Exploring univariate mixed polynomials

We consider mixed polynomials P (z, ¯ z) of the single complex variable z with complex (or real) coefficients, of degree n in z and m in ¯ z. This data is equivalent to a pair of real bivariate polynomials f (x, y) and g(x, y) obtained by separating real and imaginary parts of P. However specifying the degrees, here we focus on the case where m is small, allows to investigate interesting roots structures and roots counting; intermediate between complex and real algebra. Mixed polynomials naturally appear in the study of complex polynomial matrices and complex moment problems, harmonic maps, and in recent papers dealing with Milnor fibrations.