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Fast Reduction and Composition of Binary Quadratic Forms ISSAC , 1991. Arnold Schonhage

Fast Reduction and Composition of Binary Quadratic Forms ISSAC , 1991. Arnold Schonhage. 田錦燕 95/02/17. Binary Quadratic Forms. F (x,y)=ax 2 +bxy+cy 2 =(x,y)[a,b,c](x,y) T where [a,b,c]= Discriminant : d = b 2 -4ac = β 2 -4αγ= -4 ． det [a,b,c]. Monotone reduction of positive forms.

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Fast Reduction and Composition of Binary Quadratic Forms ISSAC , 1991. Arnold Schonhage

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  1. Fast Reduction and Composition of Binary Quadratic Forms ISSAC, 1991. Arnold Schonhage 田錦燕 95/02/17

  2. Binary Quadratic Forms • F(x,y)=ax2+bxy+cy2=(x,y)[a,b,c](x,y)T where [a,b,c]= • Discriminant : d = b2-4ac = β2-4αγ= -4．det [a,b,c]

  3. Monotone reduction of positive forms • Positive forms([a,b,c] with a,b,c >0) • [a,b,c] = L [a, b-2a,c-b+a] H ,for a<c [a,b,c] = H [a-b+c,b-2a,c] L ,for a>c • s > 0 and matrix entries a,½b,c ≧s is “minimal above s”

  4. [a,b,c] = Lt [a,B,C] Ht where B=b-2at , C=c-bt+at2 • If d+4as < 4s2 then r:=2s else r:= t:= • Full monotone reduction (the case s=½)

  5. [範例] • F(x,y)=2x2+14xy+24y2 [a,b,c]=[2,14,24] =L[2,10,12]H (L [a, b-2a,c-b+a] H ,for a<c) =LL [2,6,4] HH • 以下s取2.1.½,求得t值均一樣 d+4as=20 因為d+4as >4s2 所以 r≒5 ,t ≒2

  6. [範例] • F(x,y)=14x2+44xy+34y2 [a,b,c]=[14,44,34] =L[14,16,4]H (L [a, b-2a,c-b+a] H ,for a<c) =HL [2,8,4] LH (H [a-b+c,b-2a,c] L ,for a>c ) • d+4as=208 因為d+4as >4s2 所以 r≒15 ,t ≒1

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