On the AC^0[⊕] complexity of Andreev's Problem
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Andreev's Problem states the following: Given an integer d and a subset of S ⊆F_q ×F_q, is there a polynomial y = p(x) of degree at most d such that for every a ∈F_q, (a,p(a)) ∈ S? We show an AC^0[⊕] lower bound for this problem. This problem appears to be similar to the list recovery problem for degree d-Reed-Solomon codes over F_q which states the following: Given subsets A_1,...,A_q of F_q, output all (if any) the Reed-Solomon codewords contained in A_1×...× A_q. For our purpose, we study this problem when A_1, ..., A_q are random subsets of a given size, which may be of independent interest.
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