and by elimination we obtain the resultant
This is Euler’s method. Sylvester’s leads to the same expression, but in a simpler manner.
He forms n equations from ƒ by separate multiplication by xn–1, xn–2,...x, 1, in succession, and similarly treats φ with m multipliers xm –1, xm –2,...x, 1. From these m + n equations he eliminates the m + n powers xm +n –1, xm +n–2, x,.. 1, treating them as independent unknowns. Taking the same example as before the process leads to the system of equations
| a0x4+ | a1x3+ | a2x2+ | a3x | ＝0, | |
| a0x3+ | a1x2+ | a2x+ | a3 | ＝0, | |
| b0x4+ | b1x3+ | b2x2 | ＝0, | ||
| b0x3+ | b1x2+ | b2x | ＝0, | ||
| b0x2+ | b1x+ | b2 | ＝0, |
whence by elimination the resultant
| a0 | a1 | a2 | a3 | 0 |
| 0 | a0 | a1 | a2 | a3 |
| b0 | b1 | b2 | 0 | 0 |
| 0 | b0 | b1 | b2 | 0 |
| 0 | 0 | b9 | b1 | b2 |
which reads by columns as the former determinant reads by rows, and is therefore identical with the former. E. Bézout’s method gives the resultant in the form of a determinant of order m or n, according as m is ≷ n. As modified by Cayley it takes a very simple form. He forms the equation
ƒ(x)φ(x ′) − ƒ(x ′)φ(x)＝0,
which can be satisfied when ƒ and φ possess a common factor. He first divides by the factor x − x ′, reducing it to the degree m − 1 in both x and x ′ where m > n; he then forms m equations by equating to zero the coefficients of the various powers of x ′; these equations involve the m powers x0, x, x2,... xm−1 of x, and regarding these as the unknowns of a system of linear equations the resultant is reached in the form of a determinant of order m. Ex. gr. Put
(a0x3+a1x2+a2x +a3) (b0x ′2+b1x ′+b2) − (a0x ′3+a1x ′2+a2x ′ +a3) (b0x2+b1x+b2)＝0;
after division by x − x ′ the three equations are formed
| a0b0x2+a0b1x+a0b2 | ＝0, |
| a0b1x2+(a0b2+a1b1−a0b2)x+a1b2−a3b0 | ＝0, |
| a0b2x2+(a1b2−a3b0)x+a2b2−a3b1 | ＝0 |
and thence the resultant
| a0b0 | a0b1 | a0b2 |
| a0b1 | a0b2+a1b1−a0b2 | a1b2−a3b0 |
| a0b2 | a1b2−a3b0 | a2b2−a3b1 |
which is a symmetrical determinant.
Case of Three Variables.—In the next place we consider the resultants of three homogeneous polynomials in three variables. We can prove that if the three equations be satisfied by a system of values of the variable, the same system will also satisfy the Jacobian or functional determinant. For if u, v, w be the polynomials of orders m, n, p respectively, the Jacobian is (u1 v2 w3), and by Euler’s theorem of homogeneous functions
xu1 + yu2 + zu3＝mu
xv1 + yv2 + zv3＝nv
xw1 + yw2 + zw3＝pw;
denoting now the reciprocal determinant by (U1 V2 W3) we obtain Jx＝muU1 + nvV1 + pwW1; Jy=..., Jz=..., and it appears that the vanishing of u, v, and w implies the vanishing of J. Further, if m＝n＝ p, we obtain by differentiation
J + x∂J∂x =m (u∂U1∂x. + v∂V1∂x + u∂W1∂x + u1U1 v1V1 w1W1).
or
x∂J∂x =m – 1)J + m (u∂U1∂x. + v∂V1∂x + u∂W1∂x).
Hence the system of values also causes ∂J∂x to vanish in this case; and by symmetry ∂J∂y and ∂J∂z also vanish.
The proof being of general application we may state that a system of values which causes the vanishing of k polynomials in k variables causes also the vanishing of the Jacobian, and in particular, when the forms are of the same degree, the vanishing also of the differential coefficients of the Jacobian in regard to each of the variables.
There is no difficulty in expressing the resultant by the method of symmetric functions. Taking two of the equations
axm + (by + cz) xm–1 +... =0,
a′xn + (b′y + c′z) xn–1 +... =0,
we find that, eliminating x, the resultant is a homogeneous function of y and z of degree mn; equating this to zero and solving for the ratio of y to z we obtain mn solutions; if values of y and z, given by any solution, be substituted in each of the two equations, they will possess a common factor which gives a value of x which, combined with the chosen values of y and z, yields a system of values which satisfies both equations. Hence in all there are mn such systems. If, therefore, we have a third equation, and we substitute each system of values in it successively and form the product of the mn expressions thus formed, we obtain a function which vanishes if any one system of values, common to the first two equations, also satisfies the third. Hence this product is the required resultant of the three equations.
