Math 465 Lecture Notes

In this lecture, we give precise necessary conditions for the vanishing of re-sultants. We have already found sufficient conditions: when the polynomials have common roots. First, we review some results on polynomials with common factors: Proposition 13.1 Consider polynomials of degree m and n: f = amx m +...+ a0, g = bnx n +...+ b0 with nonzero leading coefficients am and bn. The following are equivalent: 1. f and g have a common nonconstant factor; 2. there exist nonzero polynomials A and B wth deg(A) ≤ n − 1 and deg(B) ≤ m − 1, so that Af +Bg = 0. proof Assume that f and g have a common factor h, so we may write f = Bh g = −Ah for suitable polynomials A,B with deg(B) < deg(f) = m and deg(A) < deg(g) = n. Substituting gives the relation Af +Bg = 0. Conversely, assume that Af +Bg = 0 as above, with B 6 = 0. If f, g have no common factor, we can writ