Communication complexity of point-line incidences over the reals

2026-06-23

AI Summary

This work establishes a strong separation between randomized and deterministic communication complexity for a point-line incidence problem over the real numbers. Building upon Bloom et al.'s counterexample to the real-number sum–product conjecture and leveraging algebraic tools involving totally real number fields of high degree and small discriminant, the authors construct an explicit communication problem whose randomized communication complexity is constant, while its deterministic complexity remains linear even with access to an equality oracle. This yields the maximal possible gap—between constant and linear complexity—and improves the best-known lower bound for the sign-rank of constant-sign matrices under an equality oracle from logarithmic to optimal linear, thereby resolving a central open question posed by Harms and Zamaraev.

Abstract

We construct a point-line incidence problem over the reals whose randomized communication complexity is constant, but whose deterministic communication complexity is linear even when the players have access to an equality oracle. This is the strongest possible separation between these two measures, and it improves on an earlier $O(1)$-versus-$Ω(\sqrt{n})$ separation of Göös, Harms, and Riazanov. Because point-line incidence problems have constant sign rank, our construction also bears on a question of Harms and Zamaraev, who asked whether constant sign rank together with constant randomized communication complexity forces constant equality-oracle complexity. This was already refuted by Göös, Harms, Imbach, and Sokolov with a logarithmic lower bound; our example improves the separation to linear, which is optimal. The proof draws on a construction in the recent disproof of the sum-product conjecture over the reals by Bloom, Sawin, Schildkraut, and Zhelezov, using totally real number fields of large degree and small discriminant.

Problem

• communication complexity
• point-line incidences
• deterministic complexity
• randomized complexity
• equality oracle

Innovation

• communication complexity
• point-line incidences
• sign rank
• equality oracle
• totally real number fields