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Results tagged with quadratic-forms
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user 3324
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
adelic quadratic forms
EDIT: I have never seen this one, but it is said to cover some of the same ground as Kneser (1961): A. Weil, Sur la theorie des formes quadratiques (1962).
ORIGINAL: I did not notice this earlier... …
- 25.7k
7
votes
Difference of two sums of two squares
Sure. Demand $a \geq b \geq 0$ as well as $c,d \geq 0.$
Then map
$$ (a,b,c,d) \mapsto (25a+11b+24c+13d, 11a-b+11c, 24a+11b+23c+13d, 13a + 13 c+d) $$
- 25.7k
9
votes
Accepted
Spin Representation
It was a longstanding problem to decide equivalence of indefinite forms. The showpiece of the spinor genus is that, for indefinite forms in at least three variables over the rational integers, the spi …
- 25.7k
1
vote
Isotropic ternary forms
Thursday: here is an example I proved in full detail, that illustrates the use of the mappings in one direction, along with the possible intricacy of the difference between finding all rational null v …
- 25.7k
0
votes
Isotropic ternary forms
Wednesday morning, June 3.
I will be back home tonight, probably tomorrow I can provide the specific example requested.
The fact in Cassels is correct. It goes back to the seminal book by Fricke and …
- 25.7k
0
votes
Isotropic ternary forms
Ternary Quadratic Forms and Norms edited by Olga Taussky (1982). Pages 5-30 is William Plesken, Automorphs of Ternary Quadratic Forms. The word automorph is one of the traditional terms for what would …
- 25.7k
2
votes
Fricke Klein method for isotropic ternary quadratic forms
Another example, with three of the "recipes" required. All these problems I have checked needed either $2^k$ or $3 \cdot 2^k$ such $R$ matrices.
jagy@phobeusjunior:~$ ./isotropy 1 50
A = 1 B …
2
votes
Fricke Klein method for isotropic ternary quadratic forms
Here's another one I really did prove, $x^2 + y^2 + z^2 - 5(yz+zx+xy)=0.$
This one requires just one recipe,
$$
\left(
\begin{array}{c}
5 u^2 + 9 uv + 3 v^2 \\
3 u^2 - 3 u v - v^2 \\
- u^2 + uv + 5 v …
1
vote
Counting integral points on a diagonal conic
I don't see that this need be tied to Pell's equation; I am taking your $b,c$ squarefree for ease. I have also picked the product so that there are no imprimitive forms of this discriminant. In compar …
- 25.7k
1
vote
A description of the isometry group $O(U\oplus E_8)$?
Yes. See Lattices and Codes by Wolgang Ebeling. In the second edition, this is Exercise 4.4 on page 134. I do not believe this information was in the first edition; further, there is a third edition n …
- 25.7k
2
votes
Accepted
Representation of rationals by quadratic form
Lemma B (for binary) (completing the square and a few cases to check): Given integers, $f(x,y) = a x^2 + b x y + c y^2 ,$ with discriminant $\Delta = b^2 - 4 a c $ not a square. Given a (always positi …
- 25.7k
2
votes
Positive ternary quadratic forms in the same genus that represent the same numbers
Spent a month checking, this is what I suspect is the complete list of 'sporadic' or 'exceptional' pairs. No restriction that they be in the same genus or have the same discriminant. I was able to ch …
- 25.7k
1
vote
The quadratic form $x^2+ny^2$ via prime factors
Why not. My answer at https://math.stackexchange.com/questions/229201/the-quadratic-form-x2-ny2-via-prime-factors/229270#229270
- 25.7k
1
vote
Accepted
Gram matrix modulo 4
I use the 2-adic decompositions for various tasks. I can't say I know what would be useful for you, but let me call your attention to page 141, Lemma 4.3. This refers back to Lemma 5.2 on page 123, (p …
- 25.7k
8
votes
Can a positive binary quadratic form represent 14 consecutive numbers?
I'm making this an answer to make it more visible, a suggestion of Pete L. Clark that seems correct to me.
Wadim Zudilin has been running a computer program of mine on a fast computer. Today we found …
- 25.7k