A Robust Version of Heged\H{u}s's Lemma, with Applications
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...
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| Published in: | TheoretiCS Vol. 2 |
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| Main Author: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
TheoretiCS Foundation e.V
01.03.2023
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| Subjects: | |
| ISSN: | 2751-4838, 2751-4838 |
| Online Access: | Get full text |
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| Summary: | 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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| ISSN: | 2751-4838 2751-4838 |
| DOI: | 10.46298/theoretics.23.5 |