2026-07-23T16:50:02Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:27056 2026-07-23T11:36:19Z ddc:004 open_access

A Weak Regularity Lemma for Polynomials Moshkovitz, Guy Woodruff, Dora weak regularity lemma finite-field polynomials polynomial maps structure-versus-randomness arithmetic circuits A regularity lemma for polynomials provides a decomposition in terms of a bounded number of approximately independent polynomials. Such regularity lemmas play an important role in numerous results, yet suffer from the familiar shortcoming of having tower-type bounds or worse. In this paper we design a new, weaker regularity lemma with strong bounds. The new regularity lemma in particular provides means for quantitatively studying the curves contained in the image of a polynomial map, which is beyond the reach of standard methods. The weak regularity lemma turns out to be powerful enough to yield results on arithmetic circuits and polynomial ranks that may be of independent interest: - A general upper bound on the arithmetic circuit size of low-degree polynomials based solely on their image. - An upper bound on the top fan-in of depth-4 arithmetic formulas under similar conditions. - A quantitative bound for the Green-Tao notion of rank for polynomials, significantly improving on a result of Karam. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Guy Moshkovitz and Dora Woodruff 2026 Is Part Of LIPIcs, Volume 383, 41st Computational Complexity Conference (CCC 2026) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.CCC.2026.14 urn:nbn:de:0030-drops-270569 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2026.14 eng https://creativecommons.org/licenses/by/4.0/legalcode