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Fast Reduction and Composition of Binary Quadratic Forms

ISSAC, 1991. Arnold Schonhage

田錦燕95/02/17

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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]

cb

ba

21

21

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• 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”

Monotone reduction of positive forms

1011

1101

HtransposeitsandL

,,

1det,,,,,, 22 MZMMMcba t

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• [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)

asd 4

arb 2/)(

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• 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≒ ≒

[ 範例 ]

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• 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≒ ≒

[ 範例 ]