Dense resultant of composed polynomials: mixed-mixed case
J. Symbolic Comput. 36 (2003), no. 6, 825-834
Abstract
The main question of this paper is: What is
the dense (Macaulay) resultant of composed polynomials?
By a composed polynomial f o (g_1, ..., g_n),
we mean the polynomial obtained from
a polynomial f in the variables y_1, ..., y_n
by replacing y_j by by some polynomial g_j.
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.