The ring of integers and the ring of polynomials over a finite field share a lot of properties. Using a bounded number of polynomial coefficients, any polynomial can be decomposed as a linear combination of powers of a non-constant polynomial P playing the role of the base of the numeration. Having in mind the theorem of Cobham from 1969 about recognizable sets of integers, it is natural to study P -recognizable sets of polynomials. Based on the results obtained in the Ph.D. thesis of the second author, we study the logical characterization of such sets and related properties like decidability of the corresponding first-order theory.

Under some mild assumptions, we study the state complexity of the trim minimal automaton accepting the greedy representations of the multiples of m ≥ 2 for a wide class of linear numeration systems. As an example, the number of states of the trim minimal automaton accepting the greedy representations of m N in the Fibonacci system is exactly 2m 2 .

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