Polynomials over finite fields, semigroups and semigroup rings, and their applications

有限域上的多项式由于其在编码理论、组合数学和密码学中的广泛应用而引起了人们极大的兴趣。 本研究的一个方面是对多项式的一些算术性质的理论研究，以及在适用的情况下对它们的置换行为的理论研究。探讨了它们与移位寄存器序列、组合设计和密码分析的联系。有限域上多项式的良好行为部分是由于有限域上多项式的类是良好的环类， 也就是说，唯一分解半群环这项研究的另一个方面是抽象半群环的研究，这是两个本质上不同的代数分支的聚会场所：半群理论和环理论。最重要的是理解半群和相关的半群环之间的相互作用。这项研究的另一个目标是推进这种理解，特别是关于某些算术性质。 此外，还研究了几类半群及其与组合数学的关系。

Polynomials over finite fields have attracted considerable interest due to their wide applications in coding theory, combinatorics and cryptography.  One aspect of this research is the theoretical study of some arithmetic properties of polynomials and the theoretical study of their permutation behavior when applicable. Their connections with shift register sequences, combinatorial designs, and cryptanalysis are explored.The nice behavior of  polynomials over finite fields is partially due to the fact that classes of polynomials over finite fields are nice classes of rings,   i.e., unique factorization semigroup rings. Another aspect of this research is the study of abstract semigroup rings, which is a meeting place for two essentially different branches of algebra: semigroup theory and ring theory. Of prime importance is the understanding of the interplay between semigroups and the associated semigroup rings. Another goal of this research is to advance this understanding, in particular, regarding certain arithmetic properties.  Moreover, the investigation of some classes of semigroups and their connections with combinatorics is also conducted.