That's a strange thing to do. It's more usual to use a _subscript_ to indicate an index: a₃ vs a³
Oh, we economic theorists do that too. It's typically a double-indexed array of parameters, where both rows and columns can be meaningfully be treated as vectors. So a₃ is the vector of quantities of good 3 produced by all firms, while a³ is the vector of quantities of all goods produced by firm 3. Or in analysis of international or interregional trade, there's an index indicating which country exports which good to which importing country. Some people put the good index in the superscript and the two countries in the subscript, others the opposite. IIRC, mathematical physicist use both subscript and superscript in tensor notation, nuclear physicists use one for atomic number and the other for atomic weight (and thus would expect both subscript and superscript to be treated lexically as identifier components, not as expression components).
The point is not that polynomials are not the most common use of superscript notation -- I don't care one way or the other. It's that there are many uses, important to those fields, that aren't polynomials.