データサイエンティストとして働きながら、空いた時間で数学の研究をしています
数理物質科学研究科 数学専攻 博士後期課程 博士(理学), 2017
筑波大学大学院
数理物質科学研究科 数学専攻 博士前期課程 修士(数学), 2014
筑波大学大学院
理工学群 数学類, 2012
筑波大学
Python
BigQuery
Git / Markdown / LaTex
We study quadratic approximations for two families of hyperquadratic continued fractions in the field of Laurent series over a finite field. As the first application, we give the answer to a question of the second author concerning Diophantine exponents for algebraic Laurent series. As the second application, we determine the degrees of these families in particular case.
As an analogue of Mahler’s classification for real numbers, Bundschuh introduced a classification for Laurent series over a finite field, divided into $A,S,T,U$-numbers. It is known that each of the sets of $A,S,U$-numbers is nonempty. On the other hand, the existence of $T$-numbers has been an open problem. In this paper, we prove that they exist.
In this paper, we study the properties of Diophantine exponents $w_n$ and $w_n^{∗}$ for Laurent series over a finite field. We prove that for an integer $n \geq 1$ and a rational number $w>2n-1$, there exist a strictly increasing sequence of positive integers $(k_j)_{j \geq 1}$ and a sequence of algebraic Laurent series $(\xi _j)_j$ such that $\deg \xi _j =p^{k_j}+1$ and $$ w_1(\xi _j)=w_1 ^{∗}(\xi _j)=\ldots =w_n(\xi _j)=w_n ^{∗}(\xi _j)=w $$ for any $j \geq 1$. For each $n \geq 2$, we give explicit examples of Laurent series $\xi $ for which $w_n(\xi )$ and $w_n^{∗}(\xi )$ are different.