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5
votes
1answer
117 views

Stabilizers of pairs of ternary quadratic forms

Let $A,B$ be two ternary quadratic forms with real coefficients, given by symmetric matrices $$\displaystyle 2A = \begin{pmatrix} 2a_{11} & a_{12} & a_{13} \\ a_{12} & 2a_{22} & a_{23}...
2
votes
1answer
87 views

Number of vectors of fixed norm

Let $P$ and $Q$ be two even, unimodular, positive definite quadratic forms of rank $n$. Let $r_{k}(P)$ be the number of vectors of norm $k$, in symbols: $$ r_k(P)=\textrm{cardinality of }\{v\in \...
0
votes
0answers
54 views

quadratic forms in number theory and prime numbers [duplicate]

what are the prime numbers represented by $x^2-2y^2$ ? I have seen the claim that any prime congruent to 1 or -1 modulo 8 is of this form. True ? Reference ?
5
votes
1answer
209 views

Constructing groups of Type E7 with certain Tits Index

In a new survey on $E_8$, namely Skip Garibaldi - E8 the most exceptional group , the author gives an example (Example 8.4., page 15) on how to construct a group of type E8 with a prescribed Tits-...
1
vote
1answer
117 views

cubic forms and finiteness of $k^*/(k^*)^3$

In some recent computation I came across certain cubic forms and was wondering about analogue of following result for quadratic forms. If $k^*/(k^*)^2$ is finite then there are only finitely many ...
2
votes
1answer
295 views

Irreducible variety

I asked a similar question at MSE, as the question seemed quite basic to me, but did not get any hint in 24 hours, except for one upvote for the question itself. I still think I am stuck with some ...
1
vote
0answers
46 views

Are those $2$ quadratic forms congruent over $\mathbb{Z}[1/q]$

Let $q$ be a natural number (the first cases of interest being $q = 10,12$ or $15$), and let $n = q^2+q+1$. Also, let $I_n$ be the $n\times n$ identity matrix, and let $A_n$ be the $n\times n$ ...
3
votes
0answers
86 views

Self-dual vertex algebras

Let $(V,Y)$ be a self-dual conformal vertex algebra. For instance, it could be the vertex algebra associated to a positive definite, even, unimodular quadratic form. I look for a formula to compute $$ ...
3
votes
1answer
113 views

Do all non-degenerate quadratic forms come from positive even lattices?

Let $(G,+)$ be a finite Abelian group. We say $q\colon G\to \mathbb{T}$ is a non-degenerated quadratic form, if $q(-a)=q(a)$ and the symmetric function $$ b(g,h) =q(g+h)q(g)^{-1}q(h)^{-1} $$ is a non-...
0
votes
0answers
46 views

Solving a system of quadratic equations over a subspace

Let $A \subseteq \mathbb{R}^{n}$ be subspace of dimension $d$, parametrized as $A=\{x|Vx=0\}$, where $V$ is a suitable $d \times n$ matrix. Now a system of $m$ quadratic equations $$x^{T}Q_{i}x=a_{i},...
1
vote
1answer
67 views

Positive solutions to simultaneous real quadratic equations

I have a system of $n$ quadratic equations with $n$ unknowns. It can be written as $diag(x)Ax=1$ $x$ is an $n$-vector, $A$ is $n\times n$, real, symmetric and positive definite, the diagonal ...
8
votes
2answers
354 views

What's in the genus of the cubic lattice?

I'll write $\mathbf{Z}^n$ for the integral quadratic form $x_1^2 + \cdots + x_n^2$. For which values of $n$ is $\mathbf{Z}^n$ unique in its genus, i.e. isolated in Kneser's graph? In particular can ...
5
votes
1answer
154 views

Pythagorean number in Artin's theorem on nonnegative rational fractions

Emil Artin's theorem on nonnegative rational fractions says that a rational fraction $Q$ with $n$ variables with real coefficients which is non-negative on $\mathbb R^n$ is a sum of squares of ...
12
votes
2answers
720 views

Upper bound on answer for Pell equation

A user on MSE, @martin , asked http://math.stackexchange.com/questions/1611411/pell-equations-upper-bound about an upper bound for $x$ in $x^2 - p y^2 = 1,$ when $p$ is prime. I checked, it appears ...
0
votes
1answer
142 views

Why is the Fano variety of lines on a smooth three-dimensional quadric isomorphic to $\mathbb{P}^3$?

