In this paper, we show that deciding whether a sparse polynomial does not divide another sparse polynomial exactly over finite fields is NP-hard under BPP many-one reductions. Equivalently, the sparse polynomial divisibility test over finite fields is CoNP-hard. This resolves the long-standing open problem concerning the computational complexity of the divisibility test for sparse polynomials in the setting of finite fields.

翻译：本文证明，在BPP多一归约下，判定一个稀疏多项式是否在有限域上不能整除另一个稀疏多项式是NP难的。等价地，有限域上的稀疏多项式整除性测试是CoNP难的。这一结果解决了关于有限域环境中稀疏多项式整除性测试计算复杂性的一个长期未决的公开问题。