Factorization of polynomials with analytic coefficients
  We study monic univariate polynomials whose coefficients are analytic functions of a real variable and whose roots lie in a specified analytic curve. These include characteristic polynomials of unitary and hermitian matrices whose entries are analytic functions. We use a result of Newton to prove that every polynomial in such a class is a product of degree one polynomials in the class. 
Wayne Lawton
math.AG