Let $Q \subset \mathbb{P}^4$ be a smooth three-dimensional quadric over an algebraically closed field $k$ ($\mathrm{char}(k) \neq 2$) and let $F$ be the Fano variety of lines on $Q$. In "Iskovskikh ...
1
vote
0answers
40 views

Is this QCQP convex or nonconvex

\begin{equation} \begin{split} \min_{x\in \mathbb{R}^n}\:f(x)=(1/2)x^{T}Q_0x+c_0^T x \end{split} \end{equation} s.t. $$ g_i(x)=\frac{1}{2}x^T Q_ix-lmax_i\leq0,i\in\{1,...,m/2\} $$ $$ g_i(x)=\frac{...
2
votes
2answers
237 views

A quadratic Diophantine equation

Is the following statement true? If yes, how can find the solutions? The equation $$3x^2+8xy+7y^2\equiv-1\pmod p$$ has an integral solution for every prime $p>5$.
0
votes
1answer
70 views

$p$-adic orthogonal groups in four variables

Let $p>2$ be prime. By the classification of quadratic forms, there are $8$ pairwise non-equivalent isotropic orthogonal groups in $4$ variables. Is there a concrete classification of orthogonal ...
2
votes
1answer
160 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 ...
2
votes
3answers
437 views

Fricke Klein method for isotropic ternary quadratic forms

Preface: the most natural way to take one isotropic vector for an indefinite quadratic form and find others is to use stereographic projection. This gives a parametrization in the same $n$ variables ...
9
votes
1answer
249 views

Fundamental units with norm $-1$ in real quadratic fields

If we have distinct primes $p \equiv q \equiv 1 \pmod 4,$ with Legendre $(p|q) = (q|p) = -1,$ there is a solution to $u^2 - pq v^2 = -1$ in integers and the fundamental unit of $O_{\mathbb Q(\sqrt{pq})...
2
votes
0answers
105 views

probability of a quadratic form being nonnegative at a random point

I am looking for good and explicit lower and upper bounds on the probability that $x^TGx\ge 0$, where $G$ is a symmetric matrix with zero trace and $x$ is a vector whose components are independent ...
1
vote
0answers
91 views

Bounds on the number of zeros of a quadratic form

Let $Q(x_1, \dots, x_n)$ be a non-degenerate indefinite quadratic form with integer coefficients. Let $N(Q,T)$ be the set of vectors $x=(x_1, \dots, x_n) \in {\mathbb Z}^n$ such that $|x|<T$ and $Q(...
5
votes
1answer
217 views

The number of integral solutions to $x^2+y^2-az^2=0$

I think this must be well-known (and probably not hard to prove either), but I cannot find a reference: for a (positive) rational number $a$, the number of integral solutions to the equation $$ x^2+y^...
3
votes
0answers
164 views

Distance between quadratic forms

In notes here http://math.univ-lyon1.fr/homes-www/gille/prenotes/lens.pdf on page $2$ a formulation of distance between two positive quadratic form $[q],[q']$ is given by $$d([q],[q'])=\frac{\sup_{x\...
3
votes
1answer
155 views

Bai and Silverstein's “Lemma on Quadratic Forms” - question about the constant $C_p$

In the book "Spectral Analysis of Large Dimensional Random Matrices" by Bai and Silverstein, there is the following lemma: Lemma B.26 (pg. 530) Let $A=(a_{ij})$ be an $n\times n$ nonrandom matrix ...
4
votes
0answers
91 views

Norm variety for n=5, p=2 not isomorphic to a quadric

In the paper "Motivic construction of cohomological invariants", the author displays a list of known norm varieties for several $n,p$ on page $11$. For $p=2, n=5$ it says that a norm variety is given ...
4
votes
1answer
123 views

optimal bound in diophantine representation question

Given that, with integers $t \geq 1$ and $q \geq 3,$ there are solutions to $$ x^2 - q x y + y^2 = - t q $$ with integers $x,y \geq 1,$ I was able to show that $$ q \leq 1 + \frac{324}{25} t^2. $$ ...
11
votes
2answers
374 views

“Pythagoras number” for integral matrices

It is classically known that every positive integer is a sum of at most four squares of integers, i.e. every sum of squares of integers is a sum of four squares of integers. Now consider a symmetric $...
4
votes
1answer
266 views

Listing all solutions to $n = x^2 + y^2 + z^2 $ with integers

I would like to list all ways of writing $n$ as the sum of 3 squares. This is slightly different from finding just one: Is there an algorithm for writing a number as a sum of three squares? ...
7
votes
1answer
313 views

A vanishing condition for cup products in Galois cohomology

Let $k$ be a field of characteristic $\neq 2$. For a non-zero element $a \in k^*$, let us write $[a] \in H^1(k,\mathbb{Z}/2)$ for the Galois cohomology class corresponding to the quadratic extension $...
7
votes
0answers
237 views

Question on some coverings of the euclidean space

Let $L$ be a maximal integral lattice in the euclidean $(\mathbf R^{8m},q)$ (thus the associated bilinear form $b(u,v)=q(u+v)-q(u)-q(v)$, once restricted to $L$, takes values in $2\mathbf Z$ and has ...
2
votes
0answers
151 views

Proof of Merkurjev's Theorem in “The Algebraic and Geometric Theory of Quadratic Forms”

I just have a little question about the above mentioned proof. I'm thinking for days, but I'm still not getting it. For those who have the book (or want to look it up via google books etc.), it's the ...
12
votes
2answers
658 views

Many representations as a sum of three squares

Let $r_3(n) = \left|\{(a,b,c)\in {\mathbb Z}^3 :\, a^2+b^2+c^2=n \}\right|$. I am looking for the maximum asymptotic size of $r_3(n)$. That is, the maximum number of representations that a number can ...
1
vote
0answers
75 views

Statements generalizing representability of integers by binary quadratic forms to $n$-variable higher homogeneous forms?

