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.

Abstract A recent generalization of the Erdős Unit Distance Problem, proposed by Palsson, Senger, and Sheffer, asks for the maximum number of unit distance paths with a given number of vertices in the plane and in 3-space. Studying a variant of this question, we prove sharp bounds on the number of unit distance paths and cycles on the sphere of radius $$1/{\sqrt{2}}$$ 1 / 2 . We also consider a similar problem about 3-regular unit distance graphs in $$\mathbb {R}^3$$ R 3 .