Page:EB1911 - Volume 01.djvu/679

The general term of a seminvariant of degree $\theta ,\theta '$ and weight $w$ will be

$a_{0}^{\rho _{0}}a_{1}^{\rho _{1}}a_{2}^{\rho _{2}}\dots a_{p}^{\rho _{p}}b_{0}^{\sigma _{0}}b_{1}^{\sigma _{1}}b_{2}^{\sigma _{2}}\dots b_{q}^{\sigma _{q}}$

where       ${\overset {p}{\underset {1}{\Sigma }}}\rho _{s}=\theta$, ${\overset {q}{\underset {1}{\Sigma }}}\sigma _{s}=\theta '$ and ${\overset {p}{\underset {1}{\Sigma }}}s\rho _{s}+{\overset {q}{\underset {1}{\Sigma }}}s\sigma _{s}=w$.

The number of such terms is the number of partitions of $w$ into $\theta +\theta '$ parts, the part magnitudes, in the two portions, being limited not to exceed $p$ and $q$ respectively. Denote this number by $(w;\theta ,p;\theta '.q)$. The number of linearly independent seminvariants of the given type will then be denoted by

$(w;\theta ,p;\theta ',q)-(w-1;\theta ,p;\theta ',q)$;

and will be given by the coefficient of $a^{\theta }b^{\theta '}z^{w}$ in

${\frac {1}{1-a.~1-az.1-az^{2}.~\dots ~1-az^{p}.~1-b.~1-bz.~1-bz^{2}\dots ~1-bz^{q}}}$;

that is, by the coefficient of $z^{w}$ in

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

which preserves its expression when $\theta$ and $p$ and $\theta '$ and $q$ are separately or simultaneously interchanged.

Taking the first generating function, and writing $az^{p}$, $bz^{q}$ and ${\frac {1}{z^{2}}}$ for $a$, $b$ and $z$ respectively, we obtain the coefficient of $a^{\theta }b^{\theta '}z^{p\theta +q\theta '-2w}$, that is of $a^{\theta }b\theta 'z^{\epsilon }$, in

${\frac {1-z^{-2}}{1-az^{p}.~1-az^{p-2}~\dots 1-az^{-p+2}.~1-az^{-p}.~1-bz^{q}.~1-bz^{q-2}.~\dots 1-bz^{-q+2}.~1-bz^{-q}}}$;

the unreduced generating function which enumerates the covariants of degrees $\theta ,\theta '$ in the coefficients and order $\epsilon$ in the variables. Thus, for two linear forms, $p=q=1$, we find

${\frac {1-z^{-2}}{1-az.~1-az^{-1}.~1-bz.~1-bz^{-1}}}$,

the positive part of which is

${\frac {1}{1-az.~1-bz.~1-ab}}$;

establishing the ground forms of degrees-order (1, 0; 1), (0, 1; 1), (1, 1; 0), viz:—the linear forms themselves and their Jacobian ${\text{J}}_{ab}$. Similarly, for a linear and a quadratic, $p=1$, $q=2$, and the reduced form is found to be

${\frac {1-a^{2}b^{2}z^{2}}{1-az.~1-bz^{2}.~1-abz.~1-b^{2}.~1-a^{2}b}}$,

where the denominator factors indicate the forms themselves, their Jacobian, the invariant of the quadratic and their resultant; connected, as shown by the numerator, by a syzygy of degrees-order (2, 2; 2).

The complete theory of the perpetuants appertaining to two or more forms of infinite order has not yet been established. For two forms the seminvariants of degrees 1, 1 are enumerated by ${\frac {1}{1-z}}$, and the only one which is reducible is $a_{0}b_{0}$ of weight zero; hence the perpetuants of degrees 1, 1 are enumerated by

${\frac {1}{1-z}}-1={\frac {z}{1-z}}$;

and the series is evidently

$a_{0}b_{1}-a_{1}b_{0}$,
$a_{0}b_{2}-a_{1}b_{1}+a_{2}b_{0}$,
$a_{0}b_{3}-a_{1}b_{2}+a_{2}b_{1}-a_{3}b_{0}$,

one for each of the weights 1, 2, 3,...ad infin.

For the degrees 1, 2, the asyzygetic forms are enumerated by ${\frac {1}{1-z.~1-z^{2}}}$, and the actual forms for the first three weights are

$a_{0}b_{0}^{2}$,
$(a_{0}b_{1}-a_{1}b_{0})b_{0}$,
$(a_{0}b_{2}-a_{1}b_{1}+a_{2}b_{0})b_{0}$,
$a_{0}(b_{1}^{2}-2b_{0}b_{2})$,
$(a_{0}b_{3}-a_{1}b_{2}+a_{2}b_{1}-a_{3}b_{0})b_{0}$,
$a_{0}(b_{1}b_{2}-3b_{0}b_{3})-a_{1}(b_{1}^{2}-2b_{0}b_{2})$;

amongst these forms are included all the asyzygetic forms of degrees 1, 1, multiplied by $b_{0}$, and also all the perpetuants of the second binary form multiplied by $a_{0}$; hence we have to subtract from the generating function ${\frac {1}{1-z}}$ and ${\frac {z^{2}}{1-z^{2}}}$, and obtain the generating function of perpetuants of degrees 1, 2.

