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In this note we characterize when non-classical polynomials are necessary in the inverse theorem for the Gowers $U^k$-norm. We give a brief deduction of the fact that a bounded function on $mathbb F_p^n$ with large $U^k$-norm must correlate with a classical polynomial when $kleq p+1$. To the best of our knowledge, this result is new for $k=p+1$ (when $p>2$). We then prove that non-classical polynomials are necessary in the inverse theorem for the Gowers $U^k$-norm over $mathbb F_p^n$ for all $kgeq p+2$, completely characterizing when classical polynomials suffice.
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Motivated by the concepts of the inverse Kazhdan-Lusztig polynomial and the equivariant Kazhdan-Lusztig polynomial, Proudfoot defined the equivariant inverse Kazhdan-Lusztig polynomial for a matroid. In this paper, we show that the equivariant inverse Kazhdan-Lusztig polynomial of a matroid is very useful for determining its equivariant Kazhdan-Lusztig polynomials, and we determine the equivariant inverse Kazhdan-Lusztig polynomials for Boolean matroids and uniform matroids. As an application, we give a new proof of Gedeon, Proudfoot and Youngs formula for the equivariant Kazhdan-Lusztig polynomials of uniform matroids. Inspired by Lee, Nasr and Radcliffes combinatorial interpretation for the ordinary Kazhdan-Lusztig polynomials of uniform matroids, we further present a new formula for the corresponding equivariant Kazhdan-Lusztig polynomials.
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