We introduce a generalisation of weighted automata over a field, called polynomial automata, and we analyse the complexity of the Zeroness Problem in this model, that is, whether a given automaton outputs zero on all words. While this problem is non-primitive recursive in general, we highlight a subclass of polynomial automata for which the Zeroness Problem is primitive recursive. Refining further, we identify a subclass of affine VAS for which coverability is in 2EXPSPACE. We also use polynomial automata to obtain new proofs that equivalence of streaming string transducers is decidable, and that equivalence of copyless streaming string transducers is in PSPACE.