${\frac {1}{1-z.~1-z^{2}}}-{\frac {1}{1-z}}-{\frac {z^{2}}{1-z^{2}}}={\frac {z^{3}}{1-z.~1-z^{2}}}$.

The first perpetuant is the last seminvariant written, viz.:—

$a_{0}(b_{0}b_{2}-3b_{0}b_{3})-a_{1}(b_{1}^{2}-2b_{0}b_{2})$,

or, in partition notation,

$a_{0}(21)_{b}-(1)_{a}(2)_{b}$;

and, in this form, it is at once seen to satisfy the partial differential equation. It is important to notice that the expression

$(\theta )_{a}(\theta '1^{s})_{b}-(\theta 1)_{a}(\theta '1^{s-1})_{b}+(\theta 1^{2})_{a}(\theta '1^{s-2})_{b}-\ldots \pm (\theta 1^{s})_{a}(\theta ')_{b}$

denotes a seminvariant, if $\theta ,\theta ',$ be neither of them unity, for, after operation, the terms destroy one another in pairs: when $\theta =0$, $(\theta )^{a}$ must be taken to denote $a_{0}$ and so for $\theta '$. In general it is a seminvariant of degrees $\theta ,\theta '$, and weight $\theta +\theta '+s$; for this there is an exception, viz., when $\theta =0$, or when $\theta '=0$, the corresponding partial degrees are 1 and 1. When $\theta =\theta '=0$, we have the general perpetuant of degrees 1, 1. There is a still more general form of the seminvariant; we may have instead of $\theta ,\theta '$ any collections of non-unitary integers not exceeding $\theta ,\theta '$ in magnitude respectively, Ex. gr.

⁠$(2^{\lambda _{2}}3^{\lambda _{3}}\ldots \theta ^{\lambda _{\theta }})_{a}(1^{s}2^{\mu _{2}}3^{\mu _{3}}\ldots \theta '^{\mu _{\theta '}})_{b}$
$~-(12^{\lambda _{2}}3^{\lambda _{3}}\ldots \theta ^{\lambda _{\theta }})_{a}(1^{s-1}2^{\mu _{2}}3^{\mu _{3}}\ldots \theta '^{\mu _{\theta '}})_{b}$
$~+(1^{2}2^{\lambda _{2}}3^{\lambda _{3}}\ldots \theta ^{\lambda _{\theta }})_{a}(1^{s-2}2^{\mu _{2}}3^{\mu _{3}}\ldots \theta '^{\mu _{\theta '}})_{b}$
$~~-.~.~.~.$
$(-)^{s}(1^{s}2^{\lambda _{2}}3^{\lambda _{3}}\ldots \theta ^{\lambda _{\theta }})_{a}(2^{\mu _{2}}3^{\mu _{3}}\ldots \theta '^{\mu _{\theta '}})_{b}$,

is a seminvariant; and since these terms are clearly enumerated by

${\frac {1}{1-z.~1-z^{2}.~\dots ~1-z^{\theta }.~1-z^{2}.~1-z^{3}.~\dots ~1-z^{\theta '}}}$,

an expression which also enumerates the asyzygetic seminvariants, we may regard the form, written, as denoting the general form of asyzygetic seminvariant; a very important conclusion. For the case in hand, from the simplest perpetuant of degrees 1, 2, we derive the perpetuants of weight $w$,

$a_{0}(21^{w-2})_{b}-a_{1}(21^{w-3})_{b}+a_{2}(21^{w-4})_{b}-\ldots \pm a_{w-2}(2)_{b}$,
$a_{0}(2^{2}1^{w-4}(_{b}-a_{1}(2^{2}1^{w-5})_{b}+a_{2}(2^{2}1^{w-6})_{b}-\ldots \pm a_{w-4}(2^{2})_{b}$,
$a_{0}(2^{3}1^{w-6})_{b}-a_{1}(2^{3}1^{w-7})_{b}+a_{2}(2^{3}1^{w-8})_{b}-\ldots \pm a_{w-6}(2^{3})_{b}$,

a series of ${\frac {1}{2}}(w-2)$ or of ${\frac {1}{2}}(w-1)$ forms according as $w$ is even or uneven. Their number for any weight $w$ is the number of ways of composing $w-3$ with the parts 1, 2, and thus the generating function is verified. We cannot, by this method, easily discuss the perpetuants of degrees 2, 2, because a syzygy presents itself as early as weight 2. It is better now to proceed by the method of Stroh.

We have the symbolic expression of a seminvariant.

