On finite determinacy for matrices of power series

TL;DR

This paper extends the classical finite determinacy criterion from map germs to matrices over power series rings, providing a new, computable criterion applicable in arbitrary characteristic, including positive characteristic where previous results failed.

Contribution

It introduces a general sufficient criterion for finite G-determinacy of matrices over power series rings in any characteristic, incorporating the tangent image of the orbit map.

Findings

Extended finite determinacy criterion to matrices in arbitrary characteristic.

Identified issues with orbit map separability in positive characteristic.

Provided a new computable bound for G-determinacy of matrices.

Abstract

Let $R=K[[x_1,...,x_s]]$ be the ring of formal power series with maximal ideal $\mathfrak{m}$ over a field $K$ of arbitrary characteristic. On the ring $M_{m,n}$ of $m\times n$ matrices $A$ with entries in $R$ we consider several equivalence relations given by the action on $M_{m,n}$ of a group $G$. $G$ can be the group of automorphisms of $R$, combined with the multiplication of invertible matrices from the left, from the right, or from both sides, respectively. We call $A$ finitely $G$-determined if $A$ is $G$-equivalent to any matrix $B$ with ${A-B} \in {\mathfrak{m}^k M_{m,n}}$ for some finite integer $k$, which implies in particular that $A$ is $G$--equivalent to a matrix with polynomial entries. The classical criterion for analytic or differential map germs $f:(K^s,0) \to (K^m,0)$, $K = \mathbb{R}, \mathbb{C}$, says that $f \in M_{m,1}$ is finitely determined (with respect to various group actions) iff the tangent space to the orbit of $f$ has finite codimension in $M_{m,1}$. We extend this criterion to arbitrary matrices in $M_{m,n}$ if the characteristic of K is 0 or, more general, if the orbit map is separable. In positive characteristic however, the problem is more subtle since the orbit map is in general not separable, as we show by an example. This fact had been overlooked in previous papers. Our main result is a general sufficient criterion for finite $G$-determinacy in $M_{m,n}$ in arbitrary characteristic in terms of the tangent image of the orbit map, which we introduce in this paper. This criterion provides a computable bound for the $G$-determinacy of a matrix $A$ in $M_{m,n}$, which is new even in characteristic 0.