Louis Weisner

Roots of certain classes of polynomials

Presented to the Society, September 5, 1941 ; received by the editors June 6, 1941. 1 Ch. Hermite, Nouvelles Annales de Mathématiques, vol. 5 (1866), p . 479; Pólya and Szegö, Aufgaben und Lehrsatze aus der Analysis II , p. 47, Problem 62. 2 See G. Pólya, Über trigonometrische Integrale mit nur reellen Nullstellen, Journal für die reine und angewandte Mathematik, vol. 158 (1927), pp. 6-18. 3 Replacing z by iz it follows from the above theorem of Hermite that if the roots of <f>(z) and F(z) lie on the axis of pure imaginaries, so do the roots of <f>(D)F(z).

Power series the roots of whose partial sums lie in a sector

If the roots of the partial sums of a power series f(z) =^anz n lie in a sector with vertex at the origin and aperture a < 2x, the power series cannot have a positive finite radius of convergence. But if f{z) is an entire function, the roots of its partial sums may lie in such a sector. The question arises : what restrictions are imposed on f(z) by the requirement that a be sufficiently small, say a<7r? According to a theorem of Pólya the order of f(z) must be not greater than 1 if the radius of convergence of the power series is positive. Without this assumption the investigation which follows shows that if a<Tr1f{z) is an entire function of order 0. This result was obtained by Pólya for the case in which a = 0.

Group of a set of simultaneous algebraic equations

(2) Fp(zp) = 0, (P = l, •••, w), each of which is of degree n and involves only one of the unknowns. None of these equations vanishes identically, since equations (1) are independent. We assume that none of equations (2) has a multiple root. A rational domain B which includes the coefficients of (1) includes the coefficients of (2). It will be shown in this paper that the groups of equations (2) relative to E are identical, except for the symbols which they affect.