Unformatted text preview: R 3 : 0 ≤ 2 b ≤ a ≤ c,a > ,ac > b 2 } . (1.3) By Theorem 1.2, any positive deﬁnite binary quadratic form is equivalent to a form ax 2 + 2 bxy + cy 2 , where ( a,b,c ) ∈ Ω. 1.3 Let us ﬁnd when two reduced forms are equivalent. To do this we should look at the domain Ω from a diﬀerent angle. Each positive deﬁnite 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 coeﬃcient 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 deﬁnite binary quadratic forms to the set R + ×H , where H = { z ∈ C : Im z > }...
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 Fall '09
 ONTONKONG
 Linear Algebra, Vectors, Quadratic form, positive definite, Binary Quadratic Forms, binary quadratic form

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