Monomials in arithmetic circuits: Complete problems in the counting   hierarchy
  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. We also study these questions for circuits computing multilinear polynomials. 
Herv\'e Fournier and Guillaume Malod and Stefan Mengel
cs.CC