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单词 IntegralBinaryQuadraticForms
释义

integral binary quadratic forms


An integral binary quadratic form is a quadratic formMathworldPlanetmath (q.v.) in two variables over , i.e. a polynomial

F(x,y)=ax2+bxy+cy2,a,b,c

F is said to be primitivePlanetmathPlanetmath if its coefficients are relatively prime, i.e. gcd(a,b,c)=1, and is said to represent an integer n if there are r,s such that F(r,s)=n. If gcd(r,s)=1, F is said to represent n properly. The theory of integral binary quadratic forms was developed by Gauss, Lagrange, and Legendre.

In what follows, “form” means “integral binary quadratic form”.

Following the article on quadratic forms, two such forms F(x,y) and G(x,y) are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath if there is a matrix MGL(2,) such that

G(x,y)=F(M(x,y)T)

Matrices in GL(2,) are matrices with determinant ±1. So if α,β,γ,δ and

det(αγβδ)=±1

then if

G(x,y)=F(αx+βy,γx+δy)

it follows that G is equivalent to F. If MSL(2,) (i.e. detM=1), we say that F and G are properly equivalent, written FG; otherwise, they are improperly equivalent.

Note that while both equivalence and proper equivalence are equivalence relations, improper equivalence is not. For if F is improperly equivalent to G and G is improperly equivalent to H, then the productPlanetmathPlanetmath of the transformation matrices has determinant 1, so that F is properly equivalent to H. Since proper equivalence is an equivalence relation, we will say that two forms are in the same class if they are properly equivalent.

GL(2,) is generated as a multiplicative groupMathworldPlanetmath by the two matrices

(1011),(0110)

so in particular we see that we can construct all equivalence transformationsMathworldPlanetmath by composing the following three transformations:

TransformationMatrixDeterminant
(x,y)(y,x)(0110)-1
(x,y)(y,-x)(0-110)1
(x,y)(x+dy,y)(10d1)1

Example: Let F(x,y)=x2+xy+6y2, G(x,y)=82x2+51xy+8y2. Then

G(x,y)=F(αx+βy,γx+δy)=F(4x+y,3x+y)

so

(αγβδ)=(4311)

The transformations to map F into G are

x2+xy+6y2(x,y)(x+y,y)x2+3xy+8y2(x,y)(y,x)8x2+3xy+y2(x,y)(x+3y,y)8x2+51xy+82y2(x,y)(y,x)82x2+51xy+8y2

and

(4311)=(0110)(1031)(0110)(1011)
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更新时间:2025/5/4 11:55:07