On Index Divisibility in the Orbit of Zero Generated by Polynomial Iteration



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We say that a sequence is a divisibility sequence if the n-th term divides the m-th term whenever n divides m. In 2007, Rice changed the course of papers related to divisibility sequences with his result in his undergraduate thesis at Harvey Mudd College that the orbit of zero of all polynomials without a linear term generates a rigid divisibility sequence. In 2017, Chen, Gassert, and Stange consider the index divisibility set of the orbit of zero of polynomials of monic polynomials. In a preprint, as a continuation of the previous paper, Gassert and Urbanski investigate the index divisibility set of the orbit of zero for monic polynomials with an additional term. In this thesis, we generalize many of the previous results to nonmonic polynomials. We provide full proofs of some of the previous known results where original authors supplied sketches and take into account the value of the leading coefficient. We also turn our attention toward divisibility sequences that are not rigid and conjecture how to adjust the results for rigid divisibility sequences to hold for nonrigid divisibility sequences.



Mathematics, Number Theory, Arithmetic Dynamics