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TR19-096 | 23rd July 2019 18:22
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#### On the $\text{AC}^0[\oplus]$ complexity of Andreev's Problem

**Abstract:**
Andreev's Problem asks the following: Given an integer $d$ and a subset of $S \subseteq \mathbb{F}_q \times \mathbb{F}_q$, is there a polynomial $y = p(x)$ of degree at most $d$ such that for every $a \in \mathbb{F}_q$, $(a,p(a)) \in S$? We show an $\text{AC}^0[\oplus]$ lower bound for this problem.

This problem appears to be similar to the list recovery problem for degree $d$-Reed-Solomon codes over $\mathbb{F}_q$ which asks the following: Given subsets $A_1,\ldots,A_q$ of $\mathbb{F}_q$, output all (if any) Reed-Solomon codewords contained in $A_1\times \cdots \times A_q$. For our purpose, we study this problem when $A_1, \ldots, A_q$ are random subsets of a given size, which may be of independent interest.