Shallow circuits with high-powered inputs

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 a lower bound for the permanent it would be su-cient to derandomize identity testing for polynomials of a very speciflc norm: sums of products of sparse polynomials with sparse coe-cients. This observation leads to new versions of the Shub-Smale ?-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 polynomial upper 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 fact 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 su-ce to obtain a lower bound on the size of depth 4 circuits computing the permanent. These results suggest the intriguing possibility that tools from real analysis might be brought to bear on a longstanding open problem: what is the arithmetic complexity of the permanent polynomial?

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