Whilst doing exam revision I am having trouble answering the following question from the book, "An Introduction to the Theory of Computation" by Sipser. Unfortunately there's no solution to this question in the book.

Explain why the following is not a legitimate Turing machine.

M = {

The input is a polynomial p over variables x1, ..., xn

1. Try all possible settings of x1, ..., xn to integer values
2. Evaluate p on all of these settings
3. If any of these settings evaluates to 0, accept; otherwise reject. }

This is driving me crazy! I suspect it is because the set of integers is infinite? Does this somehow exceed the alphabet's allowable size?

Thanks!