Abstract
In this paper we consider the problem of determining whether an unknown arithmetic circuit, for which we have oracle access, computes the identically zero polynomial. This problem is known as the blackbox polynomial identity testing (PIT) problem. Our focus is on polynomials that can be written in the form \(f(\bar x) = \sum\nolimits_{i = 1}^k {h_i (\bar x) \cdot g_i (\bar x)} \), where each h _{ i } is a polynomial that depends on only ρ linear functions, and each g _{ i } is a product of linear functions (when h _{ i } = 1, for each i, then we get the class of depth3 circuits with k multiplication gates, also known as ΣΠΣ(k) circuits, but the general case is much richer). When max_{ i }(deg(h _{ i } · g _{ i })) = d we say that f is computable by a ΣΠΣ(k; d;ρ) circuit. We obtain the following results.

1.
A deterministic blackbox identity testing algorithm for ΣΠΣ(k; d;ρ) circuits that runs in quasipolynomial time (for ρ=polylog(n+d)). In particular this gives the first blackbox quasipolynomial time PIT algorithm for depth3 circuits with k multiplication gates.

2.
A deterministic blackbox identity testing algorithm for readk ΣΠΣ circuits (depth3 circuits where each variable appears at most k times) that runs in time \(n^{2^{O(k^2 )} } \). In particular this gives a polynomial time algorithm for k=O(1).
Our results give the first subexponential blackbox PIT algorithm for circuits of depth higher than 2. Another way of stating our results is in terms of test sets for the underlying circuit model. A test set is a set of points such that if two circuits get the same values on every point of the set then they compute the same polynomial. Thus, our first result gives an explicit test set, of quasipolynomial size, for ΣΠΣ(k; d;ρ) circuits (when ρ=polylog(n+d)). Our second result gives an explicit polynomial size test set for readk depth3 circuits.
The proof technique involves a construction of a family of affine subspaces that have a rankpreserving property that is inspired by the construction of linear seeded extractors for affine sources of Gabizon and Raz [9], and a generalization of a theorem of [8] regarding the structure of identically zero depth3 circuits with bounded top fanin.
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Research supported by the Israel Science Foundation (grant number 439/06).
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Karnin, Z.S., Shpilka, A. Black box polynomial identity testing of generalized depth3 arithmetic circuits with bounded top fanin. Combinatorica 31, 333 (2011). https://doi.org/10.1007/s0049301125373
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Mathematics Subject Classification (2000)
 68Q17
 68Q25