A Case of Depth-3 Identity Testing, Sparse Factorization and Duality

Polynomial identity testing (PIT) problem is known to be challenging even for constant depth arithmetic circuits. In this work, we study the complexity of two special but natural cases of identity testing - first is a case of depth-3 PIT, the other of depth-4 PIT. Our first problem is a vast generalization of: verify whether a bounded top fan-in depth-3 circuit equals a sparse polynomial (given as a sum of monomial terms). Formally, given a depth-3 circuit C, having constant many general product gates and arbitrarily many semidiagonal product gates, test if the output of C is identically zero. A semidiagonal product gate in C computes a product of the form m · b i=1 ei i , where m is a monomial, i is a linear polynomial and b is a constant. We give a deterministic polynomial time test, along with the computation of leading monomials of semidiagonal circuits over local rings. The second problem is on verifying a given sparse polynomial factorization, which is a classical question (v...

• Type: Journal
• Year: 2011
• Where: ECCC
• Authors: Chandan Saha, Ramprasad Saptharishi, Nitin Saxena
• Keywords: Deterministic Polynomial Time, ECCC 2011, ECommerce, Polynomials, Sparse Polynomials