Monomials in arithmetic circuits: Complete problems in the counting hierarchy

Hervé Fournier, Guillaume Malod & Stefan Mengel
We consider the complexity of two questions on polynomials given by arithmetic circuits: testing whether a monomial is present and counting the number of monomials. We show that these problems are complete for subclasses of the counting hierarchy which had few or no known natural complete problems before. We also study these questions for circuits computing multilinear polynomials.