Questions tagged [integral-quadratic-forms]
The integral-quadratic-forms tag has no summary.
12 questions
8
votes
2
answers
631
views
Testing if a positive definite quadratic form over $\mathbb Z$ represents 1
I am looking for algorithms that can be used to test if a given positive definite $n$-ary ($n\geq 3$) quadratic form over $\mathbb Z$, whose factorization of the discriminant is known, represents 1.
I ...
- 514
1
vote
2
answers
329
views
Does $\mathbb Z^n$ contain $A_n$?
Are there any positive integer $n > 3$ such that the root lattice $A_n$ is contained in $\mathbb Z^n$?
- 656
4
votes
1
answer
260
views
Computing spinor equivalence for positive definite forms
Given an integral positive-definite rank $n$ quadratic form $f$, one can use the algorithm in Conway and Sloane (Chapter 15, SPLaG) to efficiently determine if the genus of $f$ contains more than one ...
- 323
3
votes
0
answers
136
views
Similar reduced integral matrices
Let $d\geq 1$ be an integer. I'm not assuming anything here about $d$ (it is not necessarily squarefree, for example), but if necessary I'm willing to reduce to the case where $d$ is squarefree and $-...
- 1,856
3
votes
1
answer
131
views
Normalizing trains without sign walking in compartments
Consider the quadratic form $5x^2+6y^2$. This has Conway Sloan $2$-adic symbol $[1^{-1}2^{-1}]_0$. After a sign walk from $1$ to $2$ the symbol becomes $[1^{+1}2^{+1}]_4$. However, there doesn't exist ...
- 2,489
0
votes
0
answers
144
views
p-adic numerical analysis question for a quadratic form
Let $A$ be a symmetric matrix with even diagonal and elements in $\mathbb{Z}_p$ and nonzero discriminant. (Yes, $p$ can be two) and $a$ an nonzero integer. Suppose there exists a solution to $x^{\top}...
- 2,489
5
votes
1
answer
308
views
Are stably equivalent quadratic forms over Z equivalent?
Let $Q_1, Q_2, R$ be quadratic froms over $\mathbb{Z}$ such that $Q_1 \oplus R \cong Q_2 \oplus R$ as quadratic forms. Is it necessary that $Q_1 \cong Q_2$?
I know that by Witt's theorem it is true ...
- 2,198
3
votes
1
answer
401
views
Indefinite Ternary Forms with Square Discriminant
Is there any general theory to find the numbers represented by ternary forms of the type
$q(x,y,z)=ax^2+bx^2-abz^2,$
when $a,b$ are prime?
By doing an internet search, the closest I found was the ...
- 800
3
votes
2
answers
513
views
Paper of Denis Simon on quadratic equations in dimensions 4, 5?
In several places I have come across references to a 2005-6 preprint of Denis Simon entitled
Quadratic equations in dimensions 4, 5, and more
This paper gives fast algorithms to find isotropic ...
- 2,901
0
votes
1
answer
169
views
Lower bounds on the rank of a unimodular lattice, given the binlinear pairing of a subset of basis vectors
I have some questions on lower bounds on the rank of unimodular lattices given the bilinear pairing of a subset of its basis is known.
Let $\Lambda$ be an odd, unimodular matrix of signature $(1,T)$. ...
- 524
3
votes
3
answers
301
views
A question on an involution of $E_8$ lattice
There exits an involution $\iota$ of the $E_8$ lattice such that $(E_8)^{\pm} \cong D_4$, where $(E_8)^{\pm}$ denotes the $\pm$ eigen-lattice of the involution $\iota$. Could someone kindly give me an ...
- 31
2
votes
0
answers
231
views
Orthogonal transformations with trivial spinor norm as product of reflections $r_w$ with $(w,w)=-2$
I'm trying to prove that, for a standard unimodular even lattice $\Lambda$ (by standard I mean that it is direct sum of copies of the hyperbolic plane $U$ and $E_8$) every element of $O^+(\Lambda)$, i....
- 93