Polynomial value sets

Closely related to issues of solving Diophantine equations is the question of what set of positive values a polynomial can achieve when fed all possible positive integer values for its variables. A polynomial with a single variable must always yield either be a finite set, or a simple polynomial progression of values. But already the sequence of values for x^{2} y - x y^{3} or even x (y^{2} + 1) seem quite complicated. And for example from the fact that x^{2} y^{2} + (x y ± 1) has solutions Fibonacci[n] it follows that the positive values of (2 - (x^{2} - y^{2} - x y)^{2})x are just Fibonacci[n] (achieved when {x, y} is Fibonacci[{n, n - 1}]). This is the simplest polynomial giving Fibonacci[n], and there are for example no polynomials with 2 variables, up to 4 terms, total degree less than 4, and integer coefficients between -2 and +2, that give any of 2^{n}, 3^{n} or Prime[n]. Nevertheless, from the representation for PrimeQ in the note above it has been shown that the positive values of a particular polynomial with 26 variables, 891 terms and total degree 97 are exactly the primes. (Polynomials with 42 variables and degree 5, and 10 variables and degree 10^{45}, are also known to work, while it is known that one with 2 variables cannot.) And in general the existence of a universal Diophantine equation implies that any set obtained by any finite computation must correspond to the positive values of some polynomial. The analog of doing a long computation to find a result is having to go to large values of variables to find a positive polynomial value. Note that one can imagine, say, emulating the evolution of a cellular automaton by having the t^{th} positive value of a polynomial represent the t^{th} step of evolution. That universality can be achieved just in the positive values of a polynomial is already remarkable. But I suspect that in the end it will take only a surprisingly simple polynomial, perhaps with just three variables and fairly low degree.

(See also page 1165.)