# Fun Facts about Polynomials, and Applications to Distributed Storage

Mary Wootters ( Stanford University )

In this talk I'll discuss two fun (and perhaps surprising) facts about polynomials over finite fields, and I will explain how these facts can be useful in the context of distributed storage.  The first fun fact is about polynomial interpolation.  You may know that if f(x) is a quadratic polynomial, then you can learn f(0) using any three other evaluations of f(x).  But can you learn f(0) using less information?  Fun fact: yes!  That is, over certain fields, the "standard" polynomial interpolation algorithm is not the best way to recover f(0). The second fun fact is about restrictions of multivariate polynomials.  You may know that if f(x,y) is a quadratic polynomial, then the restriction of f(x,y) to any line (for example, f(x, ax + b)) is a quadratic univariate polynomial: aka, any slice of a paraboloid is a parabola.  But is the converse true?  Fun fact: no!  That is, it was shown by Guo, Kopparty and Sudan that over certain fields, there are polynomials f(x,y) whose restrictions to all lines are low-degree, but f(x,y) itself is not -- we extend this result in a few ways to find even more such polynomials.  In the talk I'll explain these fun facts, and also explain why the questions above arise naturally in applications.  No background about finite fields will be assumed.  This talk is based on joint works with Venkat Guruswami, Luna Frank-Fischer and Ray Li.

## Speaker bio

Mary Wootters is an assistant professor of Computer Science and Electrical Engineering at Stanford University. She received a PhD in mathematics from the University of Michigan in 2014, and a BA in math and computer science from Swarthmore College in 2008; she was an NSF postdoctoral fellow at Carnegie Mellon University from 2014 to 2016. She works in theoretical computer science, applied math, and information theory; her research interests include error correcting codes and randomized algorithms for dealing with high dimensional data. She is the recipient of an NSF CAREER award and was named a Sloan Research Fellow in 2019.