Preimages of $p-$Linearized Polynomials over $\GF{p}$
This is an incremental theoretical advance in finite field mathematics, relevant for applications in coding theory and finite geometry.
The paper tackles the problem of computing preimages of p-linearized polynomials over finite fields, extending previous results to a broader class of polynomials, and provides explicit formulas for these preimages over GF(p^n) for any n.
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 $n$.