Page:EB1911 - Volume 01.djvu/677

variables. For instance, those of a ternary form involve two classes which may be geometrically interpreted as point and line co-ordinates in a plane; those of a quaternary form involve three classes which may be geometrically interpreted as point, line and plane coordinates in space.

IV. Enumerating Generating Functions

Professor Michael Roberts (Quart. Math. J. iv.) was the first to remark that the study of covariants may be reduced to the study of their leading coefficients, and that from any relations connecting the latter are immediately derivable the relations connecting the former. It has been shown above that a covariant, in general, satisfies four partial differential equations. Two of these show that the leading coefficient of any covariant is an isobaric and homogeneous function of the coefficients of the form; the remaining two may be regarded as operators which cause the vanishing of the covariant. These may be written, for the binary $n^{ic}$,

$\Sigma ka_{k-1}{\frac {d}{da_{k}}}-x_{2}{\frac {d}{dx_{1}}}=0$;
$\Sigma (n-k)a_{k+1}{\frac {d}{da_{k}}}-x_{1}{\frac {d}{dx_{2}}}=0$;

or in the form

$\Omega -x_{2}{\frac {d}{dx_{1}}}=0$, $\mathrm {O} -x_{1}{\frac {d}{dx_{2}}}=0$;

where

$\Omega =a_{0}{\frac {d}{da_{1}}}+2a_{1}{\frac {d}{da_{2}}}+\ldots +na_{n-1}{\frac {d}{da_{n}}}$,
$\mathrm {O} =na_{1}{\frac {d}{da_{0}}}+(n-1)a_{2}{\frac {d}{da_{1}}}+\ldots +a_{n}{\frac {d}{da_{n-1}}}$.

Let a covariant of degree $\epsilon$ in the variables, and of degree $\theta$ in the coefficients (the weight of the leading coefficient being $w$ and $n\theta -2w=\epsilon$), be

${\text{C}}_{0}x_{1}^{\epsilon }+\epsilon c_{1}x_{1}^{\epsilon -1}x_{2}+\ldots$.

Operating with $\Omega -x_{2}{\frac {d}{dx_{1}}}$ we find $\Omega {\text{C}}_{0}=0$; that is to say, ${\text{C}}_{0}$ 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 $\Omega =0$. Before discussing these it is best to transform the binary form by substituting $1{\text{!}}a_{1},~2{\text{!}}a_{2},~3{\text{!}}a_{3},\ldots n{\text{!}}a_{n}$, for $a_{1},~a_{2},~a_{3}\ldots a_{n}$ respectively; it then becomes

$a_{0}x_{1}^{n}+na_{1}x_{1}^{n-1}x_{2}+n(n-1)a_{2}x_{1}^{n-2}x_{2}^{2}+\ldots +n{\text{!}}a_{n}x_{2}^{n}$,

and $\Omega$ takes the simpler form

$a_{0}{\frac {d}{da_{1}}}+a_{1}{\frac {d}{da_{2}}}+a_{2}{\frac {d}{da_{3}}}+\ldots +a_{n-1}{\frac {d}{da_{n}}}$.

One advantage we have obtained is that, if we now write $a_{0}=0$, and substitute $a_{s-1}$ for $a_{s}$, when $s$ > 0, we obtain

$a_{0}{\frac {d}{da_{1}}}+a_{1}{\frac {d}{da_{2}}}+a_{2}{\frac {d}{da_{3}}}+\ldots +a_{n-2}{\frac {d}{da_{n-1}}}$

which is the form of $\Omega$ for a binary $(n-1)^{ic}$.

Hence by merely diminishing each suffix in a seminvariant by unity, we obtain another seminvariant of the same degree, and of weight $w-\theta$, appertaining to the $(n-1)^{ic}$. Also, if we increase each suffix in a seminvariant, we obtain terms, free from $a_{0}$, of some seminvariant of degree $\theta$ and weight $w+\theta$. Ex. gr. from the invariant $a_{2}^{2}-2a_{1}a_{3}+2a_{0}a_{4}$ of the quartic the diminishing process yields $a_{1}^{2}-2a_{0}a_{2}$, the leading coefficient of the Hessian of the cubic, and the increasing process leads to $a_{3}^{2}-2a_{2}a_{4}+2a_{1}a_{5}$ which only requires the additional term $-2a_{0}a_{6}$ to become a seminvariant of the sextic. A more important advantage, springing from the new form of $\Omega$, arises from the fact that if

$x^{n}-a_{1}x^{n-1}+a_{2}x^{n-2}-\ldots (-)^{n}a_{n}=(x-\alpha _{1})(x-\alpha _{2})\ldots (x-\alpha _{n})$,

the sums of powers $\Sigma \alpha ^{2},~\Sigma \alpha ^{3},~\Sigma \alpha ^{4},~\ldots \Sigma \alpha ^{n}$ all satisfy the equation $\Omega =0$. Hence, excluding $a_{0}$, we may, in partition notation, write down the fundamental solutions of the equation, viz.—

$(2),~(3),~(4),\dots (n)$,

and say that with $a_{0}$, 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.—

$a_{0},(2),~(3),~(22),~(32),\dots (2^{{\frac {1}{2}}n})$ or $(32^{{\frac {1}{2}}(n-3)})$.

