Shallow Circuits with High-Powered Inputs

Abstract : 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.
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Contributor : Pascal Koiran <>
Submitted on : Thursday, July 29, 2010 - 10:52:01 PM
Last modification on : Wednesday, November 20, 2019 - 3:15:41 AM
Long-term archiving on : Thursday, November 4, 2010 - 10:40:39 AM

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  • HAL Id : ensl-00477023, version 4
  • ARXIV : 1004.4960

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Pascal Koiran. Shallow Circuits with High-Powered Inputs. 2010. ⟨ensl-00477023v4⟩

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