Sparse Polynomial Divisibility Test over Finite Field is CoNP-hard

2026-06-10

AI Summary

This work investigates the computational complexity of determining whether one sparse polynomial divides another over a finite field. By integrating algebraic structural analysis with probabilistic reduction techniques, the study establishes—via a BPP many-one reduction—that this decision problem is CoNP-hard. This result resolves a long-standing open question in the field and rigorously characterizes the inherent computational difficulty of testing divisibility among sparse polynomials over finite fields, thereby providing a crucial theoretical foundation for its complexity classification.

Abstract

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.

Problem

• Sparse Polynomial
• Divisibility Test
• Finite Field
• CoNP-hard
• Computational Complexity

Innovation

• sparse polynomial
• divisibility test
• finite field
• CoNP-hard
• computational complexity