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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.
13
votes
Primes and $x^2+2y^2+4z^2$
You are doing something unusual here. Taking a diagonal form and insisting on the variables being nonnegative is possible but is not really natural. One counts a representation with some $x$ as distin …
- 25.7k
13
votes
When does $axy+byz+czx$ represent all integers?
Just so you know, one of Dickson's students (A. Oppenheim) finished classifying (indefinite) universal ternaries; the final family is $xy - M z^2.$ Page 161 in Modern Elementary Theory of Numbers. You …
- 25.7k
12
votes
Accepted
Primes of the form $x^2+ny^2+mz^2$ and congruences.
Mostly, you should look at a number of items at
(address updated in 2018):
http://zakuski.math.utsa.edu/~kap/
including Dickson_Diagonal_1939.pdf and Kap_Jagy_Schiemann_1997.pdf to begin with.
Now …
- 25.7k
11
votes
A Priori proof that Covering Radius strictly less than $\sqrt 2$ implies class number one
This is the more important case of "even" lattices, where all inner products are integral and all vector norms are even. We follow pages 131-134 in Wolfgang Ebeling, Lattices and Codes, available for …
- 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
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
8
votes
Accepted
Are stably equivalent quadratic forms over Z equivalent?
I would say no; an example is when $R$ is a hyperbolic plane, the relation you demand says merely that $Q_1$ and $Q_2$ are in the same genus. This is in SPLAG, page 378 in the first (1988) edition, se …
- 25.7k
8
votes
History of "no positive definite ternary integral quadratic form is universal"?
Hi, Pete. There are a few observations related to this, not widely known although basic, and that includes your colleague. First, Conway gives a quick proof on page 142 of The Sensual Quadratic Form, …
- 25.7k
8
votes
Accepted
Siegel's Mass Formula for ternary indefinite quadratic forms
Borovoi does not claim what you seem to think. Siegel's formalism discusses, for positive forms, the number of representations of an integer by an entire genus of positive ternary forms, each form wei …
- 25.7k
7
votes
Finding Generators of O( Z^3,x^2 + xy + y^2 - z^2) and integer solutions
I wouldn't call it GL, it is the orthogonal group of the lattice we are discussing. References, as i said, include Lattices and Codes by W. Ebeling, Rational Quadratic Forms by Cassels, these two bein …
- 25.7k
7
votes
Must a ring which admits a Euclidean quadratic form be Euclidean?
In an odd number of variables it is legitimate to add a final product term and make the polynomial invariant under cyclic permutations, as in seven variables and
$$ q( \vec x) = x_1^2+ x_1 x_2 + x_2^2 …
- 25.7k
7
votes
Difference of two sums of two squares
Sure. Demand $a \geq b \geq 0$ as well as $c,d \geq 0.$
Then map
$$ (a,b,c,d) \mapsto (25a+11b+24c+13d, 11a-b+11c, 24a+11b+23c+13d, 13a + 13 c+d) $$
- 25.7k
6
votes
Is there an approach to understanding solution counts to quadratic forms that doesn't involv...
Hi There,
It would help if you gave me some examples of actual positive ternary forms with specific linear dependencies. The main source of dependencies is the Siegel representation formula, which …
- 25.7k
6
votes
Accepted
Primes as the first coefficient of a reduced indefinite quadratic form
I recommend a book by Duncan A. Buell called "Binary Quadratic Forms."
First, we discard the case where $d$ is a square. In such a case the forms represent entire arithmetic progressions. For exampl …
- 25.7k
6
votes
A Priori proof that Covering Radius strictly less than $\sqrt 2$ implies class number one
EDIT, Tuesday, July 26. I have convinced myself, with my own C++ programs, that the sum of five squares does satisfy the "easier" Borcherds-Allcock condition, although it fails Pete L. Clark's criteri …
- 25.7k