Now by the theory of symmetric functions, any symmetric functions of the mn values which satisfy the two equations, can be expressed in terms of the coefficient of those equations. Hence, finally, the resultant is expressed in terms of the coefficients of the three equations, and since it is at once seen to be of degree mn in the coefficient of the third equation, by symmetry it must be of degrees np and pm in the coefficients of the first and second equations respectively. Its weight will be mnp (see Salmon’s Higher Algebra, 4th ed. § 77). The general theory of the resultant of k homogeneous equations in k variables presents no further difficulties when viewed in this manner.
The expression in form of a determinant presents in general considerable difficulties. If three equations, each of the second degree, in three variables be given, we have merely to eliminate the six products x², y², z², yz, zx, xy from the six equations
u＝v＝w＝∂J∂x＝∂J∂y＝∂J∂z＝0; if we apply the same process to these equations each of degree three, we obtain similarly a determinant of order 21, but thereafter the process fails. Cayley, however, has shown that, whatever be the degrees of the three equations, it is possible to represent the resultant as the quotient of two determinants (Salmon, l.c. p. 89).
Discriminants.—The discriminant of a homogeneous polynomial in k variables is the resultant of the k polynomials formed by differentiations in regard to each of the variables.
It is the resultant of k polynomials each of degree m–1, and thus contains the coefficients of each form to the degree (m–1)k–1; hence the total degrees in the coefficients of the k forms is, by addition, k(m–1)k–1; it may further be shown that the weight of each term of the resultant is constant and equal to m(m–1)k–1 (Salmon, l.c. p. 100).
A binary form which has a square factor has its discriminant equal to zero. This can be seen at once because the factor in question being once repeated in both differentials, the resultant of the latter must vanish.
Similarly, if a form in k variables be expressible as a quadratic function of k – 1, linear functions X1, X2, ... Xk – 1, the coefficients being any polynomials, it is clear that the k differentials have, in common, the system of roots derived from X1＝X2＝...＝Xk – 1＝0, and have in consequence a vanishing resultant. This implies the vanishing of the discriminant of the original form.
Expression in Terms of Roots.—Since x∂ƒ∂x+∂ƒ∂y＝mƒ, if we take any root x1, y1, of ∂ƒ∂x, and substitute in mf we must obtain, y1(∂ƒ∂y)
x–x1
y–y1; hence the resultant of ∂ƒ∂x and ƒ is, disregarding numerical factors, y1y2...yn–1 × discriminant of ƒ＝a0 × disct. of ƒ.
Now
ƒ＝(xy1 – x1y)(xy2 – x2y) ... (xym – xmy),
∂ƒ∂x =Σ1 y1(xym – xmy),
and substituting in the latter any root of ƒ and forming the product, we find the resultant of ƒ and ∂ƒ∂x, viz.
y1y2...ym(x1y2 – x2y1)2(x1y3 – x3y1)2...(xrys – xsyr)2...
and, dividing by y1y2...ym, the discriminant of ƒ is seen to be equal to the product of the squares of all the differences of any two roots of the equation. The discriminant of the product of two forms is equal to the product of their discriminants multiplied by the square of their resultant. This follows at once from the fact that the discriminant is
II(αr – αs)2II(βr – βs)2{II(αr – βs}2.
References for the Theory of Determinants.—T. Muir’s “List of Writings on Determinants,” Quarterly Journal of Mathematics. vol. xviii. pp. 110-149, October 1881, is the most important bibliographical article on the subject in any language; it contains 589 entries, arranged in chronological order, the first date being 1693 and the last 1880. The bibliography has been continued, and published at various dates (vol. xxi. pp. 299-320; vol. xxxvi. pp. 171-267) in the same periodical. These lists contain 1740 entries. T. Muir, History of the Theory of Determinants (2nd ed., London, 1906). School treatises are those of Thomson, Mansion, Bartl, Mollame, in English, French, German and Italian respectively.—Advanced treatises are those of William Spottiswoode (1851), Francesco Brioschi (1854), Richard Baltzer (1857), George Salmon (1859), N. Trudi (1862), Giovanni Garbieri (1874), Siegmund Gunther (1875), Georges J. Dostor (1877), Baraniecki (the most extensive of all) (1879), R. F. Scott (2nd ed., 1904), T. Muir (1881).
