2-Variable Frobenius Problem in Z[\sqrt M]

概率Frobenius-Euler多项式与退化概率Frobenius-Euler多项式的表示

Representations by probabilistic Frobenius-Euler and degenerate Frobenius-Euler polynomials

针对矩生成函数在原点邻域存在的随机变量Y，借助umbra微积分推导显式公式，实现用相关概率Frobenius-Euler多项式及其退化、高阶形式表示任意多项式，并举例说明。

Proofs of Chappelon and Alfons\'ın Conjectures On Square Frobenius Numbers and its Relationship to Simultaneous Pell's Equations

Recently, Chappelon and Alfons\'ın defined the square Frobenius number of coprime numbers $m$ and $n$ to be the largest perfect square that cannot be expressed in the form $mx+ny$ for nonnegative integers $x$ and $y$. When $m$ and $n$ differ by $1$ or $2$, they found simple expressions if neither $m$ nor $n$ is a perfect square. If either $m$ or $n$ is a perfect square, they formulated some interesting conjectures which have an unexpected close connection with a known recursive sequence, related to the denominators of Farey fraction approximations to $\sqrt{2}$. In this note, we prove these co…

Weighted Sylvester sums on the Frobenius set in more variables

Let $a_1,a_2,\dots,a_k$ be positive integers with $\gcd(a_1,a_2,\dots,a_k)=1$. Let ${\rm NR}={\rm NR}(a_1,a_2,\dots,a_k)$ denote the set of positive integers nonrepresentable in terms of $a_1,a_2,\dots,a_k$. The largest nonrepresentable integer $\max{\rm NR}$, the number of nonrepresentable positive integers $\sum_{n\in{\rm NR}}1$ and the sum of nonrepresentable positive integers $\sum_{n\in{\rm NR}}n$ have been widely studied for a long time as related to the famous Frobenius problem. In this paper by using Eulerian numbers, we give formulas for the weighted sum $\sum_{n\in{\rm NR}}λ^{n}n^μ$,…

关于三变量 Frobenius 硬币问题

On the Frobenius Coin Problem in Three Variables

本文提出基于同步欧几里得除法序列的算法求解三变量Frobenius硬币问题，复杂度为Θ(log a1)，并给出常数开销归约及Frobenius数上界。

The Frobenius number for sequences of triangular numbers associated with number of solutions

The famous linear diophantine problem of Frobenius is the problem to determine the largest integer (Frobenius number) whose number of representations in terms of $a_1,\dots,a_k$ is at most zero, that is not representable. In other words, all the integers greater than this number can be represented for at least one way. One of the natural generalizations of this problem is to find the largest integer (generalized Frobenius number) whose number of representations is at most a given nonnegative integer $p$. It is easy to find the explicit form of this number in the case of two variables. However,…

The $p$-adic Riemann Hypothesis For Expnonential Sums

The $L$-function of exponential sums associated to the generic polynomial of degree $d$ in $n$ variables over a finite field of characteristic $p$ is studied. A polygon called the Frobenius polygon of the generic polynomial of degree $d$ in $n$ variables over a finite field of characteristic $p$ is defined. A $p$-adic Riemann hypothesis is formulated. It asserts that the Newton polygon of the $L$-function coincides with the Frobenius polygon when $p$ is large enough. This $p$-adic Riemann hypothesis is proved when $n=2$ and $p\equiv-1({\rm mod }\ d)$. In general, it is proved that the Newton p…

与解的数量相关的移位几何序列的Frobenius数

The Frobenius number for shifted geometric sequences associated with the number of solutions

本文给出了移位几何序列$\{a b^n-c\}_n$的$p$-Frobenius数的闭式公式，涵盖几何、Thabit和Mersenne序列及其变体，解决了$p>0$时广义Frobenius数公式缺失的问题。

Solubility of Additive Forms of Twice Odd Degree over $\mathbb{Q}_2(\sqrt{5})$

We prove that an additive form of degree $d=2m$, $m$ odd, $m\ge3$, over the unramified quadratic extension $\mathbb{Q}_2(\sqrt{5})$ has a nontrivial zero if the number of variables $s$ satisifies $s \ge 4d+1$. If $3 \nmid d$, then there exists a nontrivial zero if $s \ge \frac{3}{2}d + 1$, this bound being optimal. We give examples of forms in $3d$ variables without a nontrivial zero in case that $3 \mid d$.

