In search of Robbins stability

We speculate on whether a certain p-adic stability phenomenon, observed by David Robbins empirically for Dodgson condensation, appears in other nonlinear recurrence relations that “unexpectedly” produce integer or nearly-integer sequences. We exhibit an example (number friezes) where this phenomenon provably occurs. This note may be viewed as an addendum to Robbins’s note [5] in this volume. Its purpose is to speculate on whether the p-adic stability phenomenon that Robbins observed empirically for Dodgson condensation appears in other nonlinear recurrence relations that “unexpectedly” produce integer or nearly-integer sequences, and to provide an example where this provably occurs. In order to carry out this speculation, we’ll phrase Robbins’s observation in a somewhat more general framework. For us, a recurrence relation over a field K will consist of a finite partially ordered set S plus, for each s ∈ S, a rational function fs = Ps/Qs over K in the indeterminate vector (xt)t<s. (The restriction to S finite does not concede any generality for our purposes: to consider an infinite recurrence, look instead at all of its finite truncations.) We also assume (for simplicity) that the partial order on S is generated by the relation in which t is less than s if fs is nonconstant as a function of xt alone. In this case, s ∈ S is