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TR18-164 | 18th September 2018 18:41

On the symmetries of design polynomials


Authors: Nikhil Gupta, Chandan Saha
Publication: 18th September 2018 19:36
Downloads: 216


In a Nisan-Wigderson design polynomial (in short, a design polynomial), the gcd of every pair of monomials has a low degree. A useful example of such a polynomial is the following:
$$\text{NW}_{d,k}(\mathbf{x}) = \sum_{h \in \mathbb{F}_d[z], ~\deg(h) \leq k}{~~~~\prod_{i = 0}^{d-1}{x_{i, h(i)}}},$$
where $d$ is a prime, $\mathbb{F}_d$ is the finite field with $d$ elements, and $k \ll d$. The degree of the gcd of every pair of monomials in $\text{NW}_{d,k}$ is at most $k$. For concreteness, let us fix $k = \lceil \sqrt{d} \rceil$. The family of polynomials $\mathcal{NW} :=$ {$\text{NW}_{d,k} : d \text{ is a prime}$} and close variants of it have been used as hard explicit polynomial families in several recent arithmetic circuit lower bound proofs. But, unlike the permanent, very little is known about the various complexity and structural aspects of $\mathcal{NW}$ beyond the fact that $\mathcal{NW} \in \text{VNP}$. Is $\mathcal{NW}$ $\text{VNP}$-complete? Is $\text{NW}_{d,k}$ characterized by its symmetries? Is it circuit-testable, i.e., given a circuit $\text{C} $ can we check efficiently if $\text{C}$ computes $\text{NW}_{d,k}$? Characterization of polynomials by their symmetries plays a central role in the geometric complexity theory program. Here, we answer the last two questions.

We show that $\text{NW}_{d,k}$ is characterized by its group of symmetries over $\mathbb{C}$. We also show that $\text{NW}_{d,k}$ is characterized by circuit identities which implies that $\text{NW}_{d,k}$ is circuit-testable in randomized polynomial time. As another implication, we obtain the ``flip theorem'' for $\mathcal{NW}$: Suppose $\text{NW}_{d,k}$ is not computable by circuits of size $s$. Then, there exist $\text{poly}(s)$ many points $\mathbf{a}_1, \ldots, \mathbf{a}_m$ such that for every circuit $\text{C} $ of size $s$, there is an $\ell \in [m]$ satisfying $\text{C} (\mathbf{a}_\ell) \neq \text{NW}_{d,k}(\mathbf{a}_\ell)$. These points can be computed deterministically in $\text{poly}(s)$ time if black-box polynomial identity testing for size-$s$ circuits can be derandomized in $\text{poly}(s)$ time. It is well-known that the permanent polynomial has the above-mentioned features.

We also show a structural similarity between the group of symmetries of the permanent and that of $\text{NW}_{d,k}$: If $A$ is in the group of symmetries of $\text{NW}_{d,k}$ then $A = D \cdot P$, where $D$ and $P$ are diagonal and permutation matrices respectively. This is proved by completely characterizing the Lie algebra of $\text{NW}_{d,k}$, and using an interplay between the Hessian of $\text{NW}_{d,k}$ and the evaluation dimension.

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