English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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 -time black-box identity test over rationals (Kayal &amp; Saraf, FOCS 2009) to one that takes d(O(k<sup>2</sup>))-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 &amp; 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.

From Sylvester-Gallai Configurations to Rank Bounds: Improved Black-Box Identity Test for Depth-3 Circuits