Summary: Classifying Polynomials and Identity Testing
August 26, 2009
One of the fundamental problems of computational algebra is to classify polynomials ac-
cording to the hardness of computing them. Recently, this problem has been related to another
important problem: Polynomial identity testing. Informally, the problem is to decide if a cer-
tain succinct representation of a polynomial is zero or not. This problem has been extensively
studied owing to its connections with various areas in theoretical computer science.
Several efficient randomized algorithms have been proposed for the identity testing problem
over the last few decades but an efficient deterministic algorithm is yet to be discovered. It
is known that such an algorithm will imply hardness of computing an explicit polynomial. In
the last few years, progress has been made in designing deterministic algorithms for restricted
circuits, and also in understanding why the problem is hard even for small depth.
In this article, we survey important results for the polynomial identity testing problem and
its connection with classification of polynomials.
The interplay between mathematics and computer science demands algorithmic approaches to var-
ious algebraic constructions. The area of computational algebra addresses precisely this. The most