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I can not understand the definition of balanced triple $(I_1,I_2,I_3)$ in a Dedekind Domain which is defined in the Higher composition laws I of Manjul Bhargava.

I would be so thankful if somebody can help me!

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This is impossible without knowing your background and the points you are having trouble with. Googling for Bhargava and composition yields lots of sources containing very detailed descriptions. –  Franz Lemmermeyer Dec 17 '11 at 18:06
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I could not find any good source. Except a note from you which is not contained the main theorem of Bhargava paper. I mean Theorem 11 of the first paper. Could you please tell me some other notes? –  Sina Dec 17 '11 at 18:16
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According to the definition, the product of these ideals must be contained in the domain (which is not assumed to be Dedekind in order to be able to work with orders in quadratic fields), and the product of their norms must be equal to 1. Where do you think Bhargava's proof goes wrong? –  Franz Lemmermeyer Dec 17 '11 at 18:43
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By this definition the cube associated to a triple is not well defined because if we consider an equivalent triple then the cube will change. –  Sina Dec 17 '11 at 19:03
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Of course the cube changes, but its equivalence class under the action of SL_2(Z)^3 does not. wstein.org/129/projects/schoenebeck/Composition.pdf –  Franz Lemmermeyer Dec 18 '11 at 17:15
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