Dr. Mrinal Kumar, Simons Institute for the Theory of Computing, Berkeley, USA.
Description
Speaker: Dr. Mrinal Kumar, Simons Institute for the Theory of Computing,
Berkeley, USA.
Time: 11 am, Friday, 11th January.
Venue: Ramanujan Hall.
Title : Some closure results for polynomial factorization and applications
Abstract : In a sequence of seminal results in the 80's, Kaltofen showed
that if an n-variate polynomial of degree poly(n) can be computed by an
arithmetic circuit of size poly(n), then each of its factors can also be
computed an arithmetic circuit of size poly(n). In other words,
the complexity class VP (the algebraic analog of P) of polynomials, is
closed under taking factors.
A fundamental question in this line of research, which has largely
remained open is to understand if other natural classes of
multivariate polynomials, for instance, arithmetic formulas, algebraic
branching programs, constant depth arithmetic circuits or the
complexity class VNP (the algebraic analog of NP) of polynomials, are
closed under taking factors. In addition to being fundamental
questions on their own, such 'closure results' for polynomial
factorization play a crucial role in the understanding of hardness
randomness tradeoffs for algebraic computation.
I will talk about the following two results, whose study was motivated
by these questions.
1. The class VNP is closed under taking factors. This proves a
conjecture of B{\"u}rgisser.
2. All factors of degree at most poly(log n) of polynomials with
constant depth circuits of size
poly(n) have constant (a slightly larger constant) depth arithmetic
circuits of size poly(n).
This partially answers a question of Shpilka and Yehudayoff and has
applications to hardness-randomness tradeoffs for constant depth
arithmetic circuits. Based on joint work with Chi-Ning Chou and Noam
Solomon.