Quartic separation between decision-tree complexity and rational degree

Construct a family of Boolean functions f for which the decision-tree complexity satisfies D(f) ≥ Ω(rdeg(f)^4), thereby establishing a quartic separation between decision-tree complexity and rational degree.

Background

The paper proves an upper bound D(f) ≤ 4·sdeg(f)2·rdeg(f)2 ≤ 16·rdeg(f)4, polynomially relating rational degree to decision-tree complexity.

They conjecture that this relation is tight in the exponent, seeking explicit functions that achieve D(f) on the order of rdeg(f)4; currently, only quadratic separations are known (e.g., balanced AND–OR trees and pointer functions).

References

We conjecture that \cref{cor:final} is optimal with respect to $D$ versus $\rdeg$. \begin{conjecture}\label{conjecture:quartic_separation} There exists a family of Boolean functions $f$ such that $D(f) \geq \Omega(\rdeg(f)4)$. \end{conjecture}

— Rational degree is polynomially related to degree  (2601.08727 - Kovacs-Deak et al., 13 Jan 2026) in Section 4.4