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Exact complexity of single-variable equations AX = B and P(X) = B over FDDS

Determine the exact computational complexity of solving single-variable equations of the forms AX = B and P(X) = B over the semiring of finite discrete-time dynamical systems (FDDS) without additional restrictions on coefficients, specifically deciding whether these problems are NP-hard or admit polynomial-time algorithms.

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Background

Beyond the specialized algorithms introduced for pseudo-injective polynomials and pseudo-cancelable coefficients, the authors highlight a broader unresolved question: the overall complexity of single-variable equations AX = B and P(X) = B in the FDDS semiring when no further structural constraints (e.g., pseudo-cancelability) are imposed.

This open problem seeks to settle whether these problems are NP-hard or solvable in polynomial time, providing a definitive complexity classification for the core equation-solving task over FDDS.

References

The exact complexity of solving equations of the form~$AX=B$ or, more generally, $P(X)=B$ of a single variable, but without other restrictions, is still open. Is it an~$\NP$-hard problem or does it admit an efficient algorithm?

— Solving "pseudo-injective" polynomial equations over finite dynamical systems (2504.06986 - Porreca et al., 9 Apr 2025) in Section “Conclusions”