Computer Science – Computational Complexity
Scientific paper
2010-04-28
Computer Science
Computational Complexity
A few typos corrected
Scientific paper
A polynomial identity testing algorithm must determine whether an input polynomial (given for instance by an arithmetic circuit) is identically equal to 0. In this paper, we show that a deterministic black-box identity testing algorithm for (high-degree) univariate polynomials would imply a lower bound on the arithmetic complexity of the permanent. The lower bounds that are known to follow from derandomization of (low-degree) multivariate identity testing are weaker. To obtain our lower bound it would be sufficient to derandomize identity testing for polynomials of a very specific norm: sums of products of sparse polynomials with sparse coefficients. This observation leads to new versions of the Shub-Smale tau-conjecture on integer roots of univariate polynomials. In particular, we show that a lower bound for the permanent would follow if one could give a good enough bound on the number of real roots of sums of products of sparse polynomials (Descartes' rule of signs gives such a bound for sparse polynomials and products thereof). In this third version of our paper we show that the same lower bound would follow even if one could only prove a slightly superpolynomial upper bound on the number of real roots. This is a consequence of a new result on reduction to depth 4 for arithmetic circuits which we establish in a companion paper. We also show that an even weaker bound on the number of real roots would suffice to obtain a lower bound on the size of depth 4 circuits computing the permanent.
No associations
LandOfFree
Shallow Circuits with High-Powered Inputs does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Shallow Circuits with High-Powered Inputs, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Shallow Circuits with High-Powered Inputs will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-68311