282
ALGEBRAIC
FORMS
weight of the function is the sum of the numbers in the bracket, and the degree the highest of those numbers. Ex. gr. The elementary functions are denoted by (1), (l2), (l3), ... (1”), are all of the first degree, and are of weights 1, 2, 3, ... n respectively. Remark.—In this notation (0) = 2a? = (i); (02) = 2aia2=(2) ;... (0®) = (s), &c. The binomial coefficients appear, in fact, as symmetric functions, and this is frequently of importance. The order of the numbers in the bracket (pp2...pf) is immaterial ; we may therefore always place them, as is most convenient, in descending order of magnitude ; the numbers then constitute an ordered partition of the weight w, and the leading number denotes the degree. The sum of the monomial functions of a given weight is called the homogeneous-product-sum or complete symmetric function of that weight; it is denoted by hw; it is connected with the elementary functions by the formula 1 1 1 - axx + ape2 - a-p? +... = 1 + hyX + h^x + hg/J 4-.. which remains true when the symbols a and h are interchanged, as is at once evident by writing - x for x. This proves, also, that in any formula connecting a^ a2, «3, ... with h^ h%, h3, ... the symbols a and h may be interchanged. any root xx, rq, of and substitute in to/ we must obtain Ex. gr. from h2=a-a3 we derive a3—h - 7i2. The function 'Logoff... off being as above denoted by a parti0/ hence the resultant of and / is, disregarding tion of the weight, viz. {p-iP2-■-Pn), it is necessary to bring under y=vi view other functions associated with the same series of numbers ; numerical factors, yiyz-.-ym-i * discriminant of f=a0 x disct. of/. such, for example, as Now Sa 1 /= (scyi - xiy){xy2 - xty)...{xym - xmy), i off^aff off... offj2 = {plVf (P2P4 • • •i’n—2). The expression just written is in fact a partition of a partition, J-=St/i(a52/2 - x*y)... {xym - xmy), and to avoid confusion of language will be termed a separation a partition. A partition is separated into separates so as to and substituting in the latter any root of/ and forming the pro- of produce a separation of the partition by writing down a set of partitions, each separate partition in its own brackets, so that duct, we find the resultant of / and viz. :— when all the parts of these partitions are reassembled in a single bracket the partition which is separated is reproduced. It is ViVi-■ -ymixiyz - x22/i)2(xi2/3 ..{xry, - x,yrf... convenient to write the distinct partitions or separates in and, dividing by yiy2---ym, the discriminant of / is seen to be descending order as regards weight. If the successive weights equal to the product of the squares of all the differences of any of the separates Wj, to2, ws, ... be enclosed in a bracket we obtain two roots of the equation. The discriminant of the product of a partition of the weight w which appertains to the separated two forms is equal to the product of their discriminants multi- partition. This partition is termed the specification of the plied by the square of their resultant. This follows at once separation. The degree of the separation is the sum of the degrees of the component separates. A separation is the symbolic from the fact that the discriminant is representation of a product of monomial symmetric functions. A n (ar - a,)2 n (/3r - ft)2 {n (ar - ft)} 2. partition, {PiPniP^PiPsl — ^ViTlPz)^ can be separated in the References for the Theory of Determinants.—T. Muir’s “ List of manner ipiP^PiP^p^-^PiPffiiPiPz), and we may take the Writings on Determinants,” Quarterly Journal of Mathematics, general form of a partition to be (p^pf^p^■ ■ ■) and that of a v. 18, pp. 110-14:9, October 1881, is the most important biblio(JX)A(J2)^2(J3)^s... when J1} J2, J3... denote the distinct graphical article on the subject in any language ; it contains 489 separation involved. entries, arranged in chronological order, the first date being 1693 separates function symbolized by (n), viz., the sum of the and the last 1880.—T. Muir. History of the Theory of Deter- nthTheorem.—The of the quantities, is expressible in terms of funcminants. London, 1890.—School treatises are those of Thom- tionspowers which are symbolized by separations of any partition son, Mansion, Bartl, Mollame, in English, French, German, and Italian respectively.—Advanced treatises are those of Spottis- (nVlnVinvf...) of the number n. The expression is— woode, 1851 ; Brioschi, 1854 ; Baltzer, 1857 ; Salmon, 1859 ( _ l'l+*,2+*'3+ • - •(>,1 + »'2 + >'3 + •••-!) !/ x Trudi, 1862 ; Garbieri, 1874 ; Gunther, 1875 ; Dostor, 1877 jq! vf. P3!... ' ^ Baraniecki (the most extensive of all), 1879 ; Scott, 1880 Muir, 1881. Att+ + (A+ i + =2< - )
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} ,/, ," II. The Theory of Symmetric Functions. (Jj/i (J2) ^2 (.. being a separation of (r/1«%/3...) and the Consider n quantities ai, a2, a3, ... an. Every rational integral function of these quantities, which summation being in regard to all such separations. For the does not alter its value however the n suffixes 1, 2, 3, ... ■h- be particular case {n'fffn*2...) — (ln) permuted, is a rational integral symmetric function of the fi+h+h+ • • ■U1+j?+j3+...-iy.{iyi (l2y2(13y3. quantities. If we write (1 + aia;)(l +a2a:)...(l + a„x) = l +aix + c s n jf-Jz'Js'---hP + ■■■ + a„x , «i, cio, ... an are called the elementary symmetric To establish this write— functions. «i ==aj + a2 + ... + a„ = 2ai 1 + y.X1 + /2X2 + /asX3 + ... = II( 1 + ficqaq +^2afx2-|-^3ajx3+ ...), a «2 + “itts + a2“3 +... = Saxa2 the product on the right involving a factor for each of the ccn = CI1&2CI3,.. an. quantities cq, a2, a3..., and y. being arbitrary. Multiplying out the right-hand side and comparing coefficients The general monomial symmetric function is Xi = (l)aq, 2 p p r Xa fa^a /...a f, X,=(2)»2+(l )*?, X3 = (3)a:3 +(21)a:2a:1-f(l32)xJ, the summation being for all permutations of the indices which X4=(4)a;4 + (Sl^aq + (2 )a:l + (212)3^:2 + (14M, result in different terms. The function is written (Pl'p2P3...Pn) Xm=^{mfhn^m!^... )x^x!fxm3- •» for brevity, and repetitions of numbers in the bracket are indicated by exponents, so that {pipip*) is written {plp2). The the summation being for all partitions of to.
minant 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 1c variables is the resultant of the 1c polynomials formed by differentiations in regard to each of the variables. It is the resultant of 1c polynomials each of degree m -1, and thus contains the coefficients of each form to the degree (to-l)*-1; hence the total degrees in the coefficients of the k forms is, by addition, k{m -1)*-1 ; it may further be shown that the weight of each term of the resultant is constant and equal to to(to — 1)*—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 Xj, X2, ...X^_1, the coefficients being any polynomials, it is clear that the k differentials have, in common, the system of roots derived from Xx = X2 =... = X*^ = 0, and have in consequence a vanishing resultant. This implies the vanishing of the discriminant of the original form. Expressioyi in Terms of Roots.—Since cc ^': to/, if we take
