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Unformatted text preview: R 3 : 0 2 b a c,a > ,ac > b 2 } . (1.3) By Theorem 1.2, any positive denite binary quadratic form is equivalent to a form ax 2 + 2 bxy + cy 2 , where ( a,b,c ) . 1.3 Let us nd when two reduced forms are equivalent. To do this we should look at the domain from a dierent angle. Each positive denite quadratic form f = ax 2 + 2 bxy + cy 2 can be factored over C into product of linear forms: f = ax 2 + 2 bxy + cy 2 = a ( xzy )( x zy ) , where z =b a + i acb 2 a . (1.4) It is clear that f is completely determined by the coecient a and the root z . Observe that Im z > . We have a bijective correspondence f = ax 2 + 2 bxy + cy 2 ( a,z ) from the set of positive denite binary quadratic forms to the set R + H , where H = { z C : Im z > }...
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 Fall '09
 ONTONKONG
 Vectors

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