Influence bound in terms of nondeterministic degree

Prove that for every Boolean function f: {0,1}^n → {0,1}, the total influence Inf[f] satisfies Inf[f] ≤ O(√n · ndeg(f)), where ndeg(f) denotes the nondeterministic degree of f.

Background

The authors propose a weaker variant of the Gotsman–Linial conjecture by replacing sign degree with nondeterministic degree, noting the general inequality sdeg(f)/2 ≤ ndeg(f).

They motivate this conjecture because, together with algebraic arguments, it would imply lower bounds on rational degree for functions depending on many variables.

References

We make the following conjecture, which is weaker\footnote{Though the conjecture, if true, would still be tight, as witnessed by the function in \cref{rem:sdeg_lower}.} since $\sdeg(f)/2 \leq \ndeg(f)$. \begin{conjecture}\label{conj:gotsman_linial} For every $f\colon {0,1}n \to {0,1}$, $\Inf[f]\leq O(\sqrt{n} \ndeg(f))$. \end{conjecture}

— Rational degree is polynomially related to degree  (2601.08727 - Kovacs-Deak et al., 13 Jan 2026) in Section 4.2