Dense resultant of composed polynomials: Mixed-mixed case

The main question of this paper is: What is the dense (Macaulay) resultant of composed polynomials? By a composed polynomial f ◦ (g1, . . . , gn), we mean the polynomial obtained from a polynomial f in the variables y1, . . . , yn by replacing yj by by some polynomial gj . Cheng, McKay and Wang and Jouanolou have provided answers for two particular subcases. The main contribution of this paper is to complete these works by providing a uniform answer for all subcases. In short, it states that the dense resultant is the product of certain powers of the dense resultants of the component polynomials and of some of their leading forms. It is expected that these results can be applied to compute dense resultants of composed polynomials with improved efficiency. We also state a lemma of independent interest about the dense resultant under vanishing of leading forms.