On the location of the roots of certain types of polynomials
Joseph L. Walsh · Transactions of the American Mathematical Society · 1922
tThe coefficients of f{z) need not be homogeneous in each of these sets of variables, but each coefficient must be a linear combination of the elementary symmetric functions of each of these sets with coefficients linear combinations of the elementary symmetric functions of the other sets.These linear combinations may, moreover, contain constant terms.% This is what actually occurs in the situation of Theorem II if we choose P inside C. 1922]* Walsh, these Transactions, vol.22 (1921), p. 102; Lemma I.Theorem II is closely connected with another more simple corollary of Theorem I, namely, that if k equaljparticles lie in a circle their center of gravity also lies in that circle.* For the corresponding fact for Theorem II, compare Walsh, these Transactions, vol.24 (1922), pp.31-69; Theorem III.t The terminology that z divides the segment (zi, z2) in the ratio (mL : m3) is usual when * The reader will easily prove from Rolle's Theorem that no point z(,l> distinct from zi and zi can be a multiple root of 0<w (z).