The Similarities (and Diierences) between Polynomials and Integers

The purpose of this paper is to examine the two domains of the integers and the polynomials, in an attempt to understand the nature of complexity in these very basic situations. Can we formalize the integer algorithms which shed light on the polynomial domain, and vice versa? When will the casting of one in the other speed up an existing algorithm? Why do some problems not lend themselves to this kind of speed-up? We give several simple and natural theorems that show how problems in one domain can be embedded in the other, and we examine the complexity-theoretic consequences of these embeddings. We also prove several results on the impossibility of solving integer problems by mimicking their polynomial counterparts.