On the Discriminant Criterion and a Generalization
M. H. Eggar · American Mathematical Monthly · 1996
or, equivalently, the square of their Vandermonde determinant. The discriminant can be written as a polynomial A(aO, . . ., a,l_l) in aO, . . ., a,l_l, since the defining product is a symmetric polynomial of the roots (xi and the main result on symmetric polynomials gives an algorithmic procedure for writing any symmetric polynomial in the exi as a polynomial in the elementary symmetric polynomials (l)ia,l_i of the ai. Clearly A vanishes if and only if at least two roots coincide. Equivalently, from the Vandermonde point of view, the non-vanishing or otherwise of the determinant is the criterion for whether or not the n simultaneous linear equations in aO,. . .,a,l_l, cti'l + Era,l_rcYi'l r = 0 for 1 < i < n, have a unique solution. For ai not all distinct there is duplication amongst these equations and so the aj are not uniquely determined, but for cYi all distinct the polynomial P is fli(z-CYi) and so the aj are uniquely determined. We note that if P(z) is replaced by the general polynomial anZn + aRl_lzZl l + *@ +aO Of degree not exceeding n, then the condition for a root of multiplicity at least 2 splits into cases according to the precise degree. The condition is /v(aO/a,l, . . ., a,l_l /a,l) = O if a,l + 0, A(aO/a,l_l, . . ., a,l_2/a,l_l ) = O if a,l = 0, aRl_l + 0, etc. The whole condition cannot be described by the vanishing of a single expression in aO, . . ., a,l. Returning to the polynomial P(z) of (1) one can with more effort count the number, N say, of pairs (i, j) such that i < j and ai = aj. One obsenes that ZN iS the highest power of z that divides the polynomial p(z) of degree (2) with zeros (a1 a2)2,(a1 ex3)2,...,(a,l_l ex,l)2. Each coefficient in p(z) can be expressed in terms of aO, . § ., a,l_l, since the coefficients are with alternating signs