Resultant and Discriminant of Polynomials

This is a collection of classical results about resultants and discriminants for polynomials, compiled mainly for my own use. All results are well-known 19th century mathematics, but I have not investigated the history, and no references are given. 1. Resultant Definition 1.1. Let f(x) = anx n + · · ·+ a0 and g(x) = bmx + · · ·+ b0 be two polynomials of degrees (at most) n and m, respectively, with coefficients in an arbitrary field F . Their resultant R(f, g) = Rn,m(f, g) is the element of F given by the determinant of the (m + n) × (m + n) Sylvester matrix Syl(f, g) = Syln,m(f, g) given by  an an−1 an−2 . . . 0 0 0 0 an an−1 . . . 0 0 0 .. .. .. .. .. .. 0 0 0 . . . a1 a0 0 0 0 0 . . . a2 a1 a0 bm bm−1 bm−2 . . . 0 0 0 0 bm bm−1 . . . 0 0 0 .. .. .. .. .. .. 0 0 0 . . . b1 b0 0 0 0 0 . . . b2 b1 b0  (1.1) where the m first rows contain the coefficients an, an−1, . . . , a0 of f shifted 0, 1, . . . ,m− 1 steps and padded with zeros, and the n last rows contain the coefficients bm, bm−1, . . . , b0 of g shifted 0, 1, . . . , n−1 steps and padded with zeros. In other words, the entry at (i, j) equals an+i−j if 1 ≤ i ≤ m and bi−j if m + 1 ≤ i ≤ m + n, with ai = 0 if i > n or i < 0 and bi = 0 if i > m or i < 0. Date: September 22, 2007; revised August 16, 2010. 1