A note on the paper arXiv:2112.14547

关于Aesthetic数与广义中心三项式系数

On Apery numbers and generalized central trinomial coefficients

我们决定将这个预印本与另外两个预印本（arXiv：1101.1946和arXiv：1008.3887）合并。关于Awesome数和多项式的结果将被添加到arxiv的新版本：1101.1946，关于广义三项式系数的结果将被添加到arxiv的新版本：1008.3887

与三项式超曲面的HK函数有关的秩计算问题

A rank computation problem related to the HK function of trinomial hypersurfaces

在这篇文章中，我提供了一个解决方案的秩计算问题有关的计算希尔伯特-昆兹函数的任何不相交项三项式超曲面，在任何领域的特征2。这个秩计算问题在Upadhyay（arXiv：1204.5417）中提出。不交项三叉超曲面的Hilbert-Kunz函数的公式可以从本文的结果中推导出来，模许多繁琐的符号。

Polynomial Identities Implying Capparelli's Partition Theorems

We propose and recursively prove polynomial identities which imply Capparelli's partition theorems. We also find perfect companions to the results of Andrews, and Alladi, Andrews and Gordon involving $q$-trinomial coefficients. We follow Kurşungöz's ideas to provide direct combinatorial interpretations of some of our expressions. We use of the trinomial analogue of Bailey's lemma to derive new identities. These identities relate triple sums and products. A couple of new Slater type identities are also noted.

A note on permutation polynomials over finite fields

Permutation polynomials over finite fields constitute an active research area and have applications in many areas of science and engineering. In this paper, two conjectures on permutation polynomials proposed recently by Wu and Li [19] are settled. Moreover, a new class of permutation trinomials of the form $x+γ\textup{Tr}_{q^n/q}(x^k)$ is also presented, which generalizes two examples of [10].

The Grassl-Rötteler cyclic and consta-cyclic MDS codes are generalised Reed-Solomon codes

We prove that the cyclic and constacyclic codes constructed by Grassl and Rötteler in arXiv:1502.05267 are generalised Reed-Solomon codes. This note can be considered as an addendum to that article. It can also be considered as an appendix to arXiv:2106.10180, where Conjecture 11 of arXiv:1502.0526, which was stated for Grassl-Rötteler codes, is proven for generalised Reed-Solomon codes. The content of this note, together with arXiv:2106.10180, therefore implies that Conjecture 11 from arXiv:1502.0526 is true.

Permutation polynomials, fractional polynomials, and algebraic curves

In this note we prove a conjecture by Li, Qu, Li, and Fu on permutation trinomials over $\mathbb{F}_3^{2k}$. In addition, new examples and generalizations of some families of permutation polynomials of $\mathbb{F}_{3^k}$ and $\mathbb{F}_{5^k}$ are given. We also study permutation quadrinomials of type $Ax^{q(q-1)+1} + Bx^{2(q-1)+1} + Cx^{q} + x$. Our method is based on the investigation of an algebraic curve associated with a {fractional polynomial} over a finite field.