The approximate non-deterministic degree of a Boolean function $f$, denoted $\mathsf{ndeg}_ε(f)$ (written $\mathsf{N}_ε(f)$ for brevity), is the minimum degree of a real polynomial $p$ such that $0 \le |p(x)| \le ε$ whenever $f(x) = 0$, and $|p(x)| \ge 1$ whenever $f(x) = 1$. Unlike exact non-deterministic degree, which only requires the polynomial to be nonzero on $1$-inputs, this measure enforces a uniform gap: the polynomial must stay close to zero on all $0$-inputs and bounded away from zero on all $1$-inputs. The rational degree conjecture, open for over three decades, was recently resolved by Kothari, Kovacs-Deak, Wang, and Yang, who showed that for every total Boolean function $f$, \[ deg(f) \le \widetilde O\!\left(\operatorname{rdeg}(f)^3\right). \] In their paper, they explicitly propose a stronger conjecture: that approximate degree is polynomially bounded by $\mathsf{N}_ε(f)$ and $\mathsf{N}_ε(\overline{f})$ jointly, i.e., for every total Boolean function $f$ and every constant $0<ε<1$, \[ \widetilde{deg}(f) \le \operatorname{poly}(\mathsf N_ε(f), \mathsf N_ε(\overline f)). \] This conjecture, if true, would imply a polynomial version of the rational degree result and bring us closer to resolving de Wolf's longstanding non-deterministic degree conjecture. In this work, we make the first systematic progress on this problem, establishing the conjecture for several broad and natural function classes: monotone and unate functions, functions of bounded alternation number, symmetric functions, $k$-uniform hypergraph properties, and read-$k$ Disjunctive Normal Form (DNF) formulas.

翻译：布尔函数 $f$ 的近似非确定度记作 $\mathsf{ndeg}_ε(f)$（简写为 $\mathsf{N}_ε(f)$），定义为满足以下条件的实多项式 $p$ 的最小次数：当 $f(x) = 0$ 时，$0 \le |p(x)| \le ε$；当 $f(x) = 1$ 时，$|p(x)| \ge 1$。与精确非确定度仅要求多项式在 $1$-输入上非零不同，该度量强制了一个一致间隙：多项式在所有 $0$-输入上需接近零，而在所有 $1$-输入上需远离零。有理度猜想已存在三十余年未解，近期由 Kothari、Kovacs-Deak、Wang 和 Yang 解决，他们证明了对每个全布尔函数 $f$，\[ deg(f) \le \widetilde O\!\left(\operatorname{rdeg}(f)^3\right). \] 在该论文中，他们明确提出了一个更强的猜想：近似度可由 $\mathsf{N}_ε(f)$ 和 $\mathsf{N}_ε(\overline{f})$ 联合多项式界定，即对每个全布尔函数 $f$ 及每个常数 $0<ε<1$，\[ \widetilde{deg}(f) \le \operatorname{poly}(\mathsf N_ε(f), \mathsf N_ε(\overline f)). \] 若该猜想成立，将推出有理度结果的多项式版本，并使我们更接近解决 de Wolf 长期未决的非确定度猜想。本文首次对该问题进行了系统性推进，为以下几类广泛而自然的函数类建立了该猜想：单调函数与单边函数、有界交错次数函数、对称函数、$k$-一致超图性质，以及读-$k$ 析取范式（DNF）公式。