No Arabic abstract
We obtain estimates on the number $|mathcal{A}_{boldsymbol{lambda}}|$ of elements on a linear family $mathcal{A}$ of monic polynomials of $mathbb{F}_q[T]$ of degree $n$ having factorization pattern $boldsymbol{lambda}:=1^{lambda_1}2^{lambda_2}cdots n^{lambda_n}$. We show that $|mathcal{A}_{boldsymbol{lambda}}|= mathcal{T}(boldsymbol{lambda}),q^{n-m}+mathcal{O}(q^{n-m-{1}/{2}})$, where $mathcal{T}(boldsymbol{lambda})$ is the proportion of elements of the symmetric group of $n$ elements with cycle pattern $boldsymbol{lambda}$ and $m$ is the codimension of $mathcal{A}$. Furthermore, if the family $mathcal{A}$ under consideration is sparse, then $|mathcal{A}_{boldsymbol{lambda}}|= mathcal{T}(boldsymbol{lambda}),q^{n-m}+mathcal{O}(q^{n-m-{1}})$. Our estimates hold for fields $mathbb{F}_q$ of characteristic greater than 2. We provide explicit upper bounds for the constants underlying the $mathcal{O}$--notation in terms of $boldsymbol{lambda}$ and $mathcal{A}$ with good behavior. Our approach reduces the question to estimate the number of $mathbb{F}_q$--rational points of certain families of complete intersections defined over $mathbb{F}_q$. Such complete intersections are defined by polynomials which are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning their singular locus, from which precise estimates on their number of $mathbb{F}_q$--rational points are established.
We obtain an estimate on the average cardinality of the value set of any family of monic polynomials of Fq[T] of degree d for which s consecutive coefficients a_{d-1},..., a_{d-s} are fixed. Our estimate holds without restrictions on the characteristic of Fq and asserts that V(d,s,bfs{a})=mu_d.q+mathcal{O}(1), where V(d,s,bfs{a}) is such an average cardinality, mu_d:=sum_{r=1}^d{(-1)^{r-1}}/{r!} and bfs{a}:=(a_{d-1},.., d_{d-s}). We provide an explicit upper bound for the constant underlying the mathcal{O}--notation in terms of d and s with good behavior. Our approach reduces the question to estimate the number of Fq--rational points with pairwise--distinct coordinates of a certain family of complete intersections defined over Fq. We show that the polynomials defining such complete intersections are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning the singular locus of the varieties under consideration, from which a suitable estimate on the number of Fq--rational points is established.
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.
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}/mathcal{O})$ of $mathcal{O}$ over $K$. Let $n_D=(mathcal{O}_{mathbb{Q}( j_D)}:mathbb{Z}[ j_D])$ denote the index of $mathbb{Z}[ j_D]$ in the ring of integers of $mathbb{Q}(j_D)$. Suppose $p$ is any prime. We completely determine the factorization of $H_D(x)$ in $mathbb{F}_p[x]$ if either $p mid n_D$ or $p mid D$ is inert in $K$ and the $p$-adic valuation $v_p(n_D)leq 3$. As an application, we analyze the key space of Oriented Supersingular Isogeny Diffie-Hellman (OSIDH) protocol proposed by Col`o and Kohel in 2019 which is the roots set of the Hilbert class polynomial in $mathbb{F}_{p^2}$.
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.
We use generating functions over group rings to count polynomials over finite fields with the first few coefficients prescribed and a factorization pattern prescribed. In particular, we obtain different exact formulas for the number of monic $n$-smooth polynomial of degree $m$ over a finite field, as well as the number of monic $n$-smooth polynomial of degree $m$ with the prescribed trace coefficient.