论文标题
在有限场上给定多项式中随机多项式中零数的渐近分布
Asymptotic distribution of the number of zeros in random polynomials in a given equivalence class over a finite field
论文作者
论文摘要
Hayes等价在有限字段$ \ fq $上定义在规定的领先系数方面，而残基类别模拟给定的一元多项式$ Q $。我们研究了给定的Hayes等效类别中有限磁场上随机多项式中零数的分布。众所周知，$ \ fq $上随机多项式的不同零的数量是渐变的泊松，平均值为1。我们表明，对于任何给定的Hayes等价类别中的随机多项式，这也是正确的。当此多项式的程度与$ Q $成正比时，此类多项式的数量也给出了渐近公式，并且$ Q $的度数和规定的领先系数的数量受$ \ sqrt {q} $界定。当$ q = 1 $时，问题等同于研究芦苇 - 固体代码中距离分布的研究。我们的渐近公式扩展了一些早期的结果，并暗示着大型芦苇 - 固体代码的所有单词都是普通的，这进一步支持了众所周知的{\ em deep-hole}猜想。
Hayes equivalence is defined on monic polynomials over a finite field $\fq$ in terms of the prescribed leading coefficients and the residue classes modulo a given monic polynomial $Q$. We study the distribution of the number of zeros in a random polynomial over finite fields in a given Hayes equivalence class. It is well known that the number of distinct zeros of a random polynomial over $\fq$ is asymptotically Poisson with mean 1. We show that this is also true for random polynomials in any given Hayes equivalence class. Asymptotic formulas are also given for the number of such polynomials when the degree of such polynomials is proportional to $q$ and the degree of $Q$ and the number of prescribed leading coefficients are bounded by $\sqrt{q}$. When $Q=1$, the problem is equivalent to the study of the distance distribution in Reed-Solomon codes. Our asymptotic formulas extend some earlier results and imply that all words for a large family of Reed-Solomon codes are ordinary, which further supports the well-known {\em Deep-Hole} Conjecture.
