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.

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The catalytic Turing machine is a model of computation defined by Buhrman, Cleve,
Kouck, Loff, and Speelman (STOC 2014). Compared to the classical space-bounded Turing
machine, this model has an extra space which is filled with arbitrary content in addition
to the clean space. In such a model we study ... more >>>

A $k$-uniform hypergraph is said to be $r$-rainbow colorable if there is an $r$-coloring of its vertices such that every hyperedge intersects all $r$ color classes. Given as input such a hypergraph, finding a $r$-rainbow coloring of it is NP-hard for all $k \ge 3$ and $r \ge 2$. ... more >>>