On finite determinacy for matrices of power series

Abstract

Let R = K[[x 1 , ..., x s ]] be the ring of formal power series with maximal ideal m over a field K of arbitrary characteristic. On the ring M m,n of m × 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 ∈ 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) → (K m , 0), K = R, C, says that f ∈ 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. OverviewThroughout this paper let K denote a field of arbitrary characteristic andthe formal power series ring over K with maximal ideal m. We denote by M m,n := M at(m, n, R) the ring of all m × n matrices of power series. We consider the group of K-algebra automorphisms of R R := Aut(R) and the semi-direct products G l := GL(m, R) ⋊ R, G r := GL(n, R) ⋊ R, and G lr := (GL(m, R) × GL(n, R)) ⋊ R.These groups act on the space M m,n as followsThroughout this paper let G denote one of the groups R, G l , G r , and G lr .For A ∈ M m,n , we denote by GA the orbit of A under the action of G on M m,n . TwoA is G-equivalent to every matrix which coincides with A up to and including terms of degree k. A is called finitely G-determined if there exists a positive integer k such that it is G k-determined.Note that the case n = 1, i.e. M m,1 , covers the case of map-germs (f 1 , ..., f m ), K[[y 1 , ..., y m ]] → R, y i → f i , where G-equivalence is called right-equivalence for G = R and contact-equivalence for G = G l ; the case m = n = 1 is the classical case of one power series. In [GP16] we give necessary and sufficient conditions for finite determinacy of map germs in arbitrary characteristic, in particular for complete intersections, also for non-separable orbit maps.

Finite determinacy of matrices and ideals

“…, a m m is the ideal generated by m × m minors of the right-hand block of Θ (G,A) and Jac(A) is a block of Θ (G,A) . The rest is well-known (see also [GP18,Proposition 4.2]).…”

“…If char(K) = 0 thenT A (GA) coincides T A (GA) ([GP18, Lemma 2.8]) and hence condition (1.1) for some k is necessary and sufficient. In general we haveT A (GA) ⊂ T A (GA), and it is important to notice that the inclusion can be strict in positive characteristic (see Example 2.9 in [GP18]). One of the main aims of this paper is to prove the conjecture for matrices in M m,1 (c.f.…”

Algorithms for group actions in arbitrary characteristic and a problem in singularity theory

“…We review theoretical results from [GP16] on tangent images and tangent spaces of the action of the group G ∈ {R, G l , G r , G lr }, with R := Aut(R), where Aut(R) is the group of K-algebra automorphisms of R. These groups act on the space M m,n as follows (φ, A) → φ(A) := [φ(a ij (x))] = [a ij (φ(x))],…”

“…and call it the tangent space at A to the orbit GA. We haveT A (GA) ⊂ T A (GA), and if K has characteristic zero then the equality holds (see [GP16]). T A (GA) is contained in mM m,n , the tangent space of M m,n .…”

“…Moreover, if A is finitely determined, we give algorithms to compute the image T A (GA) of the tangent map to the orbit map G → GA, which is contained in the tangent space T A (GA) of the orbit GA. It was discovered only recently by the authors, and announced in [GP16], that both spaces may be different if the field K has positive characteristic. One of the purposes of this paper is to give a proof of this result.…”