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May 27th, 2014, 01:40 AM   #1
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Invariant Theory

A form is a polynomial in n variables where the sum of exponents of each term is the same.

I.2 The linear transformation

We are led to essentially new and deeo properties of forms through the application of linear transformations. Let

$F^n$ ($x_1, x_2, \dots , x_m$)

be a general form. We can derive another form from it, if we replace the m variables x by other variables x' via relations of the form

$x_1 = \phi_1$ (x_1', ..., x_m'),

$x_2 = \phi_2$ (x_1', ..., x_m'),

...

$x_m = \phi_m$ (x_1', ..., x_m'),

where the $\phi$s denote forms of the same order (the sum of exponents of the variables).

I will continue this and in about 5 pages get to the question I need to ask.
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May 28th, 2014, 03:03 AM   #2
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Every invariant of a form must be homogenous in the coefficients, and every term must have degree

$g = \frac{2p}{n}$,

where p is that exponent of the transformation determinant by which the invariant changes under substitution of the transformed coefficients. Furthermore, all terms must have the same weight, which is also equal to p.

Can someone prove this please?
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