论文标题
通过离散对数的多项式图的算术约束
Arithmetic constraints of polynomial maps through discrete logarithms
论文作者
论文摘要
令$ q $为主要功率，令$ \ m马理bb f_q $是带有$ q $元素的有限字段，让$θ$为循环组$ \ mathbb f_q^*$的生成器。对于\ Mathbb f_q^*$中的每个$ a \，令$ \log_θa $为唯一的整数$ i \ in \ {1，\ ldots，q-1 \} $，这样$ a =θ^i $。给定多项式$ p_1，\ ldots，p_k \ in \ mathbb f_q [x] $ and divisors $ 1 <d_1，\ ldots，\ ldots，d_k $ of $ q-1 $，我们讨论了函数$$ f_ {i}的分布$ \ mathbb f_q \ setMinus \ cup_ {i = 1}^k \ {y \ in \ athbb f_q \，| \，p_i（y）= 0 \} $。我们的主要结果需要是，在对$（d_i，p_i）$的自然乘法条件下，功能$ f_i $在渐近独立。我们还提供了一些与过去的工作有关的应用程序。
Let $q$ be a prime power, let $\mathbb F_q$ be the finite field with $q$ elements and let $θ$ be a generator of the cyclic group $\mathbb F_q^*$. For each $a\in \mathbb F_q^*$, let $\log_θ a$ be the unique integer $i\in \{1, \ldots, q-1\}$ such that $a=θ^i$. Given polynomials $P_1, \ldots, P_k\in \mathbb F_q[x]$ and divisors $1<d_1, \ldots, d_k$ of $q-1$, we discuss the distribution of the functions $$F_{i}:y\mapsto \log_θP_i(y)\pmod {d_i}, $$ over the set $\mathbb F_q\setminus \cup_{i=1}^k\{y\in \mathbb F_q\,|\, P_i(y)=0\}$. Our main result entails that, under a natural multiplicative condition on the pairs $(d_i, P_i)$, the functions $F_i$ are asymptotically independent. We also provide some applications that, in particular, relates to past work.
