A multilinear polynomial $f\in\mathbb Z[x_1,\dots,x_t]$ has terms only of form $$b\prod_{i=1}^tx_i^{a_i}$$ where $a_i\in\{0,1\}$ and $b\in\mathbb Z$.

Denote the following problem Multilinear Hilbert Tenth Problem.

Given $f$ is there a computable bound $0<d<\infty$ such that

$$\{(x_1,\dots,x_t)\in\mathbb Z^n\wedge\|(x_1,\dots,x_n)\|_\infty\leq d\implies f(x_1,\dots,x_n)\neq0\}\iff\{\forall(x_1,\dots,x_n)\in\mathbb Z^n\mbox{ }f(x_1,\dots,x_n)\neq0\}$$ holds?

$\underline{General\mbox{ }Conjecture}$: There is such computable $d$.

Is the $\underline{General\mbox{ }Conjecture}$ for Multilinear Hilbert Tenth Problem true?

A multilinear polynomial $f\in\mathbb Z[x_1,\dots,x_t]$ has terms only of form $$b\prod_{i=1}^tx_i^{a_i}$$ where $a_i\in\{0,1\}$ and $b\in\mathbb Z$.

Is there no general purpose algorithm for finding integer roots of this class of polynomials?