Zero of a multivariate cubic equation Given a multivariate cubic equation $f(\vec{x})$ over reals where all coefficients of $f$ are integers, and a hyper-rectangle $B=\bigwedge_i x_i\in[a_i, b_i]$ where $a_i, b_i$ are constant integers, the problem is to check whether $f$ has a zero in $B$. 

What’s the complexity of this problem?

Obviously, it is in PSPACE; more precisely, it is a special case of the existential theory of the reals, hence it is in the class $\exists\mathbb R$. A related problem (denoted here as 4FEAS), viz. to check whether a quartic equation with integer coefficients (even without constraints in $B$) has a real zero is $\exists\mathbb R$-complete. Likewise, checking whether a set of quadratic (or cubic) equations has a common zero is $\exists\mathbb R$-complete. On the other hand, if one drops the constraint of $B$, a zero exists trivially if the cubic equation is nondegenerate. Hence, an obvious question is, can we reduce 4FEAS to the current problem?

A variant of 4FEAS with real coefficients is $\mathrm{NP}_\mathbb R$-complete in the Blum–Shub–Smale model, but this question is about the Boolean Turing model, not BSS.

A related question is here: Checking whether two quadratic equations have a common zero
That problem is strongly NP-hard by reduction from 3-SAT:

Each ai is 0, each bi is 1, each negative literal is represented by its propositional variable,

each positive literal is represented by 1 minus that, each clause is represented by the product

of the representations of its literals, and f is the sum of the representations of the clauses.

I don't have any evidence regarding possible ∃ℝ-completeness or membership in NP.