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?

Given $f$ is there a 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?

twomultilinear polynomials is undecidable. First, by Richard Stanley’s comment, we can reduce integer solvability of a general polynomial system to a single “multiquadratic” polynomial, and by applying his reduction again, we reduce it to solvability of a system of the form $\{f(x_1,\dots,x_n,y_1,\dots,y_n)=0,x_1=y_1,\dots,x_n=y_n\}$, where $f$ is multilinear. Moreover, we can combine $\{x_1-y_1,\dots,x_n-y_n\}$ to a single multilinear polynomial: using the fact that the only integer solution of $3xy+x+y=0$ is $(0,0)$, we can ... $\endgroup$ – Emil Jeřábek Aug 12 at 16:44