A Divergent Argument Concerning Hadamard Roots Of Rational Functions

Abstract

A compelling conjecture attributed variously to Pisot and Schutzenberger suggests that ∑a _h X ^h is the Taylor expansion of a rational function provided only that the a _h all belong to some field finitely generated over Q and there is a nonzero polynomial f so that ∑ f(ah)X ^h represents a rational function — of course only ‘coherent’ choices of the a _h yielding f(a _h ) will do. In part because the sequence of Taylor coefficients is a recurrence sequence (satisfying a linear homogeneous recurrence relation with constant coefficients), the following is a plain language example of the conjecture: A sequence of rationals (a _h ) is a recurrence sequence if (a ³_h ) is a recurrence sequence. Conversely, $\sum {\sqrt h } {X^h}$ is not rational because the √h do not all belong to a field finitely generated over Q. The conjecture can be proved in a ‘generic’ case, but seems inaccessible in general. Breaching the tradition that one should only sketch proofs one believes to be correct, I manipulate formal series divergent everywhere to ‘verify’ the conjecture and to illustrate the apparent obstructions to a convincing proof.