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In this paper we introduce the additive analogue of the index of a polynomial over finite fields. We study several problems in the theory of polynomials over finite fields in terms of their additive indices, such as value set sizes, bounds on multiplicative character sums, and characterizations of permutation polynomials.
In this paper, we present three classes of complete permutation monomials over finite fields of odd characteristic. Meanwhile, the compositional inverses of these complete permutation polynomials are also proposed.
In 2005, Kayal suggested that Schoofs algorithm for counting points on elliptic curves over finite fields might yield an approach to factor polynomials over finite fields in deterministic polynomial time. We present an exposition of his idea and then
Let K be a global field and f in K[X] be a polynomial. We present an efficient algorithm which factors f in polynomial time.
Let $D$ be a negative integer congruent to $0$ or $1bmod{4}$ and $mathcal{O}=mathcal{O}_D$ be the corresponding order of $ K=mathbb{Q}(sqrt{D})$. The Hilbert class polynomial $H_D(x)$ is the minimal polynomial of the $j$-invariant $ j_D=j(mathbb{C}/m
Let $K$ be the function field of a smooth, irreducible curve defined over $overline{mathbb{Q}}$. Let $fin K[x]$ be of the form $f(x)=x^q+c$ where $q = p^{r}, r ge 1,$ is a power of the prime number $p$, and let $betain overline{K}$. For all $ninmathb