Computing representation number quad forms

In summary, the conversation discusses the calculation of the number of solutions for a given quadratic form, denoted by ##r_A(n)##, where ##A[x]=x^tAx## and A is a positive definite and symmetric matrix of rank m. Two specific quadratic forms, Q and R, are given and their diagonalized forms are calculated. When solving for Q=1, it is concluded that u=v=0 and x,y=±1. However, the solution for R=1 involves the conditions u+v=1 and |v|=0,±1, which gives the solutions of (±1,0,0,0), (±1,0,-1,1), and (0,0,-1
  • #1
binbagsss
1,254
11

Homework Statement



## r_{A} (n) = ## number of solutions of ## { \vec{x} \in Z^{m} ; A[\vec{x}] =n} ##
where ##A[x]= x^t A x ##, is the associated quadratic from to the matrix ##A##, where here ##A## is positive definite, of rank ##m## and even. (and I think symmetric?)

I am solving for the ##r_{A}(1) ## for the two quadratic forms:

##Q(x,y,uv)= 2(x^2+y^2+u^2+v^2)+2xu+xv+yu-2yv##
##R(x,y,uv)=x^2+4(y^2+u^2+v^2)+xu+4yu+3yv+7uv##

Homework Equations



see above

The Attempt at a Solution


[/B]
diagonalized these read:

##Q=2(x+u/2+v/4)^2+2(y-v/2+u/4)^2+11u^2/8 + 11v^2/8 ##
##R=(x+1/2u)^2+4(y+1/2u+3/8v)^2+11/4(u+v)^2+11/16v^2 ##

Solving ##Q=1## with all ##x,y,u,v## integer, it is clear that ##u,v=0## is needed, and then ##x,y=\pm 1 ## gives ##r_{Q}(1)=4##.

Now looking at ## r_{R}(1) ## by the same reasoning as above I would have said that we require ##v=0## , and then I' m not sure what to do.

However the solution is:

Must have ##u+v=1 ## & ##|v|=0,\pm 1 ##, this gives ## \pm(1,0,0,0)##, ##\pm(1,0,-1,1) ## , ##\pm(0,0,-1,1)##

(the symbol that I interpreted as '&' in the solutions is a bit smudged, so looking at the solutions I'm not sure that this is supposed to be a 'or'? )

Either way, I'm really confused, unsure where these conditions come from, how to think about this in a logical way...

Many thanks for your help in advance.
 
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  • #2
binbagsss said:

Homework Statement



## r_{A} (n) = ## number of solutions of ## { \vec{x} \in Z^{m} ; A[\vec{x}] =n} ##
where ##A[x]= x^t A x ##, is the associated quadratic from to the matrix ##A##, where here ##A## is positive definite, of rank ##m## and even. (and I think symmetric?)

I am solving for the ##r_{A}(1) ## for the two quadratic forms:

##Q(x,y,uv)= 2(x^2+y^2+u^2+v^2)+2xu+xv+yu-2yv##
##R(x,y,uv)=x^2+4(y^2+u^2+v^2)+xu+4yu+3yv+7uv##

Homework Equations



see above

The Attempt at a Solution


[/B]
diagonalized these read:

##Q=2(x+u/2+v/4)^2+2(y-v/2+u/4)^2+11u^2/8 + 11v^2/8 ##
##R=(x+1/2u)^2+4(y+1/2u+3/8v)^2+11/4(u+v)^2+11/16v^2 ##

Solving ##Q=1## with all ##x,y,u,v## integer, it is clear that ##u,v=0## is needed, and then ##x,y=\pm 1 ## gives ##r_{Q}(1)=4##.

Now looking at ## r_{R}(1) ## by the same reasoning as above I would have said that we require ##v=0## , and then I' m not sure what to do.

However the solution is:

Must have ##u+v=1 ## & ##|v|=0,\pm 1 ##, this gives ## \pm(1,0,0,0)##, ##\pm(1,0,-1,1) ## , ##\pm(0,0,-1,1)##

(the symbol that I interpreted as '&' in the solutions is a bit smudged, so looking at the solutions I'm not sure that this is supposed to be a 'or'? )

Either way, I'm really confused, unsure where these conditions come from, how to think about this in a logical way...

Many thanks for your help in advance.
Are you sure there are no typos? I calculated ##R(0,0,-1,1) = \frac{61}{64}##. It could also help to multiply the equations by ##8##, resp. ##16##, which would make the comparisons easier.
 
  • #3
binbagsss said:
Solving Q=1 with all x,y,u,v integer, it is clear that u,v=0 is needed
True, but not quite trivial.
binbagsss said:
and then x,y=±1
How do you get that? Don't those give Q=4?
binbagsss said:
by the same reasoning as above I would have said that we require v=0
Then you would be wrong. I said it wasn't quite trivial.
As fresh_42 writes, it will help to multiply through the equations to eliminate the fractions.
 
  • #4
haruspex said:
True, but not quite trivial.

How do you get that? Don't those give Q=4?

Then you would be wrong. I said it wasn't quite trivial.
.

oh right, the reason is that each term needs to be ##\leq 1 ## ?
 
  • #5
binbagsss said:
oh right, the reason is that each term needs to be ##\leq 1 ## ?
Each term must be no more than 1, but I cannot say whether that is the "reason" you were wrong to conclude v=0 since I do not know how you concluded it. All I can say is that there is a solution with v not 0.
 

What is a computing representation number quad form?

A computing representation number quad form is a mathematical concept used in number theory and algebraic geometry. It is a way of representing a quadratic form as a sum of squares of integers.

How is a computing representation number quad form calculated?

The calculation of a computing representation number quad form involves finding the number of ways a given quadratic form can be written as a sum of squares of integers. This can be done using various algorithms and techniques in number theory and algebraic geometry.

What is the significance of computing representation number quad forms?

Computing representation number quad forms are significant in many areas of mathematics, including number theory, algebraic geometry, and cryptography. They have applications in solving problems related to Diophantine equations, quadratic forms, and integer factorization.

What are some examples of computing representation number quad forms?

One example of a computing representation number quad form is the sum of squares of two integers, such as x^2 + y^2. Another example is the sum of squares of three integers, such as x^2 + y^2 + z^2. These forms can be used to represent various mathematical concepts and solve problems in number theory and algebraic geometry.

How are computing representation number quad forms used in cryptography?

In cryptography, computing representation number quad forms are used in the construction of cryptographic systems, such as the RSA algorithm. They are also used in the analysis of the security of cryptographic systems, specifically in the study of the hardness of factoring large integers.

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