ALGEBRAIC
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vanishing of the covariant. These may be written, for the binary _7 d d du.k X'1dx1 ~ ^ ’ or in the form where fl = a
o^ + 2ai^2 + -+na"-1£’
dan-{ ^ + {n~1)aitLl + ---+ann'^ Let a covariant of degree e in the variables, and of degree 6 in the coefficients (the weight of the leading coefficient being w and nd - ew—2), be Cq^i +
- 2 +....
0=na
Operating with Q-a;2-^- we find OC0=:0 ; that is to say, C0 satisfies one of the two partial differential equations satisfied by an invariant. It is for this reason called a seminvariant, and every seminvariant is the leading coefficient of a covariant. The whole theory of invariants of a binary form depends upon the solutions of the equation 0 = 0. Before discussing these it is best to transform the binary form by substituting 1 ! cq, 2 ! «2) 3 ! a3,...?i! an, for ax, eq > a3...an respectively ; it then becomes , !. a x^, n x2 + ...+n atfCi +na jaqw-1x^ +n{n - )a^Xi -22, n and 0 takes the simpler form d d d dax dcL>s dan One advantage we have obtained is that, if we now write aQ=0, and substitute as_i for as when s>0, we obtain d a d d A. Zn-j— dax + T~ da2 + dci^ + ...+ #n—> 'dan- ic which is the form of O for a binary n - l . Hence, by merely diminishing each suffix in a seminvariant by unity, we obtain another seminvariant ofic the same degree, and of weight w- d, appertaining to the n - . Also, if we increase each suffix in a seminvariant, we obtain terms, free from a0, of some seminvariant of degree d and weight w + 9. Ex. gr. from the invariant a- 2a1a3 + 2a0a4 of the quartic the diminishing process yields a - 2a0a2> the leading coefficient of the Hessian of the cubic, and the increasing process leads to a| -2a2a4 + 2a1a5 which only requires the additional term - 2a0rt6 to become a seminvariant of the sextic. A more important advantage, springing from the new form of 0, arises from the fact that if xn - eqa:”-1 + a^xn ■ • ( - )nan = (x-a1)(x- a2)...(z-an), the sums of powers Sa2,Sa3,2a4, ...SaM all satisfy the equation II = 0. Hence, excluding a0, we may, in partition notation, write down the fundamental solutions of the equation, viz.— (2), (3), (4), ...(n), and say that, with a0, we have an algebraically complete system. Every symmetric function denoted by partitions, not involving the figure unity (say a non-unitary symmetric function), which remains unchanged by any increase of n, is also a seminvariant, and we may take if we please another fundamental system, viz.— ®o>(2)»(3),(22), (32),...(2iw) or (32K,l"3)). Observe that, if we subject any symmetric function []PP3Pz-..) to the diminishing process, it becomes av^~v,i{p2p3...). Next consider the solutions of 0 = 0 which are of degree 6 and weight w. The general term in a solution involves the product a^0a^1a^2".a"n wherein 27r = 0,2sirs= w; the number of such products that may appear depends upon the number of partitions of w into 9 or fewer parts limited not to exceed n in magnitude. Let this number be denoted by (w ; 9, n). In order to obtain the seminvariants we would write down the (w ; 9, n) terms each associated with a literal coefficient; if we now operate with fi we obtain a linear function of (w--,9,n) products, for the vanishing of which the literal coefficients must satisfy (w-1 -,9,%) linear equations ; hence (m> •,9,ri)-(w- ; 9, n) of these coefficients may be assumed arbitrarily, and the number of linearly independent solutions of 12 = 0, of the given degree and weight, is precisely (w -,9,%)-(w- -,9, n). This theory is due to Cayley; its validity depends upon showing that the (w- •, 9, n) linear equations satisfied by the literal coefficients are independent; this has only
FORMS
recently been established by E. B. Elliott. These seminvariants are said to form an asyzygetic system. It is shown in the Article on Combinatokial Analysis that (w; 9, n) is the coefficient of a6 zw in the ascending expansion of the fraction 1 1-a.l-az.l - az2.... 1 - azn' Hence (w ; 9, n) - (w - 1 ■, 9, n) is given by the coefficient of aezw in the fraction l_-z 1 -a.l -az.l -az1 1 -azn.> the enumerating generating function of asyzygetic seminvariants. We may, by a well-known theorem, write the result as a coefficient of zw in the expansion of 1 — z”4-4.1 — Zn'^’2.... 1 — zn+e l-z2.l-z3....l-z9 ’ and since this expression is unaltered by the interchange of n and 9 we prove Hermite’s Law of Reciprocity, which states that the asyzygetic forms of degreeic 9 for the nic are equinumerous with those of degree n for the 9 . The degree of the covariant in the variables is e — n9-2w consequently we are only concerned with positive terms in the developments and (w>, 9,n)-{w- 9, n) will be negative unless n9 -2w^0. It is convenient to enumerate the seminvariants of degree 9 and order e—n9- 2w by a generating function ; so, in the first written generating function for seminvariants, write for z and azn for a ; we obtain 1 - azn.l -azn-2.l -az"-4. ...1 - as-n+4.1 - az~n+2. - az-n in which we have to take the coefficient of aezn8~2w, the expansion being in ascending powers of a. As we have to do only with that part of the expansion which involves positive powers of z, we must try to isolate that portion, say Aw(z). For n — 2 we can prove that the complete function may be written a2(s)-Ja2(1), where A2(z) — z1 -az-.l 2 z - ao1 ) and this is the reduced generating function which tells us, by its denominator factors, that the complete system of the quadratic is composed of the form itself of degree order 1, 2 2shown by az2, and of the Hessian of degree order 2, 0 shown by a . Again, for the cubic, we can find 1 _ ft6..6 A3(sO = r ■ az1.1 . 1 - asz3.1 - a,4 ’ where the ground forms are indicated by the denominator factors, viz. : these are the cubic itself of degree order 1, 3 ; the Hessian of degree order 2, 2; the cubi-covariant G of degree order 3, 3, and the quartic invariant of degree order 4, 0. Further, the numerator factor establishes that these are not all algebraically independent, but are connected by a syzygy of degree order 6, 6. Similarly for the quartic . 4, , l-a6s12 ~ 1 - <zz4.l - a2.1 -a2z4.l - a3.l - a3z6’ establishing the 5 ground forms and the syzygy which connects them. The process is not applicable with complete success to quintic and higher ordered binary forms. This arises from the circumstance that the simple syzygies between the ground forms are not all independent, but are connected by second syzygies, and these again by third syzygies, and so on ; this introduces new difficulties which have not been completely overcome. As regards invariants a little further progress has been made by Cayley, who established the two generating functions for the quintic 1 - a3 4 1 - ct .1 - a8.1 - a12.1 - a18’ and for the sextic 1 - a30 1 -a2.l -<z4.l -cr.1 -a.l -a15 Accounts of further attempts in this direction will be found in Cayley’s Memoirs on Quantics (Collected Papers), in the papers of Sylvester and Franklin (Amer. J. i.-iv.), and in Elliott’s Algebra of Quantics, chap. viii. Perpetuants.—M.a,ny difficulties, connected with binary forms of finite order, disappear altogether when we come to consider the form of infinite order. In this case the ground forms, called also perpetuants, have been enumerated and actual representative
