On the location of the roots of certain types of polynomials

When we study the dependence of a variable on k other variables which vary independently, our problem may be very much simplified if we can consider all or some of these independent variables to coincide and thus study the dependence of our original dependent variable on one new variable or at least on a number of new independent variables less than k. The present writer has recently published a theorem (Theorem II, below) which enables us to make a reduction of this sort in the study of the relations between the roots of certain types of polynomials. The present paper aims to prove Theorem I (below), which is a much more general result of the same nature, and to indicate various applications of that theorem. The applications given are extremely simple and follow from Theorem I with practically no further machinery. t The most interesting application is Theorem VI. Our problem is, more explicitly, to study the geometric relationship of the roots of a polynomial