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 $$L(X)=\sum_{i=0}^t \alpha_i X^{p^i}, \alpha_i\in\GF{p}.$$ 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 $n$.

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$ \ GF {p} $上的$ p- $线性多项式的原像

在过去的几十年中，对有限域上的线性多项式进行了深入研究。近年来，线性化多项式在编码理论和有限几何上的有趣新应用也得到了强调。设$ p $为素数。最近，$ p- $线性化多项式$ \ sum_ {i = 0} ^ {\ frac kl-1} X ^ {p ^ {li}} $和$ \ sum_ {i = 0} ^ {\ frac的原像对于任何$ n $，在$ \ GF {p ^ n} $上显式计算出kl-1}（-1）^ i X ^ {p ^ {li}} $。本文将研究扩展到$ \ GF {p} $上的$ p- $线性多项式，即形状为$$ L（X）= \ sum_ {i = 0} ^ t \ alpha_i X ^ {p ^ i}，\ alpha_i \ in \ GF {p}。$$给定$ k $使得$ L（X）$除以$ XX ^ {p ^ k} $，$ L（X）$的原像可以是在$ \ GF {p ^ n} $上显式计算任何$ n $。