Zero of a multivariate cubic equation
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.