Page:EB1911 - Volume 01.djvu/679

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

where &emsp; &emsp; &emsp; $$\overset{p}{\underset{1}{\Sigma}}\rho_s = \theta$$, $$\overset{q}{\underset{1}{\Sigma}}\sigma_s = \theta'$$ 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

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

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

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

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

the positive part of which is

establishing the ground forms of degrees-order (1, 0; 1), (0, 1; 1), (1, 1; 0), viz:&mdash;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

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 $$a_0b_0$$ of weight zero; hence the perpetuants of degrees 1, 1 are enumerated by

and the series is evidently

$a_0b_1-a_1b_0$, $a_0b_2-a_1b_1+a_2b_0$, $a_0b_3-a_1b_2+a_2b_1-a_3b_0$,

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

$a_0b_0^2$, $(a_0b_1-a_1b_0)b_0$, $(a_0b_2-a_1b_1+a_2b_0)b_0$, $a_0(b_1^2-2b_0b_2)$, $(a_0b_3-a_1b_2+a_2b_1-a_3b_0)b_0$, $a_0(b_1b_2-3b_0b_3)-a_1(b_1^2-2b_0b_2)$;

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.

The first perpetuant is the last seminvariant written, viz.:&mdash;

or, in partition notation,

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

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.

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

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 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.

where

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

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

where $$\text{A}_1\text{B}_1=0$$.

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

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 $$a_0b_1-a_1b_0 = a_0(1)_b-(1)_a b_0$$; 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

$$\lambda_1$$ and $$\mu_2$$ each assuming all integer (including zero) values. For the case $$\theta=\theta'=2$$, the condition is

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,