${\frac {1}{w{\text{!}}}}(\sigma _{1}\alpha _{1}+\sigma _{2}\alpha _{2}+\ldots +\sigma _{\theta }\alpha _{\theta }+\tau _{1}\beta _{1}+\tau _{2}\beta _{2}+\ldots +\tau _{\theta '}\beta _{\theta '})^{w}$

where

${\frac {\alpha _{1}^{s}}{s{\text{!}}}}={\frac {\alpha _{2}^{s}}{s{\text{!}}}}=\ldots =a_{s}$; ${\frac {\beta _{1}^{s}}{s{\text{!}}}}={\frac {\beta _{2}^{s}}{s{\text{!}}}}=\ldots =b_{s}$;

and ⁠$\sigma _{1}+\sigma _{2}+\ldots +\sigma _{\theta }+\tau _{1}+\tau _{2}+\ldots +\tau _{\theta }=0$.

Proceeding as we did in the case of the single binary form we find that for a given total degree $\theta +\theta '$, the condition which expresses reducibility is of total degree $2^{\theta +\theta '-1}-1$ in the coefficients $\sigma$ and $\tau$; combining this with the knowledge of the generating function of asyzygetic forms of degrees $\theta$, $\theta '$, we find that the perpetuants of these degrees are enumerated by

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

and this is true for $\theta +\theta '=2$ as well as for other values of $\theta +\theta '$ (compare the case of the single binary form).

Observe that, if there be more than two binary forms, the weight of the simplest perpetuant of degrees $\theta ,\theta ',\theta '',\dots$ is $2^{\theta +\theta '+\theta ''+\ldots -1}-1$, as can be seen by reasoning of a similar kind.

To obtain information concerning the actual forms of the perpetuants, write

$(1+\sigma _{1}x)(1+\sigma _{2}x)\ldots (1+\sigma _{\theta }x)=1+{\text{A}}_{1}x+{\text{A}}_{2}x^{2}+\ldots +{\text{A}}_{\theta }x^{\theta }$
$(1+\tau _{1}x)(1+\tau _{2}x)\ldots (1+\tau _{\theta '}x)=1+{\text{B}}_{1}x+{\text{B}}_{2}x^{2}+\ldots +{\text{B}}_{\theta '}x^{\theta '}$

where ⁠ ${\text{A}}_{1}{\text{B}}_{1}=0$.

For the case $\theta =1$, $\theta '=1$, the condition is

$\sigma _{1}\tau _{1}={\text{A}}_{1}{\text{B}}_{1}=0$,

which since ${\text{A}}_{1}+{\text{B}}_{1}=0$, is really a condition of weight unity. For $w=1$ the form is ${\text{A}}_{1}a_{1}+{\text{B}}_{1}b_{1}$, which we may write $a_{0}b_{1}-a_{1}b_{0}=a_{0}(1)_{b}-(1)_{a}b_{0}$; the remaining perpetuants, enumerated by ${\frac {z}{1-z}}$, have been set forth above.

For the case $\theta =1$, $\theta '=2$, the condition is $\sigma _{1}\tau _{1}\tau _{2}={\text{A}}_{1}{\text{B}}_{2}=0$; and the simplest perpetuant, derived directly from the product ${\text{A}}_{1}{\text{B}}_{2}$, is $(1)_{a}(2)_{b}-(21)_{b}$; the remainder of those enumerated by ${\frac {z^{3}}{1-z.~1-z^{2}}}$ may be represented by the form

$(1^{\lambda _{1}+1})_{a}(2^{\mu _{2}+1})_{b}-(1^{\lambda _{1}})_{a}(2^{\mu _{2}+1}1)_{b}+\ldots \pm (2^{\mu _{2}+1}1^{\lambda _{1}+1})_{b}$;

$\lambda _{1}$ and $\mu _{2}$ each assuming all integer (including zero) values. For the case $\theta =\theta '=2$, the condition is

$\sigma _{1}\sigma _{2}\tau _{1}\tau _{2}(\sigma _{1}+\sigma _{2})(\sigma _{1}+\tau _{1})(\sigma _{1}+\tau _{2})={\text{A}}_{2}^{2}{\text{B}}_{1}{\text{B}}_{2}-{\text{A}}_{1}{\text{A}}_{2}{\text{B}}_{2}^{2}=0$.

To represent the simplest perpetuant, of weight 7, we may take as base either ${\text{A}}_{2}^{2}{\text{B}}_{1}{\text{B}}_{2}$ or ${\text{A}}_{1}{\text{A}}_{2}{\text{B}}_{2}^{2}$, and since ${\text{A}}_{1}+{\text{B}}_{1}=0$ the former is equivalent to ${\text{A}}_{1}{\text{A}}_{2}^{2}{\text{B}}_{2}$ and the latter to ${\text{A}}_{2}{\text{B}}_{1}{\text{B}}_{2}^{2}$; so that we have,