AI 中文总结
研究椭圆曲线平移除多项式序列，证明有限域时其为纯周期，利用周期性证其子序列是\(\mathbb{Z} _{p}\)-柯西，进而证明索莫斯4和索莫斯5序列的类似结果。
AI 中文摘要
本文研究椭圆曲线\(E/K\)在点\(\mathbf{P}\in E(K)^{2}\)处求值的平移除多项式序列\((\Psi _{n}(\mathbf{P}))_{n\geq 0}\)、\((\Phi _{n}(\mathbf{P}))_{n\geq 0}\)和\((\overline{\Omega }\,_{n}(\mathbf{P}))_{n\geq 0}\)。证明当\(K\)为有限域时这些序列是纯周期的。利用序列的周期性性质证明其某些子序列是\(\mathbb{Z} _{p}\)-柯西的。最后用此结果证明索莫斯4和索莫斯5序列的类似结果。
英文摘要
In this paper we consider the sequences $(Ψ_{n}(\mathbf{P}))_{n\geq 0}$, $(Φ_{n}(\mathbf{P}))_{n\geq 0}$ and $(\overline{Ω}\,_{n}(\mathbf{P}% ))_{n\geq 0}$ of values of the translated division polynomials of an elliptic curve $E/K$ evaluated at a point $\mathbf{P}\in $ $E(K)^{2}$. We prove that these sequences are purely periodic when $K$ is a finite field. Then we use the periodicity properties of these sequences to prove that certain subsequences of these sequences are $% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion _{p}$-Cauchy. Finally, we use this result to prove analogous results for Somos $4$ and Somos $5$ sequences.
Comments21 pages
