[Paper Review] Shallow Circuits with High-Powered Inputs

Pascal Koiran|arXiv (Cornell University)|Apr 28, 2010

Complexity and Algorithms in GraphsComputer Science24 references17 citations

TL;DR

This paper shows that a deterministic black-box identity testing algorithm for univariate polynomials formed as sums of products of sparse polynomials with sparse coefficients would imply a lower bound on the arithmetic circuit complexity of the permanent, specifically that the permanent is not in VP⁰. The key insight connects derandomization of identity testing for this restricted class to fundamental lower bounds in algebraic complexity theory.

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.

Motivation & Objective

• To establish a connection between derandomization of polynomial identity testing and lower bounds for the permanent in algebraic complexity theory.
• To show that a hitting set for a specific class of univariate polynomials—sums of products of sparse polynomials with sparse coefficients—would imply that the permanent is not computable by constant-free polynomial-size arithmetic circuits.
• To propose a new, potentially more tractable version of the Shub-Smale τ-conjecture focused on real roots of such polynomials.
• To explore whether upper bounds on the number of real roots of these polynomials can yield strong circuit lower bounds, even if only slightly superpolynomial.
• To investigate the feasibility of proving or refuting a real τ-conjecture for sums of products of sparse polynomials, with implications for permanent complexity.

Proposed method

• Reduces the problem of proving lower bounds for the permanent to bounding the number of real roots of univariate polynomials of the form ∑_{i=1}^m ∏_{j=1}^k f_{ij}(x), where each f_{ij} is a sparse polynomial with sparse coefficients.
• Uses the equivalence between black-box identity testing and hitting set construction: a hitting set H for a family of polynomials F ensures that no nonzero f ∈ F vanishes on all points in H.
• Applies a strengthened version of the completeness of the permanent in VNP⁰ to show that if such polynomials had too many real roots, they could simulate hard families like the algebraic number generator.
• Employs the algebraic number generator f_i(x) = x - i and constructs a polynomial g_n(x) = ∏_{i=1}^{2^n} (x - i) to derive a contradiction under the assumption that the number of real roots is sub-exponential.
• Leverages results from real algebraic geometry, particularly fewnomial theory, to analyze the number of real roots of sparse-like polynomials.
• Establishes that even slightly superpolynomial upper bounds on the number of real roots (e.g., q(s) ≤ 2^{(log s)^{1+c}} for c < 1) are sufficient to imply that the permanent is not in VP⁰, via a reduction to depth-4 circuits.

Experimental results

Research questions

• RQ1 Can a deterministic black-box identity testing algorithm for sums of products of sparse polynomials with sparse coefficients imply that the permanent is not in VP⁰?
• RQ2 Is it possible to prove a slightly superpolynomial upper bound on the number of real roots of such polynomials, sufficient to yield a circuit lower bound?
• RQ3 Does the real τ-conjecture hold for polynomials that are sums of products of sparse polynomials, even when k=2?
• RQ4 Can tools from real algebraic geometry, such as fewnomial theory, be used to bound the number of real roots of these structured polynomials?
• RQ5 What is the minimal upper bound on the number of real roots of such polynomials that still implies a lower bound for the permanent?

Key findings

• A deterministic black-box identity testing algorithm for univariate polynomials that are sums of products of sparse polynomials with sparse coefficients would imply that the permanent is not in VP⁰.
• Even a slightly superpolynomial upper bound on the number of real roots—specifically, q(s) ≤ 2^{(log s)^{1+c}} for some c < 1—would suffice to prove that the permanent is not in VP⁰.
• A weaker bound of the form q(s) = 2^{s^{o(1)}} on the number of real roots of such polynomials would still imply that the permanent is not computable by polynomial-size depth-4 circuits using polynomial-size integer constants.
• The conjecture that such polynomials have at most q(kmt) real roots with q(s) = 2^{s^{o(1)}} is plausible, as it holds for k=1 by Descartes’ rule of signs.
• Known examples of polynomials with short arithmetic circuits and many real roots (e.g., Chebyshev polynomials) do not fall into the class of sums of products of sparse polynomials, suggesting the class may be well-behaved.
• A refutation of the real τ-conjecture for this class would yield new examples of sparse-like polynomials with many real roots, while a proof would imply a strong lower bound for the permanent.