arXiv subjects

Bhitali Kousik

Publications and source records attributed to Bhitali Kousik.

3 recordsLinked to original sources

arXiv2026

A further study of quandles and quandle rings

We investigate core quandles and idempotents in quandle rings of core quandles. We answer several questions on the rank of core quandles and nontrivial idempotents in quandle rings. We also present solutions to two questions raised in a recent paper about non-trivial idempotents in quandle rings $\mathbb{Z}[R_5]$ and $\mathbb{Z}[C_5]$, where $\mathbb{Z}$, $R_5$ and $C_5$ are the ring of integers, the dihedral quandle of order 5, and the commutative quandle of order 5, respectively. We then study units in extended quandle rings of a trivial quandle and the Joyce quandle, and nilpotent elements in extended quandles rings of a trivial quandle, where the ground ring is an integral domain. As a consequence, we show that the quandle ring and the extended quandle ring of a trivial quandle are not nil clean rings. We also explore prime rings and semi-prime rings among quandle rings. We introduce zero-divisor graphs of quandle rings and find an intriguing mirror symmetry among the in-degree and out-degree of the vertices. Moreover, we find a bivariate polynomial in the ring $(\mathbb{Z}_{2n+1}[Q])[X,Y]$ that determines the commutative quandle of order $2n+1$, where $2n+1$ is prime.

math.RA↗

arXiv2026

A further study of polynomial $g_{n,q}$ over finite fields

Let $n\geq 0$ be an integer and $q$ a prime power. The polynomial $g_{n,q}$ was introduced in [10] with the purpose of finding new classes of permutation polynomials over finite fields. We investigate the permutation behaviour of the polynomial $g_{n,q}(X)$ over finite fields of even characteristic. We introduce the multivariate case of the polynomial $g_{n,q}$, and study the permutation polynomials in several variables and local permutation polynomials resulting from the polynomials $g_{n,q}(X_1,X_2,\ldots , X_k)$. We also present several new identities of $g_{n,q}(X)$, and present some open questions on the permutation property of $g_{n,q}(X)$.

math.NT↗

arXiv2026

Orthomorphism Polynomials of degree $7$ over finite fields

In 2019, Xiang Fan \cite{xfan} classified all permutation polynomials of degree $7$ over finite fields of odd characteristics. In this paper, we use this classification to determine the complete list of degree $7$ orthomorphism polynomials over finite fields of order $q\in\{11,~13,~17,~19,~25,~49\}.$ In addition, the non-existence of these polynomials is established for certain fields.

math.NT↗