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Results tagged with quadratic-forms
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user 11142
Algebraic and geometric theory of quadratic forms and symmetric bilinear forms, e.g., values attained by quadratic forms, isotropic subspaces, the Witt ring, invariants of quadratic forms, the discriminant and Clifford algebra of a quadratic form, Pfister forms, automorphisms of quadratic forms.
16
votes
Accepted
Do almost all systems of quadratic equations have solutions?
Yes. The magic words are "elimination theory" and "resultant". In essence, the system has a solution unless some determinant (the iterated resultant) vanishes.
- 96.4k
10
votes
Asymptotic formula for sums of four squares?
Note that
$$L(x) = \sum_{n\leq x} r_4(n)$$ is the number of lattice points in the ball of radius $\sqrt{x}.$
It is known that $$L(x) = \frac{\pi^2}2 x^2 + O(x \log(x)).$$ (the error term can be impr …
- 96.4k
7
votes
Listing all solutions to $n = x^2 + y^2 + z^2 $ with integers
The number of representation can be as big as $\sqrt{n},$ so this is a lower bound on the complexity of any algorithm. Now, the algorithm is to iterate through all $k\leq \sqrt{n},$ and try to represe …
- 96.4k
7
votes
Accepted
Simultaneous quadratic equations
The magic word is "elimination". The obvious thing to do is to solve the first equation for $\mu_2$ in terms of $\mu_1.$ You will have two roots. Plug in each one into the second equation, expressing …
- 96.4k
3
votes
A nice necessary and sufficient condition on positive semi-definiteness of a matrix with a s...
I very much doubt that a necessary and sufficient condition exists. For example, for finite element approximations to the Laplacian in a planar domain, you will get the (potentially non-zero) off-diag …
- 96.4k
3
votes
maximizing convex quadratic form over the intersection of unit sphere and positive orthant
The magic words are quadratically constrained quadratic programming. And semidefinite programming.
EDIT I misread the question (it is a convex maximization problem). This is still somewhat tractable …
- 96.4k
0
votes
on the determination of a quadratic form from its isotropy group in char. 2
The theory in general characteristic is covered by these nice notes of Casselman's
- 96.4k