A Note on the Number of Zeros of Polynomials and Exponential Polynomials

?0. The negative solution of Hilbert's Tenth Problem brought with it a number of unsolvable Diophantine problems. Moreover, by actually providing a Diophantine characterization of recursive enumerability, the proof of the negative solution opened the door to the techniques of recursion theory. In this note, we wish to apply several recursion-theoretic facts and an improvement on the exponential Diophantine representation to refine the exponential case of a result of Davis [1972] regarding the difficulty of determining the number of zeros of a polynomial. P, Q, etc. will denote polynomials or exponential polynomials-exactly which will be clear from the context. Let # (P) denote the number of distinct nonnegative zeros of P. Further, let C = {0, 1, * * *, N4o} be the set of possible values of # (P). For A C C, we define A * to be