A weak regularity lemma for polynomials

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 to quantitatively study the curves contained in the image of a polynomial map, which is beyond the reach of standard methods. Applications include strong bounds for a problem of Karam on generalized rank, as well as a new method to obtain upper bounds for fan-in parameters in arithmetic circuits. For example, we show that if the image of a polynomial map $\mathbf{P} \colon \mathbb{F}^n \to \mathbb{F}^m$ of degree $d$ does not contain a line, then $\mathbf{P}$ can be computed by a depth-$4$ homogeneous formula with bottom fan-in at most $d/2$ and top fan-in at most $(2m)^{C(d)}$ (with $C(d)=2^{(1+o(1))d}$). Such a result was previously known only with at least a tower-type bound on the top fan-in. One implication of our work is a certain ``barrier'' to arithmetic circuit lower bounds, in terms of the smallest degree of a polynomial curve contained in the image of the given polynomial map.