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| Format: | Preprint |
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2026
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| Online-Zugang: | https://arxiv.org/abs/2605.22624 |
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| _version_ | 1866917520335699968 |
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| author | Caprace, Pierre-Emmanuel Vast, Justin |
| author_facet | Caprace, Pierre-Emmanuel Vast, Justin |
| contents | 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 $(α_1, \dots, α_n)$ the coefficient of the monomial $x_1^{α_1} \dots x_n^{α_n}$ 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 $p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_22624 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the self-similarity of rational power series with matrix coefficients Caprace, Pierre-Emmanuel Vast, Justin Combinatorics Rings and Algebras 28A80, 11B85, 11T06, 13F25, 52C20, 52C22 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 $(α_1, \dots, α_n)$ the coefficient of the monomial $x_1^{α_1} \dots x_n^{α_n}$ 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 $p$. |
| title | On the self-similarity of rational power series with matrix coefficients |
| topic | Combinatorics Rings and Algebras 28A80, 11B85, 11T06, 13F25, 52C20, 52C22 |
| url | https://arxiv.org/abs/2605.22624 |