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Results tagged with quadratic-forms
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user 89064
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.
11
votes
1
answer
783
views
How to describe all integer solutions to $x^2+y^2=3z^2+1$?
The question is in the title. Here is a short motivation. The general quadratic Diophantine equation is
$$
x^TAx+bx+c=0,
$$
where $x$ is a vector of $n$ variables, $A$ is $n \times n$ matrix with inte …
- 4,905
1
vote
0
answers
102
views
Polynomial parametrization for solutions of quadratic Diophantine equations
A previous Mathoverflow question asks if there is an algorithm that would determine all integer solutions to a given quadratic Diophantine equation.
To make this question more formal, we need to agree …
- 4,905
1
vote
1
answer
272
views
Integers representable as binary quadratic forms
It is known that odd prime $p$ can be represented as $p=x^2+y^2$ if and only if $p \equiv 1$ mod $4$, represented as $p=x^2+2y^2$ if and only if $p \equiv 1$ or $3$ mod $8$, represented as $p=x^2+3y^2 …
- 4,905
6
votes
3
answers
603
views
A cubic equation, and integers of the form $a^2+32b^2$
I am trying to determine whether there are any integers $x,y,z$ such that
$$
1+2 x+x^2 y+4 y^2+2 z^2 = 0. \quad\quad\quad (1)
$$
It is clear that $x$ is odd. We can consider this equation as quadratic …
- 4,905