Samrith Ram
We give a new proof of Fitzgerald's criterion for primitive polynomials over a finite field. Existing proofs essentially use the theory of linear recurrences over finite fields. Here, we give a much shorter and self-contained proof which does not use the theory of linear recurrences.
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Proof of a Conjecture on Primitive Quartic Polynomials over Finite Fields
针对有限域上本原四次多项式的猜想，本文在q足够大时，借助Fu和Wan的定理，通过AI辅助完成了证明。
We prove that for all $q>61$, every non-zero element in the finite field $\mathbb{F}_{q}$ can be written as a linear combination of two primitive roots of $\mathbb{F}_{q}$. This resolves a conjecture posed by Cohen and Mullen.
By definition primitive and $2$-primitive elements of a finite field extension $\mathbb{F}_{q^n}$ have order $q^n-1$ and $(q^n-1)/2$, respectively. We have already shown that, with minor reservations, there exists a primitive element and a $2$-primitive element $ξ\in \mathbb{F}_{q^n}$ with prescribed trace in the ground field $\mathbb{F}_q$. Here we amend our previous proofs of these results, firstly, by a reduction of these problems to extensions of prime degree $n$ and, secondly, by deriving an exact expression for the number of squares in $\mathbb{F}_{q^n}$ whose trace has prescribed value …
Simple Proof of the Primitive Root Conjecture
本文通过简单证明，给出了原始根猜想的无条件渐进行式，明确了密度常数δ(u)和误差项的改进。
The celebrated Primitive Normal Basis Theorem states that for any $n\ge 2$ and any finite field $\mathbb F_q$, there exists an element $α\in \mathbb F_{q^n}$ that is simultaneously primitive and normal over $\mathbb F_q$. In this paper, we prove some variations of this result, completing the proof of a conjecture proposed by Anderson and Mullen (2014). Our results also imply the existence of elements of $\mathbb F_{q^n}$ with multiplicative order $(q^n-1)/2$ and prescribed trace over $\mathbb F_q$.
Using a Zariski topology associated to a finite field extensions, we give new proofs and generalize the primitive and normal basis theorems.
In this article, we give a largely self-contained proof that the quartic extension $\mathbb{F}_{q^4}$ of the finite field $\mathbb{F}_q$ contains a primitive element $α$ such that the element $α+α^{-1}$ is also a primitive element of ${\mathbb{F}_{q^4}},$ and $Tr_{\mathbb{F}_{q^4}|\mathbb{F}_{q}}(α)=a$ for any prescribed $a \in \mathbb{F}_q$. The corresponding result for finite field extensions of degrees exceeding 4 has already been established by Gupta, Sharma and Cohen.
A proof of the Erdős primitive set conjecture
本文证明了 Erdős 本原集猜想，即级数 $f(A) = \sum_{a\in A}1/(a \log a)$ 的上界在素数集合上取得，并将该方法应用于推进多个相关的数论问题。
A famous conjecture of Artin states that there are infinitely many prime numbers for which a fixed integer $g$ is a primitive root, provided $g \neq -1$ and $g$ is not a perfect square. Thanks to work of Hooley, we know that this conjecture is true, conditional on the truth of the Generalized Riemann Hypothesis. Using a combination of Hooley's analysis and the techniques of Maynard-Tao used to prove the existence of bounded gaps between primes, Pollack has shown that (conditional on GRH) there are bounded gaps between primes with a prescribed primitive root. In the present article, we provide …
We prove that for any prime power $q\notin\{3,4,5\}$, the cubic extension $\mathbb{F}_{q^3}$ of the finite field $\mathbb{F}_q$ contains a primitive element $ξ$ such that $ξ+ξ^{-1}$ is also primitive, and $\textrm{Tr}_{\mathbb{F}_{q^3}/\mathbb{F}_q}(ξ)=a$ for any prescribed $a\in\mathbb{F}_q$. This completes the proof of a conjecture of Gupta, Sharma, and Cohen concerning the analogous problem over an extension of arbitrary degree $n\ge3$.
