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XXXII. On the symmetric functions of the roots of certain systems of two equations

1857; Royal Society; Volume: 147; Linguagem: Inglês

10.1098/rstl.1857.0033

2053-9223

Arthur Cayley,

Functional Equations Stability Results

Suppose in general that ϕ = 0, Ψ = 0, See.denote a system of ( n — 1) equations between the n variables ( x, y, z ,..), where the functions ϕ , Ψ , &c. are quantics ( i. e . rational and integral homogeneous functions) of the variables. Any values ( x 1 , y 1 , z 1 , ..) satisfying the equations, are said to constitute a set of roots of the system; the roots of the same set are, it is clear, only determinate to a common factor Prés, i. e . only the ratios inter se and not the absolute magnitudes of the roots of a set are determinate. The number of sets, or the degree of the system, is equal to the product of the degrees of the component equations. Imagine a function of the roots which remains unaltered when any two sets ( x 1 , y 1 , z 1 , ..) and ( x 2 , y 2 , z 2 , ..) are interchanged (that is, when x 1 and x 2 , y 1 and y 2 , &c. are simultaneously interchanged), and which is besides homogeneous of the same degree as regards each entire set of roots, although not of necessity homogeneous as regards the different roots of the same set; thus, for example, if the sets are ( x 1 , y 1 ), ( x 2 , y 2 ), then the functions x 1 x 2 , x 1 y 2 + x 2 y 1 , y 1 y 2 are each of them of the form in question; but the first and third of these functions, although homogeneous of the first degree in regard to each entire set, are not homogeneous as regards the two variables of each set. A function of the above-mentioned form may, for shortness, be termed a symmetric function of the roots; such function (disregarding an arbitrary factor depending on the common factors which enter implicitly into the different sets of roots) will be a rational and integral function of the coefficients of the equations, i. e . any symmetric function of the roots may be considered as a rational and integral function of the coefficients. The general process for the investigation of such expression for a symmetric function of the roots is indicated in Professor Schläfli’s Memoir, “Ueber die Resultante eines Systemes mehrerer algebraischer Gleichungen,” Vienna Transactions, t. iv. (1852). The process is as follows:—Suppose that we know the resultant of a system of equations, one or more of them being linear; then if ϕ = 0 be the linear equation or one of the linear equations of the system, the resultant will be of the form ϕ 1 ϕ 1 .., where ϕ 1 ϕ 1 &c. are what the function ϕ becomes upon substituting therein the different sets ( x 1 , y 1 , z 1 ..), ( x 2 , y 2 , z 2 ..) of the remaining ( n — 1) equations Ψ = 0 X = 0, &c.; comparing such expression with the given value of the resultant, we have expressed in terms of the coefficients of the functions Ψ X , &c.., certain symmetric functions which may be called the fundamental symmetric functions of the roots of the system Ψ = 0 X = 0, &c., these are in fact the symmetric functions of the first degree in respect to each set of roots. By the aid of these fundamental symmetric functions, the other symmetric functions of the roots of the system Ψ = 0 X = 0, &c.. maybe expressed in terms of the coefficients, and then combining with these equations a non-linear equation Ф = 0, the resultant of the system Ф = 0, Ψ = 0, X = 0, &c. will be what the function Ф 1 Ф 2 becomes, upon substituting therein for the different symmetric functions of the roots of the system Ψ = 0, X = 0, &c. the expressions for these functions in terms of the coefficients. We thus pass from the resultant of a system ϕ = 0, Ψ = 0, X = 0, &c, to that of a system Φ = 0, Ψ = 0, X = 0, &c, in which the linear function ϕ is replaced by the non-linear function Φ. By what has preceded, the symmetric functions of the roots of a system of ( n — 1) equations depend on the resultant of the system obtained by combining the ( n — 1) equations with an arbitrary linear equation; and moreover, the resultant of any system of n equations depends ultimately upon the resultant of a system of the same number of equations, all except one being linear; but in this case the linear equations determine the ratios of the variables or (disregarding a common factor) the values of the variables, and by substituting these values in the remaining equation we have the resultant of the system. The process leads, therefore, to the expressions for the symmetric functions of the roots of any system of ( n — 1) equations, and also to the expression for the resultant of any system of n equations. Professor Schläfli discusses in the general case the problem of showing how the expressions for the fundamental symmetric functions lead to those of the other symmetric functions, but it is not necessary to speak further of this portion of his investigations. The object of the present Memoir is to apply the process to two particular cases, viz. I propose to obtain thereby the expressions for the simplest symmetric functions (after the fundamental ones) of the following systems of two ternary equations; that is, first, a linear equation and a quadric equation; and secondly, a linear equation and a cubic equation.