  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.
