The relative distribution of the real roots of a system of polynomials

Abstract

The problem about to be discussed may be looked upon as a generalization of the classical problem solved by Sturm in 1829 with regard to the real roots of a polynomial.t In Sturm's theorem it is shown how the number of distinct roots of a single polynomial which fall within a given real interval mnay be determined through a process rational in the coefficients. In the present paper we shall study a system of two or more polynomials in a single variable, and our aim will be to develop a rational process by which the order of succession of the roots of the several polynomials in a given real interval may be discovered. Let us confine ourselves, at least for the present, to the case where the endvalues of the interval are not roots and where the polynomials have only simple roots in the interval. It is clear that in such a case Sturm's theorem determines the only relations of the real roots of a single polynomial to the interval that remain invariant under continuous transformation of the real number system into itself. A similar remark applies to the theory about to be developed with respect to the real root system of several polynomials; so that we are undertaking the study of a problem which, from the point of view of a onedimensional analysis situs, may be said to be the fundamental problem of the system under consideration. The greater part of the paper will be devoted to the case of two polynomials. If the roots of the first are denoted generally by a and those of the second by f, and if we write down the roots within the interval considered in increasing numerical order (as aaa af3f3c43oa), the O's effect a certain partition of the a's (in the present case into groups of 3, 0, 1, O, O, 1, O,). The solution of the problem will consist in the determination of the numbers of a's in the successive groups.