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In this note we show (for a large enough dimension of the underlying field) a conjecture of [C. Beierle, C. Carlet, G. Leander, L. Perrin, {\em A further study of quadratic APN permutations in dimension nine}, Finite Fields Appl. 81 (2022), 102049] o
Externí odkaz:
http://arxiv.org/abs/2410.23097
In this paper we use algebraic curves and other algebraic number theory methods to show the validity of a permutation polynomial conjecture regarding $f(X)=X^{q(p-1)+1} +\alpha X^{pq}+X^{q+p-1}$, on finite fields $\mathbb{F}_{q^2}, q=p^k$, from [A. R
Externí odkaz:
http://arxiv.org/abs/2410.22692
Autor:
Bary-Soroker, Lior, Shmueli, Roy
We study a random polynomial of degree $n$ over the finite field $\mathbb{F}_q$, where the coefficients are independent and identically distributed and uniformly chosen from the squares in $\mathbb{F}_q$. Our main result demonstrates that the likelih
Externí odkaz:
http://arxiv.org/abs/2410.16814
Autor:
Ehrenborg, Richard
Using the cyclotomic identity we compute sums over d-tuples of monic polynomials in F_q[x] weighted by the multiplicity of their irreducible factors. As consequences we determine explicit expressions for the number of d-tuples of polynomials such tha
Externí odkaz:
http://arxiv.org/abs/2410.12058
A function from $\Bbb F_{2^n}$ to $\Bbb F_{2^n}$ is said to be {\em $k$th order sum-free} if the sum of its values over each $k$-dimensional $\Bbb F_2$-affine subspace of $\Bbb F_{2^n}$ is nonzero. This notion was recently introduced by C. Carlet as,
Externí odkaz:
http://arxiv.org/abs/2410.10426
Autor:
Song, Yufeng, Luo, Jinquan
Projective Reed-Muller codes(PRM codes) are constructed from the family of projective hypersurfaces of a fixed degree over a finite field $\F_q$. In this paper, we completely determine the minimal distance of the hull of any Projective Reed-Muller co
Externí odkaz:
http://arxiv.org/abs/2410.07217
Autor:
Salia, Nika, Tóth, Dávid
This paper establishes an analog of the Erd\H{o}s-Ko-Rado theorem to polynomial rings over finite fields, affirmatively answering a conjecture of C. Tompkins. A $k$-uniform family of subsets of a set of finite size $n$ is $l$-intersecting if any two
Externí odkaz:
http://arxiv.org/abs/2409.17821
We determine necessary and sufficient conditions for unicritical polynomials to be dynamically irreducible over finite fields. This result extends the results of Boston-Jones and Hamblen-Jones-Madhu regarding the dynamical irreducibility of particula
Externí odkaz:
http://arxiv.org/abs/2409.10467
Publikováno v:
Finite Fields Appl. 92(2023), Paper No. 102281, 22 pp
Let $k \geq 1$ be a natural number and $f \in \mathbb{F}_q[t]$ be a monic polynomial. Let $\omega_k(f)$ denote the number of distinct monic irreducible factors of $f$ with multiplicity $k$. We obtain asymptotic estimates for the first and the second
Externí odkaz:
http://arxiv.org/abs/2409.08559
Autor:
Rahaman, Habibur
Following the work of Castillo-Hall-Oliver-Pollack-Thompson who extended Maynard-Tao theorem on admissible tuples to number fields and function fields for tuples with monic linear forms, here we obtain the Maynard-Tao theorem for admissible tuples of
Externí odkaz:
http://arxiv.org/abs/2409.04705