Checking Polynomial Identities over any Field: Towards a Derandomization?

We present a Monte Carlo algorithm for testing multivariate polynomial identities over any field using fewer random bits than other methods. To test if a polynomial P(x1 ::: xn) is zero, our method uses Pn i=1dlog(di + 1)e random bits , where di is the degree of xi in P, to obtain any inverse polynomial error in polynomial time. The algorithm applies to polynomials given as a black box or in some implicit representation such as a straight-line program. Our method works by evaluating P at truncated formal power series representing square roots of irreducible polynomials over the field. This approach is similar to that of Chen and Kao [CK97], but with the advantage that the techniques are purely algebraic and apply to any field. We also prove a lower bound showing that the number of random bits used by our algorithm is essentially optimal in the black-box model.

• Type: Conference
• Year: 1998
• Where: STOC
• Authors: Daniel Lewin, Salil P. Vadhan
• Keywords: Algorithms, Polynomials, Random Bits, STOC 1998, Testing Multivariate Polynomial