Questions tagged [integral-quadratic-forms]

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4 votes
1 answer
160 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 ...
  • 303
3 votes
0 answers
98 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 $-...
  • 1,489
3 votes
1 answer
100 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 ...
  • 2,341
0 votes
0 answers
102 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}...
  • 2,341
5 votes
1 answer
244 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 ...
  • 1,811
2 votes
1 answer
285 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 ...
  • 790
3 votes
2 answers
403 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 ...
  • 2,609
0 votes
1 answer
104 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 $(1,T)$. ...
3 votes
3 answers
211 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
1 vote
0 answers
171 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....
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