论文标题
$ p- $线性化的多项式的预映率在$ \ gf {p} $上
Preimages of $p-$Linearized Polynomials over $\GF{p}$
论文作者
论文摘要
在过去的几十年中，对有限领域的线性化多项式进行了深入研究。近年来，线性化多项式对编码理论和有限几何形状的有趣新应用也得到了强调。 让$ p $成为任何素数。最近，$ p- $线性化的多项式的预图，$ \ sum_ {i = 0}^{\ frac kl-1} x^{p^{li}} $和$ \ sum_ {i = 0} $ \ gf {p^n} $对于任何$ n $。本文将该研究扩展到$ p- $线性化的多项式，超过$ \ gf {p} $，即形状$ l（x）= \ sum_ {i = 0}^tα_ix^^^^^^{p^i}的多项式$ x-x^{p^k} $，可以在任何$ n $的$ \ gf {p^n} $上明确计算$ l（x）$的预图。
Linearized polynomials over finite fields have been intensively studied over the last several decades. Interesting new applications of linearized polynomials to coding theory and finite geometry have been also highlighted in recent years. Let $p$ be any prime. Recently, preimages of the $p-$linearized polynomials $\sum_{i=0}^{\frac kl-1} X^{p^{li}}$ and $\sum_{i=0}^{\frac kl-1} (-1)^i X^{p^{li}}$ were explicitly computed over $\GF{p^n}$ for any $n$. This paper extends that study to $p-$linearized polynomials over $\GF{p}$, i.e., polynomials of the shape $$L(X)=\sum_{i=0}^t α_i X^{p^i}, α_i\in\GF{p}.$$ Given a $k$ such that $L(X)$ divides $X-X^{p^k}$, the preimages of $L(X)$ can be explicitly computed over $\GF{p^n}$ for any $n$.
