Certified computation in algebraic geometry using interval arithmetic

Algebra Seminar
Monday, December 4, 2023 - 1:00pm for 1 hour (actually 50 minutes)
Skiles 006
Kisun Lee – Clemson University
Changxin Ding

Please Note: There will be a pre-seminar (aimed toward grad students and postdocs) from 11 am to 11:30 am in Skiles 006.

The homotopy continuation is a widely recognized method for finding solutions to polynomial systems by tracking the homotopy paths of solutions. However, the current implementation of homotopy continuation relies on heuristics, and hence it requires certification to verify its correctness. We discuss two modalities of certification in algebraic geometry exploiting interval arithmetic. The first is certified homotopy tracking using the Krawczyk method which guarantees correct tracking without path jumping. The second is Smale’s alpha theory over regions for faster certification. We discuss experimental results to demonstrate the effectiveness of these new methods. This talk is a preliminary report of two separate ongoing works.