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# Appendix 1: The three variables case

In this section we show that any form in three variables modulo the cubics is equivalent to xyz, , or . Using the notation introduced earlier we may assume that f is

If all coefficients are 0 except , we are done. This means one of the other terms is not 0 and we may assume and by scaling assume . Now set to get an equivalent form with . We now divide into two cases depending on whether or not . Case 1: . Now set . Let and be a root of . The resulting form has .

If , the form is . If , this is which is on the list. Otherwise scale to get . This is which by interchanging variables is which is on the list.

If , scale to get . The resulting form is . Transforming xyz by gives this form.

Case 2: . Scale to get . Now transform this by . Set and a root of to get . The form is now . If , this is Case 1 with the variables permuted according to . If not it can be scaled to give . If , this is . Now transform by and then by to get it.

Assume then that . The form is now . Let a,b,c be the three distinct roots of where . Now transform the form xyz by to get the form . By choosing appropriately these are then all equivalent. This transformation is found as in by finding a linear transformation which under conjugation transforms the Lie algebras for the centralizers of the forms. The coeffients for the transformed xyz are obtained using the facts that abc=1, a+b+c=0, and also that . All but the last follow immediately because a,b,c are the roots of the equation . The last follows because and are the two roots of . The sum of these terms, 0, can be found from computing the coefficient of in . The product, , can be evaluated using the expressions above.

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