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math 2004-04-27 05:10

number theory help
 
a homogeneous polynomial in 2 variables x,y, of degree 2, say, f(x,y)=ax^2+bxy+cy^2 with a,b,c, all integers is called a quadratic form over the integers. the discriminant of the above quadratic form is d=b^2-4ac. a change of variables u,v, is x=αu+βv, y=λu+δv, α, β, λ, δ are all integers, αδ-βλ= ±1. thus you get g(u,v)=f(x,y)=f(αu+βv, λu+δv).


--->show that if f(x,y) is a quadratic form with positive discriminant, then the equation f(x,y)=n may have infinitely many solutions by exhibiting an example.

-thanks.

juergen 2004-05-02 14:29

[QUOTE=math]a homogeneous polynomial in 2 variables x,y, of degree 2, say, f(x,y)=ax^2+bxy+cy^2 with a,b,c, all integers is called a quadratic form over the integers. the discriminant of the above quadratic form is d=b^2-4ac. a change of variables u,v, is x=αu+βv, y=λu+δv, α, β, λ, δ are all integers, αδ-βλ= ±1. thus you get g(u,v)=f(x,y)=f(αu+βv, λu+δv).


--->show that if f(x,y) is a quadratic form with positive discriminant, then the equation f(x,y)=n may have infinitely many solutions by exhibiting an example.

-thanks.[/QUOTE]

isn't this also what is called an ellyptic curve?

xilman 2004-05-02 18:09

[QUOTE=juergen]isn't this also what is called an ellyptic curve?[/QUOTE]
Nope.

An elliptic curve, E(x,y) is a polynomial which is cubic in x and quadratic in y.

The formula given is indeed a quadratic form. The way in which it is phrased makes me almost certain that the original post was an attempt to get assistance with a homework problem.

Paul


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