Trajan Hammonds
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View article: $k$-Diophantine $m$-tuples in Finite Fields
$k$-Diophantine $m$-tuples in Finite Fields Open
In this paper, we define a $k$-Diophantine $m$-tuple to be a set of $m$ positive integers such that the product of any $k$ distinct positive integers is one less than a perfect square. We study these sets in finite fields $\mathbb{F}_p$ fo…
View article: The completed standard $L$-function of modular forms on $G_2$
The completed standard $L$-function of modular forms on $G_2$ Open
The goal of this paper is to provide a complete and refined study of the standard $L$-functions $L(π,\operatorname{Std},s)$ for certain non-generic cuspidal automorphic representations $π$ of $G_2(\mathbb{A})$. For a cuspidal automorphic r…
View article: Rank and bias in families of hyperelliptic curves via Nagao's conjecture
Rank and bias in families of hyperelliptic curves via Nagao's conjecture Open
View article: Modified Erd\H{o}s-Ginzburg-Ziv Constants for $(\mathbb{Z}/n\mathbb{Z})^2$
Modified Erd\H{o}s-Ginzburg-Ziv Constants for $(\mathbb{Z}/n\mathbb{Z})^2$ Open
For an abelian group $G$ and an integer $t > 0$, the modified Erd\H{o}s-Ginzburg-Ziv constant $s'_t(G)$ is the smallest integer $\ell$ such that any zero-sum sequence of length at least $\ell$ with elements in $G$ contains a zero-sum subse…
View article: Modified Erdős-Ginzburg-Ziv Constants for $(\mathbb{Z}/n\mathbb{Z})^2$
Modified Erdős-Ginzburg-Ziv Constants for $(\mathbb{Z}/n\mathbb{Z})^2$ Open
For an abelian group $G$ and an integer $t > 0$, the modified Erdős-Ginzburg-Ziv constant $s'_t(G)$ is the smallest integer $\ell$ such that any zero-sum sequence of length at least $\ell$ with elements in $G$ contains a zero-sum subsequen…
View article: Counting Roots of Polynomials over $\mathbb{Z}/p^2\mathbb{Z}$
Counting Roots of Polynomials over $\mathbb{Z}/p^2\mathbb{Z}$ Open
Until recently, the only known method of finding the roots of polynomials over prime power rings, other than fields, was brute force. One reason for this is the lack of a division algorithm, obstructing the use of greatest common divisors.…
View article: Counting Roots of Polynomials over Z/p 2 Z.
Counting Roots of Polynomials over Z/p 2 Z. Open