AC^0[p] Lower Bounds Against MCSP via the Coin Problem

Golovnev, Alexander; Ilango, Rahul; Impagliazzo, Russell; Kabanets, Valentine; Kolokolova, Antonina; Tal, Avishay

doi:10.4230/lipics.icalp.2019.66

Cited by 7 publications

(1 citation statement)

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“…This led to the rst class of explicit functions for which we have tight (up to polynomial factors) AC 0 [⊕] lower bounds. These bounds were in turn used by Golovnev, Ilango, Impagliazzo, Kabanets, Kolokolova and Tal [20] to resolve a long-standing open problem regarding the complexity of MCSP in the AC 0 [⊕] model, and by Potukuchi [36] to prove lower bounds for Andreev's problem.…”

Section: Applicationsmentioning

confidence: 99%

A Robust Version of Heged\H{u}s's Lemma, with Applications

Srinivasan¹

2023

TheoretiCS

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Heged\H{u}s's lemma is the following combinatorial statement regarding polynomials over finite fields. Over a field $\mathbb{F}$ of characteristic $p > 0$ and for $q$ a power of $p$, the lemma says that any multilinear polynomial $P\in \mathbb{F}[x_1,\ldots,x_n]$ of degree less than $q$ that vanishes at all points in $\{0,1\}^n$ of some fixed Hamming weight $k\in [q,n-q]$ must also vanish at all points in $\{0,1\}^n$ of weight $k + q$. This lemma was used by Heged\H{u}s (2009) to give a solution to \emph{Galvin's problem}, an extremal problem about set systems; by Alon, Kumar and Volk (2018) to improve the best-known multilinear circuit lower bounds; and by Hrube\v{s}, Ramamoorthy, Rao and Yehudayoff (2019) to prove optimal lower bounds against depth-$2$ threshold circuits for computing some symmetric functions. In this paper, we formulate a robust version of Heged\H{u}s's lemma. Informally, this version says that if a polynomial of degree $o(q)$ vanishes at most points of weight $k$, then it vanishes at many points of weight $k+q$. We prove this lemma and give three different applications.

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Section: Applicationsmentioning

confidence: 99%