部分二次数域的理想类群与部分二次多项式值的因式分解

Ideal class groups of some quadratic number fields and factorization of values of some quadratic polynomials

本文填补了A. Gica关于二次多项式f_d(x)=d+x²素因子个数相关结论的空白，确定了满足对应素因子个数约束的奇偶正整数d，初步探究了f_d(x)=d-x²的同类问题。

Diophantine exponents for systems of linear forms in two variables

We improve on Jarn\'ık's inequality between uniform Diophantine exponent $α$ and ordinary Diophantine exponent $β$ for a system of $ n\ge 2$ real linear forms in two integer variables. Jarn\'ık (1949, 1954) proved that $β\ge α(α-1)$. In the present paper we give a better bound in the case $α>1$. We prove that β\ge 1/2(α^2-α+1+\sqrt{(α^2-α+1)^2 +4α^2(α-1)}) if 1\le α\le 2 1/2(α^2-1+\sqrt{(α^2-1)^2+4α(α-1)}) if α\ge 2

Periodic points of algebraic functions and Deuring's class number formula

The exact set of periodic points in $\overline{\mathbb{Q}}$ of the algebraic function $\widehat{F}(z)=(-1\pm \sqrt{1-z^4})/z^2$ is shown to consist of the coordinates of certain solutions $(x,y)=(π, ξ)$ of the Fermat equation $x^4+y^4=1$ in ring class fields $Ω_f$ over imaginary quadratic fields $K=\mathbb{Q}(\sqrt{-d})$ of odd conductor $f$, where $-d \equiv 1$ (mod $8$). This is shown to result from the fact that the $2$-adic function $F(z)=(-1+ \sqrt{1-z^4})/z^2$ is a lift of the Frobenius automorphism on the coordinates $π$ for which $|π|_2<1$, for any $d \equiv 7$ (mod $8$), when consider…

Generating functions for power moments of elliptic curves over $\mathbb{F}_p$

Seminal works by Birch and Ihara gave formulas for the $m$th power moments of the traces of Frobenius endomorphisms of elliptic curves over $\mathbb{F}_{p}$ for primes $p \geq 5$. Recent works by Kaplan and Petrow generalized these results to the setting of elliptic curves that contain a subgroup isomorphic to a fixed finite abelian group $A$. We revisit these formulas and determine a simple expression for the zeta function $Z_p(A; t)$, the generating function for these $m$th power moments. In particular, we find that \[ Z_p(A;t) = \frac{\widehat{Z}_p(A; t)}{\displaystyle \prod_{a \in \textrm{…

Relations between the random variable $w_x$ and the Dirichlet divisor problem

We have developed a heuristic showing that in the Dirichlet divisor problem for the almost all $n \in \mathbb{N}^{+}$: $$ R(n) \leq O(ψ(n)n^{\frac{1}{4}}) $$ where $$ R(n) = \Big\lvert \sum_{x=1}^{n}\Big\lfloor\frac{n}{x}\Big\rfloor - n\log{n} - (2γ-1)n \Big\rvert $$ and $ ψ(n) $ - any positive function that increases unboundedly as $ n \to \infty $. The result is achieved under the hypothesis: $$ \Big \{\frac{n}{x} \Big \} \sim w_x $$ where $ w_x $ is uniformly distributed over $ [0,1) $ random variable with a values set $ \{0, \frac {1} {x}, \ldots, \frac{x-1}{x} \} $ and the value accepting…

Solubility of Additive Forms of Twice Odd Degree over Ramified Quadratic Extensions of $\mathbb{Q}_2$

We determine the minimal number of variables $Γ^*(d, K)$ which guarantees a nontrivial solution for every additive form of degree $d=2m$, $m$ odd, $m \ge 3$ over the six ramified quadratic extensions of $\mathbb{Q}_2$. We prove that if $K$ is one of $\{\mathbb{Q}_2(\sqrt{2}), \mathbb{Q}_2(\sqrt{10}), \mathbb{Q}_2(\sqrt{-2}), \mathbb{Q}_2(\sqrt{-10})\}$, $Γ^*(d,K) = \frac{3}{2}d$, and if $K$ is one of $\{\mathbb{Q}_2(\sqrt{-1}), \mathbb{Q}_2(\sqrt{-5})\}$, $Γ^*(d,K) = d+1$. The case $d=6$ was previously known.

On algebraic integers all conjugates of which belong to a given compact subset of the complex plane

The study of Frobenius endomorphism provides numerous information about its corresponding Abelian variety. To understand the action of the Frobenius endomorphism, one may be interested in its eigenvalues. According to Weil's third conjecture ("Riemann hypothesis over finite fields"), they all have absolute value less than or equal to $2g\sqrt{p}$. Thus, the eigenvalues of the Frobenius endomorphism all belong to the same compact subset of the complex plane, and are roots of the same monic polynomial with integer coefficients (the characteristic polynomial of the Frobenius endomorphism). Such c…

The two-point correlation function of the fractional parts of \sqrt{n} is Poisson

Elkies and McMullen [Duke Math.J.~123 (2004) 95--139] have shown that the gaps between the fractional parts of \sqrt n for n=1,\ldots,N, have a limit distribution as N tends to infinity. The limit distribution is non-standard and differs distinctly from the exponential distribution expected for independent, uniformly distributed random variables on the unit interval. We complement this result by proving that the two-point correlation function of the above sequence converges to a limit, which in fact coincides with the answer for independent random variables. We also establish the convergence o…

算术数列三元组的 $p$-数值半群

The $p$-numerical semigroup of the triple of arithmetic progressions

本文研究算术数列三元组的 $p$-Frobenius 数，通过确定 $p$-Apéry 集的元素，给出了此前未知的 $p>0$ 情形下显式公式及相关值。