arXiv subjects
Publications and source records attributed to Kenichi Takemura.
Let $n\geq 1$ be an odd integer, set $m=(n-1)/2$ and let $M$ be an $n\times n$ matrix whose coefficients are of the form $M_{i,j}=aij+bi+cj+d$ where $0\leq i,j\leq n-1$. Then we prove that for all squares centered at the central coefficient $M_{m,m}$, the mean value of the square equals $M_{m,m}$.
We investigate the construction of circulant matrices derived from primitive roots over finite fields. Our approach reduces exponential sums to Jacobi sums, thereby establishing explicit connections between character theory and matrix structures. The results provide new insights into the interaction between additive and multiplicative characters, and demonstrate how circulant configurations encode arithmetic information in a highly symmetric form. These findings contribute to a deeper understanding of structured matrices in finite fields and open further directions for applications in number theory and combinatorics.
In this paper, we introduce a newly defined algebraic invariant for square matrices termed the \emph{Alternating Power Difference (APD)}. The APD is defined as the signed sum of the powers of diagonal sums along permutations of the symmetric group, distinguishing between even and odd permutations. It serves as a measure of the broken even-odd symmetry inherent in a matrix through higher-order moments. We applied this invariant to all 880 essentially different normal $4\times4$ magic squares (excluding symmetries) and defined the \emph{First Appearance Degree} $m_1$ as the minimum power at which the APD first becomes non-zero. Through an exhaustive computational search, we found that these magic squares are categorized into three clearly separated classes: $m_1=3$ (240 squares), $m_1=4$ (624 squares), and $m_1=\infty$ (16 squares). In particular, the case $m_1=\infty$ identifies exceptionally rare magic squares for which the APD vanishes at all degrees. We refer to these as \emph{Complete Alternating Magic Squares} and demonstrate that they possess a strong algebraic symmetry undetectable by conventional geometric classifications or link-line patterns. Furthermore, we reveal that the APD-based classification refines the classical link-line classification based on complementary sum pairs, showing that each geometric type is clearly distinguished by its first appearance degree. All results in this paper are based on exhaustive computations and are fully reproducible. Our findings suggest that the APD is an effective new invariant for detecting hidden algebraic structures in magic squares and related combinatorial matrices.
This paper focuses on an integer-valued function $f_A(σ) := \operatorname{tr}(A P_σ)$ defined uniformly from a specific square matrix $A$ of order $n$ and a permutation $σ$ on the symmetric group $S_n$. The main objective of this study is to investigate in detail the algebraic behavior of the Alternating Power Difference (APD), denoted as $APD_m(f_A)$, and its first appearance degree $m_1(f_A)$ for this function $f_A$ across various matrix classes. Specifically, we address special matrices such as shifted $r$-th power lattices, Vandermonde matrices, and circulant matrices, analyzing the phenomenon where the value of $APD_m(A)$ remains zero as $m$ increases until a specific degree (the first appearance phenomenon). In particular, we explore closed-form formulas for the first appearance degree $m_1(A)$ and the first appearance value $APD_{m_1}(A)$, presenting Conjectures that hold across multiple matrix classes. These results suggest a deep relationship between the structure of matrices and the analytical properties of functions on the symmetric group, providing new perspectives in matrix theory and combinatorics.