The arithmetic rank of a variety is the minimal number of equations needed to define it set theoretically, i.e., the smallest number of polynomials generating the defining ideal upto radical. Computing this invariant is notoriously difficult: the minimal generators up to radical often bear little relation to the given ideal generators and can vary unpredictably across characteristics. The residual intersections provide a natural extension of the classical notion of algebraic links. In a different direction, this residual intersection also arises as the defining ideal of a variety of complexes, a notion introduced by Buchsbaum and Eisenbud. We establish a general upper bound for the arithmetic rank of any residual intersection of a complete intersection ideal in an arbitrary Noetherian ring, and we show that this bound is sharp under specific characteristic assumptions. This work is joint with Manav Batavia (Purdue), Taylor Murray (UNL), and Vaibhav Pandey (Purdue).
