论文标题
有限场和椭圆曲线的乘法和线性依赖性模型素数
Multiplicative and linear dependence in finite fields and on elliptic curves modulo primes
论文作者
论文摘要
对于积极的整数$ k $和$ l $，我们介绍并研究了$ k $ - 千篇一律的依赖性，而不是代数关闭$ \ overline {\ mathbb {f}} _ p $的有限prime prime field $ \ mathbb {f} _p} _p $，以及$ l $ l $ l $ limip culime con的概述。我们的主要结果之一表明，给定非零有理功能$φ_1，\ ldots，φ_m，\ varrho_1，\ varrho_1，\ ldots，\ varrho_n \ in \ in \ in \ mathbb {q}}（q}（x）$和椭圆形的culve $ e $ ymath $ $ \ mathbb} $，有限的许多$α\ in \ in \ intlline {\ mathbb {f}} _ p $，以下两者中最多可以发生：$φ_1（α），\ ldots，φ_m（α）$是$ k $ as $ k $ - multipplicatifeplicatife contricationallicaly依赖或点$（\ varrho_1（\ varrho_1（α）（α），\ cdrho_1（α），\ cdcot） \ ldots，（\ varrho_n（α），\ cdot）$是$ l $ - 固定地依赖于$ e $ e $ modulo $ p $的减少。作为我们的主要工具之一，我们证明了关于拆分的semiabelian品种中不可减至的曲线相交的一般性声明，$ \ mathbb {g} _ {\ mathrm {m}}^m \ times e^n $与代数的代数子组至少至少2美元。 作为我们的结果的应用，我们改善了M. C. Chang的结果，并扩展了J. F. Voloch的结果，在某些特殊情况下，关于有限领域中大阶的要素。
For positive integers $K$ and $L$, we introduce and study the notion of $K$-multiplicative dependence over the algebraic closure $\overline{\mathbb{F}}_p$ of a finite prime field $\mathbb{F}_p$, as well as $L$-linear dependence of points on elliptic curves in reduction modulo primes. One of our main results shows that, given non-zero rational functions $φ_1,\ldots,φ_m, \varrho_1,\ldots,\varrho_n\in\mathbb{Q}(X)$ and an elliptic curve $E$ defined over the integers $\mathbb{Z}$, for any sufficiently large prime $p$, for all but finitely many $α\in\overline{\mathbb{F}}_p$, at most one of the following two can happen: $φ_1(α),\ldots,φ_m(α)$ are $K$-multiplicatively dependent or the points $(\varrho_1(α),\cdot), \ldots,(\varrho_n(α),\cdot)$ are $L$-linearly dependent on the reduction of $E$ modulo $p$. As one of our main tools, we prove a general statement about the intersection of an irreducible curve in the split semiabelian variety $\mathbb{G}_{\mathrm{m}}^m \times E^n$ with the algebraic subgroups of codimension at least $2$. As an application of our results, we improve a result of M. C. Chang and extend a result of J. F. Voloch about elements of large order in finite fields in some special cases.
