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# Solving quadratic form , representation number watch

1. 1. The problem statement, all variables and given/known data

number of solutions of

where , is the associated quadratic from to the matrix ,

where here is positive definite, of rank and even.

(and I think symmetric?)

I am solving for the for the two quadratic forms:

2. Relevant equations

see above

3. The attempt at a solution

Solving with all integer, it is clear that is needed, and then gives .

Now looking at by the same reasoning as above I would have said that we require , and then I' m not sure what to do.

However the solution is:

Must have & , this gives , ,

(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...

2. (Original post by xfootiecrazeesarax)
Usual caveat - not an expert.

Can't say I follow your reasoning for Q, since u,v = 0 reduces your diagonalised form to which has no integer solutions. Unless I'm being dense.

Regarding R.
Since R=1, and is the sum of squares, then each term must be <=1.

So,

For u,v integer this implies u+v=0, not 1 as given in the solutions.

We also need (11/16)v^2 <=1, hence v must be one of -1,0,1.
3. (Original post by ghostwalker)
Usual caveat - not an expert.

Can't say I follow your reasoning for Q, since u,v = 0 reduces your diagonalised form to which has no integer solutions. Unless I'm being dense.

Regarding R.
Since R=1, and is the sum of squares, then each term must be <=1.

So,

For u,v integer this implies u+v=0, not 1 as given in the solutions.

We also need (11/16)v^2 <=1, hence v must be one of -1,0,1.
thank you ! all makes alot more sense now

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