Echelons of power series and Gabrielov’s counterexample to nested linear Artin Approximation

Gabrielov’s famous example for the failure of analytic Artin approximation in the presence of nested subring conditions is shown to be due to a growth phenomenon in standard basis computations for echelons, a generalization of the concept of ideals in power series rings. hh july 31, 2017 Introduction In the Séminaire Henri Cartan of 1960/61, Grothendieck posed the question whether analytically independent analytic functions are also formally independent [Gr]. It came as a surprise when Gabrielov answered the question in 1971 in the negative. He constructed four analytic functions e, f, g, h in three variables admitting one formal relation but no analytic one [Gb1]. To our knowledge, this is essentially the only known counterexample to Grothendieck’s question. In an opposite direction, Pawłucki constructed analytic functions and a subset Z of the reals for which there do exist analytic relations for parameter values outside Z but there do not exist formal relations for parameters in Z [Pa1]. In a later paper, Gabrielov gave a sufficient condition for a positive answer to Grothendieck’s question in terms of the rank of the Jacobian matrix of the analytic functions [Gb2], see also [Pa2]. Much more generally, Popescu proved in 1985 a difficult approximation theorem which contains as a particular case a positive answer whenever the analytic functions are algebraic power series [Po1, Po2, Sp, Te]. Gabrielov’s counterexample is based on an example of Osgood [Os] from 1916, complemented by a tricky construction and calculation. The deeper reason for the existence of formal divergent relations between analytically independent analytic functions remained mysterious over the years. In this note we explain the genesis of the phenomenon in Gabrielov’s example and provide a systematic way to construct many more counterexamples: It turns out that the existence of formal but not analytic relations is caused by accumulated growth occurrences in standard basis computations for echelons (an echelon is a generalization of an ideal in a power series ring, see below). Such a growth behaviour is well known for standard bases of ideals, but does not do any harm there due to the finiteness of the basis (which is ensured by the Noetherianity of the power series ring.) Standard bases of echelons need no longer be finite, and the iterated growth occurrence in their construction may indeed force divergence. We illustrate in the paper how this phenomenon is related to the presence of sufficiently fast converging coefficients of the (analytic) input series. In the example, the coefficients converge faster than exponentially. For algebraic power series, the phenomenon does not happen. The echelon standard basis may still be infinite, but the convergence of the coefficients of the involved series seems to be sufficiently slow so as to ensure a positive answer to Grothendieck’s question: whenever there is a formal linear relation respecting the scopes, there is also a convergent one (actually, even an algebraic one). The assertion for (1) This work was done in part during a Research-in-Teams program at the Erwin-Schrödinger Institute at Vienna, and the special semester on Artin approximation within the Chaire Jean Morlet at CIRM, Luminy-Marseille. F.J.C.-J. was supported by MTM201340455-P and MTM201675024-P, H.H. and C.K. by the Austrian Science Fund FWF, within the projects P-25652 and AI-0038211, respectively W-1214. (2) Artin attributes in [Ar] Grothendieck’s question to Abhyankar.