Observe that, if we subject any symmetric function $(p_{1}p_{2}p_{3}\dots )$ to the diminishing process, it becomes $a_{0}^{p_{1}-p_{2}}(p_{2}p_{3}\dots )$.

Next consider the solutions of $\Omega =0$ which are of degree $\theta$ and weight $w$. The general term in a solution involves the product $a_{0}^{\pi _{0}}a_{1}^{\pi _{1}}a_{2}^{\pi _{2}}\dots a_{n}^{\pi _{n}}$ wherein $\Sigma \pi =\theta$, $\Sigma s\pi _{s}=w$; the number of such products that may appear depends upon the number of partitions of $w$ into $\theta$ or fewer parts limited not to exceed $n$ in magnitude. Let this number be denoted by $(w;~\theta ,~n)$. In order to obtain the seminvariants we would write down the $(w;~\theta ,~n)$ terms each associated with a literal coefficient; if we now operate with $\Omega$ we obtain a linear function of $(w-1;~\theta ,~n)$ products, for the vanishing of which the literal coefficients must satisfy $(w-1;~\theta ,~n)$ linear equations; hence $(w;~\theta ,~n)-(w-1;~\theta ,~n)$ of these coefficients may be assumed arbitrarily, and the number of linearly independent solutions of $\Omega =0$, of the given degree and weight, is precisely $(w;~\theta ,~n)-(w-1;~\theta ,~n)$. This theory is due to Cayley; its validity depends upon showing that the $(w-1;~\theta ,~n)$ linear equations satisfied by the literal coefficients are independent; this has only recently been established by E. B. Elliott. These seminvariants are said to form an asyzygetic system. It is shown in the article on Combinatorial Analysis that $(w;~\theta ,~n)$ is the coefficient of $a^{\theta }z^{w}$ in the ascending expansion of the fraction

${\frac {1}{1-a.~1-az.~1-az^{2}.~\ldots 1-az^{n}}}$.

Hence $(w;~\theta ,~n)-(w-1;~\theta ,~n)$ is given by the coefficient of $a^{\theta }z^{w}$ in the fraction

${\frac {1-z}{1-a.~1-az.~1-az^{2}.~\ldots 1-az^{n}.}}$,

the enumerating generating function of asyzygetic seminvariants. We may, by a well-known theorem, write the result as a coefficient of $z^{w}$ in the expansion of

${\frac {1-z^{n+1}.~1-z^{n+2}.~\dots 1-z^{n+\theta }}{1-z^{2}.~1-z^{3}.~\dots 1-z^{\theta }}}$;

and since this expression is unaltered by the interchange of $n$ and $\theta$ we prove Hermite’s Law of Reciprocity, which states that the asyzygetic forms of degree $\theta$ for the $n^{ic}$ are equinumerous with those of degree $n$ for the $\theta ^{ic}$.

The degree of the covariant in the variables is $\epsilon =n\theta -2w$; consequently we are only concerned with positive terms in the developments and $(w;~\theta ,~n)-(w-1;~\theta ,~n)$ will be negative unless $n\theta -2w\geq 0$. It is convenient to enumerate the seminvariants of degree $\theta$ and order $\epsilon =n\theta -2w$ by a generating function; so, in the first written generating function for seminvariants, write ${\tfrac {1}{z^{2}}}$ for $z$ and $az^{n}$ for $a$; we obtain

${\frac {1-z^{-2}}{1-az^{n}.~1-az^{n-2}.~1-az^{n-4}.\dots 1-az^{-n+4}.~1-az^{-n+2}.~1-az^{-n}}}$

in which we have to take the coefficient of $a^{\theta }z^{n\theta -2w}$, 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 $A_{n}(z)$. For $n=2$ we can prove that the complete function may be written

${\text{A}}_{2}(z)-{\frac {1}{z^{2}}}{\text{A}}_{2}\left({\frac {1}{z}}\right)$,

where

${\text{A}}_{2}(z)={\frac {1}{1-az^{2}.~1-a^{2}}}$;

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 shown by $az^{2}$, and of the Hessian of degree order 2, 0 shown by $a^{2}$.

Again, for the cubic, we can find

${\text{A}}_{3}(z)={\frac {1-a^{6}z^{6}}{1-az^{3}.~1-a^{2}z^{2}.~1-a^{3}z^{3}.~1-a^{4}}}$,

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

${\text{A}}_{4}(z)={\frac {1-a^{6}z^{12}}{1-az^{4}.~1-a^{2}.~1-a^{2}z^{4}.~1-a^{3}.~1-a^{3}z^{6}}}$,

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

${\frac {1-a^{36}}{1-a^{4}.~1-a^{8}.~1-a^{12}.~1-a^{18}}}$,

and for the sextic

${\frac {1-a^{30}}{1-a^{2}.~1-a^{4}.~1-a^{6}.~1-a^{10}.~1-a^{15}}}$.

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.—Many difficulties, connected with binary forms of finite order, disappear altogether when we come to consider the