The rebirth of resultants, especially through the work of Gelfand, Kapranov, and Zelevinsky on the sparse resultant, has lead to many important recent improvements within computational algebraic geometry. We illustrate a few of these techniques, focusing on a new toric variety version of the classical u-resultant. The object in the title of this talk is an extension of an idea due to Canny for handling degeneracies in the u-resultant.
A corollary of our results is an algorithm for solving n by n polynomial systems (over any algebraically closed field), with complexity close to quadratic in the number of roots.