Subresultants Under Composition

It is a well known fact that the resultants are invariant under translation. We extend this fact to arbitrary composition (where a translation is a particular composition with a linear monic polynomial), and to arbitrary subresultants (where the resultant is the 0-th subresultant). 1 Introduction It is a well known fact that the resultants 1 are invariant under translation. For example, let A(x) = x 2 + 2x + 1 and B(x) = x \Gamma 1. Then their resultant is 4. Let us translate the two polynomials by one: A (x) = A(x + 1) = x 2 + 4x + 4 and B (x) = B(x + 1) = x. One can verify that their resultant remains to be 4. In general, let A(x) and B(x) be two polynomials and R be their resultant. Let A (x) = A(x + c) and B (x) = B(x + c) and let R be their resultant. Then R = R . Note that a translation (by c) is a particular composition, that is, A = A ffi C where ffi is a composition and C = x + c. In this notation, the invariance under translation can be written a..