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Results tagged with quadratic-forms
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user 47795
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.
2
votes
Fricke Klein method for isotropic ternary quadratic forms
The same formal record. If we look for a parameterization not in 2 and in 3 option the problem can be solved quite simply.
For the equation.
$$aX^2+bY^2+cZ^2=dXY+eXZ+fYZ$$
If you know any solution …
- 480
0
votes
Isotropic ternary forms
Not entirely clear where one detail which is not mentioned. Ternary quadratic form always amounts to a Pell equation. For example if you take a fairly simple equation. And set some conditions for the …
- 480
1
vote
Isotropic ternary forms
The mention of another quadratic form. You can use the standard approach.
In equation $$aX^2+bY^2+cZ^2=qXY+dXZ+tYZ$$
$a,b,c,q,d,t$ integer coefficients which specify the conditions of the proble …
- 480
1
vote
Indefinite quadratic form universal over negative integers
Representation of a number we write.
$$aX^2+bY^2=cZ^2+q$$
I think that the only way to record the desired polynomial is to use the solutions of any equation.
$$ax^2+by^2=cz^2$$
Knowing the solutio …
- 480
2
votes
Does the Diophantine equation $(x^2+ay^2)(u^2+bv^2) = p^2+cq^2$ admit a complete solution?
I think that this method of calculation it is necessary to separately draw.
As I have repeatedly said formula in General looks pretty bulky. And still remain questions about the completeness of the …
- 480
-2
votes
Does the Diophantine equation $(x^2+ay^2)(u^2+bv^2) = p^2+cq^2$ admit a complete solution?
Still not clear what you want. If you need a method, the method of finding solutions I have already in a previous post wrote. If regarding the parameterization.
Then for the equation:
$$(x^2+ay^2) …
- 480
-2
votes
Does the Diophantine equation $(x^2+ay^2)(u^2+bv^2) = p^2+cq^2$ admit a complete solution?
Once these equations are referred to - it is necessary to write and solutions.
For the equation:
$$X_1^2+X_2^2=Y_1^2+Y_2^2+Y_3^2$$
Solutions have the form:
$$X_1=t^2+2(p+s-k)t+2k^2+2p^2+4ps-4pk-2s …
- 480
0
votes
On certain solutions of a quadratic form equation
For such quadratic forms.
$$ax^2-bxy+cy^2=a$$
If we consider all the equations of Pell. The resulting factorization of the number. $4a=tq$
And use these equations Pell.
$$p^2-(b^2-4ac)s^2=\pm{t}$ …
- 480
-1
votes
Quadratic diophantine equations and geometry of numbers
It's more of a comment than an answer.
Make such a change.... $x=X+q$
$$w^2-ax^2-by^2+abz^2=1$$
$$w^2-aq^2+abz^2=by^2+aX^2+2aqX+1$$
If we use this difference as Pell's equation. $w^2-aq^2=1$
Equ …
- 480