#P- and $\oplus$P- completeness of counting roots of a sparse polynomial

TL;DR

This paper proves that counting roots of sparse polynomials over finite fields is both #P- and ⊕P-complete, establishing its computational difficulty and simplifying previous results in the area.

Contribution

It improves and simplifies existing results by demonstrating the #P- and ⊕P-completeness of counting roots of sparse polynomials over finite fields.

Findings

Counting roots of sparse polynomials over ^n is P- and P-complete.

The proof uses deterministic reductions, strengthening previous complexity results.

The work simplifies earlier proofs and clarifies the computational complexity of this problem.

Abstract

We improve and simplify the result of the part 4 of "Counting curves and their projections" (Joachim von zur Gathen, Marek Karpinski, Igor Shparlinski) by showing that counting roots of a sparse polynomial over $\mathbb{F}_{2^n}$ is #P- and $\oplus$P-complete under deterministic reductions.