We make some toric geometric observations which help improve the reliability of certain sparse resultant based equation solving algorithms. In particular, we show how to do univariate projection via monomial maps and generalize Canny's generalized characteristic polynomial to toric varieties (not to mention sparse systems). We also observe that the sparse u-resultant is doomed to failure in certain cases (even after using the toric GCP) so we outline a generalization: twisted Chow forms.
We then give a refinement of Hilbert's Tenth Problem and show how the above constructions can give a single exponential time algorithm for solving certain Diophantine systems. We conclude with a problem with a $500 (U.S.) reward.