Can anyone explain in more simple terms what rank and signature mean in terms of quadratics in x,y,z in lagrange reductions. My lecturers notes arent too clear and I missed this section. I can do the computation but i dont get what the rank and signature mean? something about the radicals of the squares i assume?
=2(x + 2y − 3z)^2− 3(y − 7z)^2+ 148z^2.
Thus rank = 3 and signature 2 − 1 = 1.
Therank of a quadratic form is the number of squares (here
the rank is 3), and the signature is the number of positive
squares minus the number of negative squares (here the
signature is 2 − 1 = 1.
Could you just explain what it actually means, I'm probably being stupid lol but I'd appreciate it, thanks
If you look at the equation there are THREE terms which are squared. ie . So that is the rank. Signature is the number of these which are positive minus the number of these which are negative. ie .
It's as simple as that really. Your expression can be boiled down to:
Where:
Hence there are two positive squared terms and 1 negative squared term. Signature is 1.
Yes well... thinking intuitively I would assume that the rank and signature for an expression should be the same whether you multiply it out or not. I'm not really familiar with the material, but it seems to be the case that the rank and signature only apply once you have factorised fully.