Antimicrobial irrigation of implant pockets fails to reduce the propensity for CC. The authors recommend that further prospective multicenter trials be conducted to further elucidate the role of antibiotic irrigation in CC.

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.

The main aim of this paper is to characterize ideals I in the power series ring R = K[[x 1 , . . . , x s ]] that are finitely determined up to contact equivalence by proving that this is the case if and only if I is an isolated complete intersection singularity, provided dim(R/I) > 0 and K is an infinite field (of arbitrary characteristic). Here two ideals I and J are contact equivalent if the local Kalgebras R/I and R/J are isomorphic. If I is minimally generated by a 1 , . . . , a m , we call I finitely contact determined if it is contact equivalent to any ideal J that can be generated by b 1 , . . . , b m with a i −b i ∈ x 1 , . . . , x s k for some integer k. We give also computable and semicontinuous determinacy bounds.The above result is proved by considering left-right equivalence on the ring M m,n of m × n matrices A with entries in R and we show that the Fitting ideals of a finitely determined matrix in M m,n have maximal height, a result of independent interest. The case of ideals is treated by considering 1-column matrices. Fitting ideals together with a special construction are used to prove the characterization of finite determinacy for ideals in R. Some results of this paper are known in characteristic 0, but they need new (and more sophisticated) arguments in positive characteristic partly because the tangent space to the orbit of the left-right group cannot be described in the classical way. In addition we point out several other oddities, including the concept of specialization for power series, where the classical approach (due to Krull) does not work anymore. We include some open problems and a conjecture.

The well-known Mather-Yau theorem says that the isomorphism type of the local ring of an isolated complex hypersurface singularity is determined by its Tjurina algebra. It is also well known that this result is wrong as stated for power series f in K [[x]] over fields K of positive characteristic. In this note we show that, however, also in positive characteristic the isomorphism type of an isolated hypersurface singularity f is determined by an Artinian algebra, namely by a "higher Tjurina algebra" for sufficiently high index, for which we give an effective bound. We prove also a similar version for the "higher Milnor algebra" considered as K[[f ]]-algebra.