Georgia Institute of Technology College of Sciences

At its core, numerical algebraic geometry is the business of solving zero-dimensional polynomial systems over the complex numbers. Thanks to incredibly fast state-of-the-art software implementations, the bottleneck in these algorithms has shifted from computation time to memory usage.

To address this, recent work has introduced iterator datatypes for solution sets. An iterator represents a list by storing a single element and providing a mechanism to obtain the next one, thereby reducing memory overhead.

In this talk, we present our design of 'homotopy iterators' and 'monodromy coordinates', two iterator datatypes based on the most widely used numerical methods for solving polynomial systems. We highlight the substantial benefits of this low-memory perspective through several iterator-friendly adaptations of existing algorithms, including parameter space searches, data compression, and certification.

This talk features joint work with subsets of Paul Breiding, Hannah Friedman, and David K. Johnson.