On the self-similarity of rational power series with matrix coefficients

TL;DR

This paper proves that the coefficient map of certain matrix-valued rational power series exhibits self-similarity, extending known properties of binomial coefficients modulo a prime to a matrix setting.

Contribution

It establishes a new self-similarity property for coefficient maps of matrix-valued rational functions, generalizing classical binomial coefficient results.

Findings

The coefficient map forms a self-similar tiling of  space.

Self-similarity is characterized by invariance under specific substitutions.

Special case recovers the self-similarity of binomial coefficients modulo p.

Abstract

Let $p$ be a prime, let $d \geq 1$ be an integer and $A$ be the algebra of square matrices of size $d$ over the field of order $p$. Let $P, Q \in A[x_1, \dots x_n]$ be polynomials in $n$ indeterminates with coefficients in $A$, such that $Q$ is invertible in $ A[\![x_1, \dots, x_n]\!]$. Let also $\mathcal M \colon \mathbf Z^n \to A$ be the map associating to the $n$-tuple of integers $(\alpha_1, \dots, \alpha_n)$ the coefficient of the monomial $x_1^{\alpha_1} \dots x_n^{\alpha_n}$ in the development of the rational fraction $PQ^{-1}$ as a power series (the support of $\mathcal M$ is contained in $\mathbf N^n$). Our main result ensures that the map $\mathcal M$, viewed as a tiling of $\mathbf R^n$ by unit cubes with color set $A$, is self-similar. The self-similarity is expressed in terms of invariance under substitutions. By specializing to $d=1$, $n=2$, $P=1$ and $Q =1-x_1-x_2$, we recover the well-known self-similarity feature of the binomial coefficients modulo $p$.