Mixing
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Dirichlet composition of quadratic forms.
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Lemma: $f(x,y)=ax^2+bxy+cy^2$ and $g(x,y)=a'x^2+b'xy+c'y^2$ have discriminant $D$ and satisfy $gcd(a,a',fracb+b'2)=1$ (b and b' have the same parity, $fracb+b'2$ is an integer). Then there is a unique integer B modulo 2aa' such that $$Bequiv bpmod2a$$ $$Bequiv b'pmod2a'$$ $$B^2 equiv Dpmod4aa'$$.
I know the existence and uniqueness of B of this Lemma, I want to know how to explicitly calculate this constant B given any two quadratic forms. for eg. $f(x,y)=x^2+2xy+3y^2$ and $g(x,y)=4x^2+5xy+6y^2$. Then their comsposition will be $aa'x^2+Bxy+fracB^2-D4aa'y^2$ where $ D (discriminant) = b^2-4ac$.
number-theory modular-arithmetic quadratic-forms
asked 2 days ago
antony james
105
add a comment |Â
up vote
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Lemma: $f(x,y)=ax^2+bxy+cy^2$ and $g(x,y)=a'x^2+b'xy+c'y^2$ have discriminant $D$ and satisfy $gcd(a,a',fracb+b'2)=1$ (b and b' have the same parity, $fracb+b'2$ is an integer). Then there is a unique integer B modulo 2aa' such that $$Bequiv bpmod2a$$ $$Bequiv b'pmod2a'$$ $$B^2 equiv Dpmod4aa'$$.
I know the existence and uniqueness of B of this Lemma, I want to know how to explicitly calculate this constant B given any two quadratic forms. for eg. $f(x,y)=x^2+2xy+3y^2$ and $g(x,y)=4x^2+5xy+6y^2$. Then their comsposition will be $aa'x^2+Bxy+fracB^2-D4aa'y^2$ where $ D (discriminant) = b^2-4ac$.
number-theory modular-arithmetic quadratic-forms
asked 2 days ago
antony james
105
you have given discriminants $-8$ and $-71.$ Nothing to be done. Recommend Buell, Binary Quadratic Forms. Also in Cox, Primes of the form $x^2 + n y^2$ which seems to be your source.
– Will Jagy
2 days ago
@WillJagy Why did you say nothing to be done since both are positive definite quadratic forms since $D<0$ and $a>0$? What if I give you 2 quadratic forms with same D, for eg, $f(x,y)=x^2+xy+4y^2$ and $g(x,y)=2x^2+xy+2y^2$ with $D= -15$.
– antony james
yesterday
With the same $D$ and primitive forms you do get composition. Since the middle coefficients already match most of the work is done. Note that you still need to know about equivalence of forms... if you compose $g$ with itself you get $4x^2 + xy + y^2.$ This is, however, equivalent to $x^2 + xy + 4 y^2.$
– Will Jagy
yesterday
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Lemma: $f(x,y)=ax^2+bxy+cy^2$ and $g(x,y)=a'x^2+b'xy+c'y^2$ have discriminant $D$ and satisfy $gcd(a,a',fracb+b'2)=1$ (b and b' have the same parity, $fracb+b'2$ is an integer). Then there is a unique integer B modulo 2aa' such that $$Bequiv bpmod2a$$ $$Bequiv b'pmod2a'$$ $$B^2 equiv Dpmod4aa'$$.
I know the existence and uniqueness of B of this Lemma, I want to know how to explicitly calculate this constant B given any two quadratic forms. for eg. $f(x,y)=x^2+2xy+3y^2$ and $g(x,y)=4x^2+5xy+6y^2$. Then their comsposition will be $aa'x^2+Bxy+fracB^2-D4aa'y^2$ where $ D (discriminant) = b^2-4ac$.
number-theory modular-arithmetic quadratic-forms
asked 2 days ago
antony james
105
Lemma: $f(x,y)=ax^2+bxy+cy^2$ and $g(x,y)=a'x^2+b'xy+c'y^2$ have discriminant $D$ and satisfy $gcd(a,a',fracb+b'2)=1$ (b and b' have the same parity, $fracb+b'2$ is an integer). Then there is a unique integer B modulo 2aa' such that $$Bequiv bpmod2a$$ $$Bequiv b'pmod2a'$$ $$B^2 equiv Dpmod4aa'$$.
