Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Roots in the substitution group and in the group of Riordan matrices with ones in the main diagonal

Jorge Calero-Sanz, Luis Felipe Prieto-Martínez

TL;DR

The paper investigates the existence and uniqueness of iterative roots in the substitution group $\mathcal{J}(Z)$ and in Riordan matrix groups by establishing a Riordan-group type criterion that translates $\omega^{[n]}=g$ into a system of matrix equations $\bigl(R_m(1,\omega)\bigr)^n=R_m(1,g)$ for all $m$. It proves that over characteristic-0 fields, $\mathcal{J}(\mathbb{K})$ and $\mathcal{R}'(\mathbb{K})$ are algebraically complete URE-groups, yielding unique roots for every order; in positive characteristic, these properties fail and root behavior becomes nuanced, with parity and modular constraints playing key roles. The work also links root existence to eigenvectors and stabilisers in the Riordan framework, providing computable coefficient relations and iterative procedures, and it discusses extensions to conjugacy classes and related problems, outlining several open questions for non-field rings. Overall, the paper connects the algebraic structure of $\mathcal{J}(Z)$ and $\mathcal{R}'(Z)$ to explicit matrix-analytic criteria for iterative roots and highlights the impact of the base ring's arithmetic on root behavior.

Abstract

We investigate the existence and uniqueness of iterative roots of order $n$ within the substitution group of formal power series $\mathcal J(Z)$ -- with coefficients in a commutative ring with unity $Z$ -- employing a matrix-based framework grounded in the Riordan group. We analyse the relationship between the substitution group and the Lagrange subgroup -- a group of Riordan matrices -- and explore some classic questions concerning algebraic completeness and uniqueness of root extractions. This approach allows us to obtain various results about the roots in $\mathcal J(Z)$ for different choices of $Z$. Furthermore, the examination of the substitution group facilitates the analysis of roots within the Riordan matrix group.

Roots in the substitution group and in the group of Riordan matrices with ones in the main diagonal

TL;DR

The paper investigates the existence and uniqueness of iterative roots in the substitution group and in Riordan matrix groups by establishing a Riordan-group type criterion that translates into a system of matrix equations for all . It proves that over characteristic-0 fields, and are algebraically complete URE-groups, yielding unique roots for every order; in positive characteristic, these properties fail and root behavior becomes nuanced, with parity and modular constraints playing key roles. The work also links root existence to eigenvectors and stabilisers in the Riordan framework, providing computable coefficient relations and iterative procedures, and it discusses extensions to conjugacy classes and related problems, outlining several open questions for non-field rings. Overall, the paper connects the algebraic structure of and to explicit matrix-analytic criteria for iterative roots and highlights the impact of the base ring's arithmetic on root behavior.

Abstract

We investigate the existence and uniqueness of iterative roots of order within the substitution group of formal power series -- with coefficients in a commutative ring with unity -- employing a matrix-based framework grounded in the Riordan group. We analyse the relationship between the substitution group and the Lagrange subgroup -- a group of Riordan matrices -- and explore some classic questions concerning algebraic completeness and uniqueness of root extractions. This approach allows us to obtain various results about the roots in for different choices of . Furthermore, the examination of the substitution group facilitates the analysis of roots within the Riordan matrix group.

Paper Structure

This paper contains 17 sections, 18 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Main groups of interest
  3. Roots in groups
  4. What is it done in this paper?
  5. Characterization of roots in J(Z) via Riordan matrices
  6. Roots in J(Z) for some particular examples of the ring Z
  7. Roots in J(K), for K being a field of characteristic 0
  8. Roots in J(Z), for J(Z) being a general ring of positive characteristic.
  9. Roots in J(Z)
  10. Roots in R'(Z)
  11. Roots in R'(Z), for some particular examples of the ring Z
  12. Connections to some other problems
  13. Stabiliser of an infinite column vector
  14. Conjugacy classes in J(Z)
  15. Roots in A(K)
  16. ...and 2 more sections

Key Result

Theorem 1

Let $Z$ be any commutative ring with unity and $g=x+g_2x^2+g_3x^3+\ldots$ and $\omega=x+\omega_2x^2+\omega_3x^3+\ldots$ be two elements in $\mathcal{J}(Z)$. Let us denote $\bm c_k=[\omega_2,\ldots, \omega_{k-1}]^T$ and $\bm r_k$ to the the row vector consisting in the elements in the positions $(k,2

Theorems & Definitions (38)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 1
  • proof
  • Proposition 2
  • proof
  • Theorem 3
  • proof
  • ...and 28 more