Let $Q(n)$ denote the count of the primitive subsets of the integers $\{1,2\ldots n\}$. We give a new proof that $Q(n) = α^{(1+o(1))n}$ which allows us to give a good error term and to improve upon the lower bound for the value of this constant $α$. We also show that the method developed can be applied to many similar problems that can be stated in terms of the divisor graph, including other questions about primitive sets, geometric-progression-free sets, and the divisor graph path-cover problem.
A proof of Sugawara's conjecture on Hasse-Weber ray class invariants
本文证明了Sugawara于1936年提出的猜想：虚二次域K上导子f的射线类域可由单个原始的f-分点τ函数值在K上生成。
In this paper we prove an explicit formula for the arithmetic intersection number (CM(K).G1)_{\ell} on the Siegel moduli space of abelian surfaces, generalizing the work of Bruinier-Yang and Yang. These intersection numbers allow one to compute the denominators of Igusa class polynomials, which has important applications to the construction of genus 2 curves for use in cryptography. Bruinier and Yang conjectured a formula for intersection numbers on an arithmetic Hilbert modular surface, and as a consequence obtained a conjectural formula for the intersection number (CM(K).G1)_{\ell} under str…
Effective equidistribution of primitive rational points on expanding horospheres
本文利用矩阵 Kloosterman 和界、Hecke 点有效等分布及 Rogers 积分公式，证明扩张闭 horosphere 上本原有理点的有效等分布，并给出线性同余小解与有限域非方阵计数应用。
The Primitive Generalized Fermat Equation x^3+y^5=z^7: A computer-assisted proof
本文通过计算机辅助方法，结合Fano预解式、模性下降和有限Hecke比较，完整证明了广义费马方程x^3+y^5=z^7无非零互素整数解，并提供了可验证的精确计算证书。
Monochromatic Sums and Products of Polynomials
本文证明了模式 $\{x,x+y,xy\}$ 在常数项为零且次数至少为一次的形式整数多项式空间上具有划分正则性及原始递归界，并为该模式在自然数上的划分正则性提供了首个原始递归界的新证明。
A long-standing conjecture in quantum field theory due to Broadhurst and Kreimer states that the amplitudes of the zig-zag graphs are a certain explicit rational multiple of the odd values of the Riemann zeta function. In this paper we prove this conjecture by constructing a certain family of single-valued multiple polylogarithms. The zig-zag graphs therefore provide the only infinite family of primitive graphs in $ϕ^4_4$ theory (in fact, in any renormalisable quantum field theory in four dimensions) whose amplitudes are now known.
Let $A$ be an abelian variety defined over a number field $K$. We say that a point $P \in A(\overline{\mathbb{Q}})$ is primitive if there is no $Q \in A(\overline{\mathbb{Q}})$ defined on the field of definition of $P$ over $K$ such that $[N]Q=P$ for some positive integer $N \ge 2$. For any primitive point $P \in A(\overline{\mathbb{Q}})$, positive integer $N$ and point $Q \in A(\overline{\mathbb{Q}})$ such that $[N]Q=P$, we prove an effective lower bound on the degree of the field of definition of $Q$ over $K$ of the form $N^δ$ that depends only on $A,K$ and the degree of the field of definit…
Integral periodic orbits on affine spaces
本文给出初等证明，表明仿射空间中单个自同态的整数周期轨道大小有仅依赖维数的统一上界，并推导了原始周期与局部周期的关系，特别证明了仿射平面中轨道大小不超过24。
The Schur polynomials in all primitive $n$th roots of unity
本文证明了当 $n$ 至多含两个不同奇素因子时，所有本原 $n$ 次单位根中的 Schur 多项式取值为 $1$、$0$ 或 $-1$，推广了分圆多项式系数性质，并通过幺模向量系统概念完成证明。
On Artin's Primitive Root Conjecture for Function Fields over $\mathbb{F}_{q}$
本文将整体函数域上的 Artin 本原根猜想推广至 $\mathbb{F}_q$ 上的 $r$ 变量函数域，移除元素 $g$ 为几何元素的限制，并借助 Weil 定理的指数和估计完成证明，同时给出非几何情形的修正因子。
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