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2010 IEEE 51st Annual Symposium on Foundations of Computer Science (2010)
Las Vegas, Nevada USA
Oct. 23, 2010 to Oct. 26, 2010
ISSN: 0272-5428
ISBN: 978-0-7695-4244-7
pp: 21-29
We study the problem of identity testing for depth-3 circuits of top fanin k and degree d. We give a new structure theorem for such identities. A direct application of our theorem improves the known deterministic d^{k^k}-time black-box identity test over rationals (Kayal & Saraf, FOCS 2009) to one that takes d^{k^2}-time. Our structure theorem essentially says that the number of independent variables in a real depth-3 identity is very small. This theorem affirmatively settles the strong rank conjecture posed by Dvir & Shpilka (STOC 2005). We devise a powerful algebraic framework and develop tools to study depth-3 identities. We use these tools to show that any depth-3 identity contains a much smaller nucleus identity that contains most of the "complexity" of the main identity. The special properties of this nucleus allow us to get almost optimal rank bounds for depth-3 identities.
depth-3 circuit, identities, Sylvester-Gallai, incidence configuration, Chinese remaindering, ideal theory

N. Saxena and C. Seshadhri, "From Sylvester-Gallai Configurations to Rank Bounds: Improved Black-Box Identity Test for Depth-3 Circuits," 2010 IEEE 51st Annual Symposium on Foundations of Computer Science(FOCS), Las Vegas, Nevada USA, 2010, pp. 21-29.
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