TITLE: Is Multilinear Hilbert's tenth problem version undecidable?
QUESTION [16 upvotes]: 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?

REPLY [3 votes]: This is not an answer but a longish comment.
There is no Hasse Principle for multilinear polynomials. Take for example the polynomial equation $$(5x+2)(5y+3)=11.$$ Evidently the equation has no integer solutions. I'll show that it has $p$-adic solutions for every prime $p$.
First take $x=0$ in the displayed equation. We then require a $p$-adic integer $y$ such that $2(5y+3)=11$. The latter equation is equivalent to $10y=5$, which has a $p$-adic integer solution for all $p\ne2$. Next, take $y=0$ in the displayed equation. We then require a $p$-adic integer $x$ such that $3(5x+2)=11$. This is equivalent to $15x=5$, which has a p-adic integer solution for all $p\ne3$. Therefore the displayed equation has $p$-adic integer solutions for all $p$. Clearly there are real solutions. So the  Hasse Principle fails for multilinear polynomials.