# On the $\text{AC}^0[\oplus]$ complexity of Andreev's Problem

## TL;DR

This paper establishes an b0[igoplus] lower bound for Andreev's Problem, linking it to list recovery in Reed-Solomon codes over finite fields, with implications for complexity theory.

## Contribution

It provides the first b0[igoplus] complexity lower bound for Andreev's Problem and explores its relation to Reed-Solomon list recovery with random subsets.

## Key findings

- Proves b0[igoplus] lower bound for Andreev's Problem
- Connects Andreev's Problem to Reed-Solomon list recovery
- Analyzes list recovery with random subsets of finite fields

## Abstract

Andreev's Problem states 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 states the following: Given subsets $A_1,\ldots,A_q$ of $\mathbb{F}_q$, output all (if any) the 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.