Characterization of the Structure-Generating Functions of Regular Sets and the DOL Growth Functions

The structure-generating functions of regular sets and the DOL growth functions are characterized. Our result is: A rational function f(z) with integral coefficients is a structure-generating function of a regular set if and only if (1) the constant term of its denominator is 1 and that of its numerator is 0, (2) every coefficient an of its Taylor series expansion is nonnegative, and (3) every pole of the minimal absolute value of fi(z) = ∑n=0∞ anM+i zn is of the form re, where r > 0 and e is a root of unity for any integer M ⩾ 1 and i = 0, 1,…, M − 1. Also stated are a result on the star height problem and an analogous characterization of the growth functions of DOL systems.

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