gaps of Generalized Numerical Semigroups
Frobenius Allowable Gaps of Generalized Numerical Semigroups Open
A generalized numerical semigroup 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 …
View article: Publisher Correction to: Distribution of genus among numerical semigroups with fixed Frobenius number
Publisher Correction to: Distribution of genus among numerical semigroups with fixed Frobenius number Open
View article: Distribution of genus among numerical semigroups with fixed Frobenius number
Distribution of genus among numerical semigroups with fixed Frobenius number Open
A numerical semigroup is a sub-monoid of the natural numbers under addition 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 t…
View article: Numerical semigroups of small and large type
Numerical semigroups of small and large type Open
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 [Formula: see text], genus [Formula: see text] and type [Formula: se…
View article: Frobenius allowable gaps of Generalized Numerical Semigroups
Frobenius allowable gaps of Generalized Numerical Semigroups Open
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 Frob…
View article: Complementary Numerical Sets
Complementary Numerical Sets Open
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. …
View article: Distribution of genus among numerical semigroups with fixed Frobenius number
Distribution of genus among numerical semigroups with fixed Frobenius number Open
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 nu…
View article: Associated semigroups of numerical sets with fixed Frobenius number
Associated semigroups of numerical sets with fixed Frobenius number Open
A numerical set is a co-finite Subset of the natural numbers that contains zero. Its Frobenius number is the largest number in its compliment. A numerical semigroup is a numerical set that is closed under addition. Each numerical set has a…