On Non-Deterministic Representation of Boolean Functions

We study non-deterministic representation of Boolean functions by polynomials, introduced by (de Wolf, 2000). A polynomial $p(x)$ is said to be a non-deterministic polynomial for a Boolean function f if for all a∈\0,1\ⁿ, p(a)≠ 0 iff $f(a)=1$. The non-deterministic degree of f, denoted ndeg(f), is the minimal degree of a non-deterministic polynomial for f. In this work, we show the following main results: - Almost all Boolean functions on n variables have non-deterministic degree at most n/2 + O(√nlog n). This confirms a conjecture of (Iyer et al., 2023). Together with the lower bound of (Iyer et al., 2023) it establishes a tight bound on the rational degree of a random Boolean function up to lower order terms. - We disprove a conjecture of (Kothari et al., 2026) about effective hypercube nullstellensatz for more than two polynomials. On the positive side, we also identify a constraint under which an effective hypercube nullstellensatz holds for many polynomials.