Hilbert's Tenth Problem: Refinements and Variants

Hilbert's 10th problem, stated in modern terms, is: Find an algorithm that will, given $p \in \mathbb{Z}[x_1,\ldots,x_n]$ determine if there exists $a_1, a_2, \ldots, a_n \in \mathbb{Z}$ such that $p(a_1,\ldots,a_n)=0$. Davis, Putnam, Robinson, and Matijasevic showed that there is no such algorithm. We look at what happens (1) for fixed degree and number of variables, (2) for particular equations, and (3) for variants which reduce the number of variables needed for undecidability results.