I know the existence and uniqueness of B of this Lemma, I want to know how to explicitly calculate this constant B given any two quadratic forms. for eg. $f(x,y)=x^2+2xy+3y^2$ and $g(x,y)=4x^2+5xy+6y^2$. Then their comsposition will be $aa'x^2+Bxy+fracB^2-D4aa'y^2$ where $ D (discriminant) = b^2-4ac$.
number-theory modular-arithmetic quadratic-forms
asked 2 days ago
antony james
105
edited 2 days ago
asked 2 days ago
antony james
105
asked 2 days ago
antony james
105
asked 2 days ago
antony james
105
105
you have given discriminants $-8$ and $-71.$ Nothing to be done. Recommend Buell, Binary Quadratic Forms. Also in Cox, Primes of the form $x^2 + n y^2$ which seems to be your source.
– Will Jagy
2 days ago
@WillJagy Why did you say nothing to be done since both are positive definite quadratic forms since $D<0$ and $a>0$? What if I give you 2 quadratic forms with same D, for eg, $f(x,y)=x^2+xy+4y^2$ and $g(x,y)=2x^2+xy+2y^2$ with $D= -15$.
– antony james
yesterday
With the same $D$ and primitive forms you do get composition. Since the middle coefficients already match most of the work is done. Note that you still need to know about equivalence of forms... if you compose $g$ with itself you get $4x^2 + xy + y^2.$ This is, however, equivalent to $x^2 + xy + 4 y^2.$
– Will Jagy
yesterday
add a comment |Â
you have given discriminants $-8$ and $-71.$ Nothing to be done. Recommend Buell, Binary Quadratic Forms. Also in Cox, Primes of the form $x^2 + n y^2$ which seems to be your source.
– Will Jagy
2 days ago
@WillJagy Why did you say nothing to be done since both are positive definite quadratic forms since $D<0$ and $a>0$? What if I give you 2 quadratic forms with same D, for eg, $f(x,y)=x^2+xy+4y^2$ and $g(x,y)=2x^2+xy+2y^2$ with $D= -15$.
– antony james
yesterday
With the same $D$ and primitive forms you do get composition. Since the middle coefficients already match most of the work is done. Note that you still need to know about equivalence of forms... if you compose $g$ with itself you get $4x^2 + xy + y^2.$ This is, however, equivalent to $x^2 + xy + 4 y^2.$
– Will Jagy
yesterday
you have given discriminants $-8$ and $-71.$ Nothing to be done. Recommend Buell, Binary Quadratic Forms. Also in Cox, Primes of the form $x^2 + n y^2$ which seems to be your source.
– Will Jagy
2 days ago
you have given discriminants $-8$ and $-71.$ Nothing to be done. Recommend Buell, Binary Quadratic Forms. Also in Cox, Primes of the form $x^2 + n y^2$ which seems to be your source.
– Will Jagy
2 days ago
@WillJagy Why did you say nothing to be done since both are positive definite quadratic forms since $D<0$ and $a>0$? What if I give you 2 quadratic forms with same D, for eg, $f(x,y)=x^2+xy+4y^2$ and $g(x,y)=2x^2+xy+2y^2$ with $D= -15$.
– antony james
yesterday
@WillJagy Why did you say nothing to be done since both are positive definite quadratic forms since $D<0$ and $a>0$? What if I give you 2 quadratic forms with same D, for eg, $f(x,y)=x^2+xy+4y^2$ and $g(x,y)=2x^2+xy+2y^2$ with $D= -15$.
– antony james
yesterday
With the same $D$ and primitive forms you do get composition. Since the middle coefficients already match most of the work is done. Note that you still need to know about equivalence of forms... if you compose $g$ with itself you get $4x^2 + xy + y^2.$ This is, however, equivalent to $x^2 + xy + 4 y^2.$
– Will Jagy
yesterday
With the same $D$ and primitive forms you do get composition. Since the middle coefficients already match most of the work is done. Note that you still need to know about equivalence of forms... if you compose $g$ with itself you get $4x^2 + xy + y^2.$ This is, however, equivalent to $x^2 + xy + 4 y^2.$
– Will Jagy
yesterday
add a comment |Â
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you have given discriminants $-8$ and $-71.$ Nothing to be done. Recommend Buell, Binary Quadratic Forms. Also in Cox, Primes of the form $x^2 + n y^2$ which seems to be your source.
– Will Jagy
2 days ago
@WillJagy Why did you say nothing to be done since both are positive definite quadratic forms since $D<0$ and $a>0$? What if I give you 2 quadratic forms with same D, for eg, $f(x,y)=x^2+xy+4y^2$ and $g(x,y)=2x^2+xy+2y^2$ with $D= -15$.
– antony james
yesterday
With the same $D$ and primitive forms you do get composition. Since the middle coefficients already match most of the work is done. Note that you still need to know about equivalence of forms... if you compose $g$ with itself you get $4x^2 + xy + y^2.$ This is, however, equivalent to $x^2 + xy + 4 y^2.$
– Will Jagy
yesterday