Some Formulae in Elimination 论文

摘要

1.The object of the following paper is to investigate the properties of the determinants which arise in the theory of elimination when conducted according to the methods of Bezout, and, in particular, to find a simple expression for the resultant.The equations are supposed homogeneous, of different orders, and general, that is, complete in all their terms with unconnected literal coefficients.Cayley* has given (without proof) an extremely general expression for the resultant of n equations in the form B/B l /... /B n .it or BB % ... /J9jD 8 ..., where D is any non-vanishing determinant of the complete matrix corresponding to the function 0, S x +... + G n 8 n of order t n (c/.§3 below), and B lt B 3 , ..., D,,_ 2 are other determinants.The simpler, but less general, expression for the resultant found below is D/A, where B is a determinant selected arbitrarily in accordance with a certain rule ( § 6a) from the same matrix, and A is a minor of B.For three equations it can be verified that the two results B/B t and D/A are the same; B t and A are not, however, composed entirely of the same elements for the same B, but each is independent of the elements in which they differ.To verify the identity of the two results for more than three equations would be difficult, and of little use.The advantage of the simpler form D/A lies in the fact that A can be at once written down from B, whereas B u B. 2 , ... are only obtained by a complicated process, which Cayley does not fully explain.The theory suggested by Cayley has been developed in considerable detail by K. Bes.fHe discusses at length the case of three equations, from which he infers the result for n equations.He does not prove that B, B v ... can be so chosen that no one of them vanishes identically; and he is scarcely justified in describing his method as a new process, since it does not appear to differ in any essential feature from that of Cayley.• The method is described generally in the Camb.and Dub.Math.Jour., Vol.in., 1848, p. 116, and is explained more in detail in Salmon's Higher Algebra (4th edition, 1885), p. 87.t '' Theorie generale de l'Elimination, d'apres la methods Bezout, suivant un nouveau procede," Verhandelingen der KoninkUjlce Akadenue van Wvtenschappcn tc Amsterdam (Sectie 1), Deel vi., No. 7, 1899, 8vo, pp.1-121.B 2 * "L'Elimination," Seientia,I'fiy8.-Math.,'No.7,1900, pp.1-75.This monograph, although curious and interesting, is rendered practically valueless in what relates to equations in Reveral unknowns by its unreliable methods and conclusions.The resultant is nowhere defined and is regarded as an indefinite fractional expression.The following are some of the principal omissions and errors :-(1) The proof ( §15) of the theorem that, when two (non-homogeneous) polynomials in two variables are given as moduli, one variable can be expressed as an integral function of the other is incomplete, since two general assumptions are made proofs of which are not supplied.(2) The proof ( § 16) that the s-eliminant C of three equations in x, y, z of orders m, n, p is of-order mnp in z is faulty, since the author's method for expressing C leads to a fraction instead of an integral function of s.The same error appears still more prominently in § 17.(3) The proof ( § 18) of Bezout's reduced form of a given polynomial with respect to other given polynomials as moduli completely fails when it passes beyond reduction in one variable.(4) The statement ( § 20) that n 2 / n / [the author uses D for J\ cf.§ 10 ( 14) of this paper] is a determinant with all its elements to the left of the diagonal zero is an eiTor, but an unimportant one, since it breaks up into a product of determinants in the diagonal.The statement that tf/nJ depends only on the coefficients of the terms of highest order in the several equations is correct, but the proof is lacking.The author's proof of the same result in the Nouv.Ann. de Math., Series 3, Vol.ii., 1883, p. 147, is not valid.In the same place, p. 149, he is in error in stating that fl cannot vanish unless the equations have a double solution, from which he deduces incorrect conclusions.Again, in § 20 of the monograph, the author states that n"/llJ is independent of the roots of the equations.He does not explain what the statement means; but it is certainly untrue.If it were true, then the ratio of n to any other expression fl' formed in like manner would also be independent of the roots, which can easily be tested and found incorrect for the case of a linear and a quadratic equation in two unknowns.Netto, in referring to Laurent, says that O 9 /n/is a constant, without further explanation (Encyklopddie d.Math.Wiss., Teili., Band i., Heft 3, 1899, p. 274).It would seem that both writers have been misled by an assumed, but false, analogy with an equation in a single unknown.(5) In § 23 IB contained the so-called explicit expression for the resultant referred to above; but the author is in error in supposing this expression "independant des ay," and in supposing it to be the resultant, or to contain the resultant as a factor.(6) In § 26 the author implies that in order to calculate the resultant of n homogeneous equations in n unknowns it is of advantage to make the orders equal by multiplying the equations of inferior order by powers of the same unknown, overlooking the fact that, if two of the equations have a common factor, the resultant vanishes identically.Multiplying by powers of different unknowns is also of no advantage.In contrast with the abo\e we may mention § 19, which gives a proof of Jacobi's theorem, and §22, which proves that, if the vanishing points (or solutions) of n given polynomials/in n variables are distinct, finite, and complete, then any polynomial which vanishes at all these points is of the form 2<fc/i i.e., vanishes identically with respect to the/'s as moduli.