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Old 2004-04-27, 05:10   #1
math
 

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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= ยฑ1. 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.
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Old 2004-05-02, 14:29   #2
juergen
 
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Quote:
Originally Posted by 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= ยฑ1. 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.
isn't this also what is called an ellyptic curve?
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Old 2004-05-02, 18:09   #3
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Quote:
Originally Posted by juergen
isn't this also what is called an ellyptic curve?
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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