Robust Sunflower lemmas imply that any large enough monotone DNF of width $w$ contains a sunflower, i.e., a DNF equivalent to the conjunction of a common core with a DNF that is heavily biased towards $1$. While these lemmas are typically proved in the context of the uniform distribution or i.i.d. distributions, they also hold for $O(w^2)$-wise independent distributions by Bazzi's theorem (Bazzi (SICOMP 2009); Tal (CCC 2017)), which states that width-$w$ DNFs cannot distinguish between such distributions and the uniform distribution.

We prove a derandomization of robust sunflower lemmas for $O(w)$-wise independent distributions, which recovers the derandomization above with slightly weaker parameters. Notably, our result also holds for distributions whose marginals are "locally independent" and "approximately identically" distributed. It is not known whether such distributions fool general DNFs, which makes the derandomization via DNF-fooling techniques unavailable. Our proof of this derandomization is via the method of moments, which might be of independent interest.

We apply this result to obtain improved separations between non-monotone arithmetic circuits and the radical of monotone circuits. Specifically, we construct an \emph{explicit} family of multilinear polynomials $P_n$ that have small non-monotone arithmetic circuits such that any power of $P_n$ cannot be computed by monotone arithmetic circuits of \emph{sub-exponential} size. This improves on a recent result of Cavalar, Fabris, Mukhopadhyay, Srinivasan and Yehudayoff (STOC 2026), who proved a non-explicit separation that was quasipolynomial. Our construction uses Nisan-Wigderson designs with an additional linear-algebraic expansion property.