Generalize the Effective Hypercube Nullstellensatz to m polynomials
Establish, for every integer m ≥ 2 and polynomials g1,…,gm ∈ ℝ[X1,…,Xn] that have no common zeros on the hypercube {0,1}^n and satisfy g1(x)⋯gm(x) = 0 for all x ∈ {0,1}^n, the existence of polynomials h1,…,hm ∈ ℝ[X1,…,Xn] such that h1(x)g1(x)+⋯+hm(x)g_m(x) = 1 for all x ∈ {0,1}^n and max_i deg( overline{h_i g_i} ) ≤ poly( deg(g1),…,deg(gm) ), where overline{•} denotes multilinearization using the relations X1^2 = X1,…,Xn^2 = Xn.
References
In view of existing Nullstellensatz results, we conjecture that a natural generalization of \cref{null_cube} to any number of polynomials holds. For all integers $m\geq 2$, the following holds. Let $g_1, \dots, g_m\in \mathbb{R}[X_1,\dots,X_n]$. Suppose $g_1, \dots, g_m$ do not share any common zeros on the hypercube ${0,1}n$. Further suppose $g_1(x) \cdots g_m(x) = 0$ for all $x\in {0,1}n$. Then there exist $h_1,\dots, h_m\in \mathbb{R}[X_1,\dots,X_n]$ such that \begin{equation} h_1(x) g_1(x)+\cdots + h_m(x) g_m(x)=1 \quad \text{for all $x\in {0,1}n$}, \end{equation} and \begin{equation} \max_{i\in [m]}(\deg(\overline{h_i g_i}))\leq \textnormal{poly}(\deg(g_1),\dots,\deg(g_m)). \end{equation}