Representing integers through the theory of binary quadratic forms is a well studied topic. We know that given $a,b,c\in\Bbb N$, based on discrimant $b^2-4ac$, we can study the representability of ...
12
votes
3answers
511 views

2-dimensional sublattices with all vectors having very big square (in absolute value)

QUESTION: Let $\Lambda\times\Lambda\rightarrow {\Bbb Z}$ be a lattice, that is, ${\Bbb Z}^n$ with a non-degenerate integer quadratic form, not definite, not necessarily unimodular, $n>2$. I want ...
5
votes
1answer
118 views

Isotropic subvarieties of $ V(x_1^2+\dots+x_n^2-t_1^2-\dots-t_n^2) $

Let $ K $ be a field, $ \operatorname{char} K = 0, $ let $ Q = x_1^2+\dots+x_n^2-t_1^2-\dots-t_n^2 $ be the totally isotropic form, then the maximal linear isotropic subspaces of $ V(Q) \subset K^{...
1
vote
1answer
82 views

how to determine a biquadratic form is positive-definite

A biquadratic form $\sum_{i,j,k,l}b_{i,j,k,l}x_{i}x_{j}y_{k}y_{l}$, how to determine whether it is positive-definite? A necessary and sufficient condition? In fact, I have a matrix $B=\sum_{1\leq i,...
4
votes
2answers
234 views

Counting fundamental units of real quadratic fields

For a given real quadratic field $K$, the group of units of its ring of integers is $\mathcal{O}_K^{\times}\cong(\pm1)\times \mathbb{Z}$ by the Dirichlet unit theorem. For each $\mathcal{O}_K$, pick ...
5
votes
4answers
217 views

From a (not positive definite) Gram matrix to a (Kac-Moody) Cartan matrix

Suppose I am given a symmetric matrix $G_{ij}$ with $G_{ii} = 2$. Can I always find an invertible integer matrix $S$ such that $(S^T G S)_{ii}=2$ and $(S^T G S)_{ij} \leq 0$ for $i \neq j$? Is there a ...
2
votes
6answers
667 views

Isotropic ternary forms

It is well known that some questions about isotropic ternary forms reduces to the study of the special case $f_0(X)=xz-y^2, X=(x,y,z)$, see page 301 of Cassel's "Rational quadratic forms" (Dover, 2008)...
2
votes
1answer
90 views

Real slices of Minkowski space, using a complex quadratic form

Ordinary Minkowski space is $\mathbb{R}^{3,1}:=(\mathbb{R}^4,\phi)$ where $\phi:\mathbb{R}^4\rightarrow\mathbb{R}$ is a quadratic form of signature $(3,1)$. Lying within this is a hyperboloid model ...
4
votes
1answer
185 views

Numbers represented by inhomogeneous forms

I have a family of Diophantine equations that I am trying to solve, and I am trying to figure out what methods could be used to prove existence of solutions. Unfortunately, the equations are ...
5
votes
3answers
388 views

genus 2 Siegel theta series of 3-dimensional lattices

Let $(V,f)$ be a $3$-dimensional positive definite quadratic space over $\mathbf Q$. Let $G(V)$ be a set of representatives of the isometry classes of maximal integral lattices on $V$. To an ...
3
votes
2answers
207 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 ...
4
votes
2answers
184 views

Orbits of the maximal compact subgroup on the light cone for $p$-adic groups

It is known that if $Q$ is an indefinite non-degenerate quadratic form on $ \mathbb{R}^n$ with $n \ge 3$, then any maximal compact subgroup $K$ of the orthogonal group $SO(Q)$ acts transitively on the ...
8
votes
3answers
370 views

Deciding a quadratic diophantine equation

Given $a,b\in\Bbb Q_+$, is there an easy way to decide if $$S_{a,b}=\{(x,y)\in\Bbb Z^2:ax^2 + by^2=1\}=\emptyset?$$ I am more interested in seeing if there is a quick way to test for case when ...
4
votes
1answer
224 views

When do two non-degenerate quadratic forms give rise to isomorphic Lie algebras?

Let $V$ be a vector space over some number field $k$. (I'm fine with $\mathbb{Q}$.) Let $\phi \colon V \to k$ be a non-degenerate quadratic form. Associated with $\phi$ is the orthogonal group $\...
0
votes
1answer
110 views

How to minimize this sparse quadratic function?

There is a problem when I'm reading a paper. Equation: $min_p|p-p^*|^2+\alpha |R(p)|^2 + \beta |D(p)-\delta|^2$, where $p, p^*, R(p), D(p), \delta$ are all $M\times N$ matrices, and $p^*, R(), D(), ...
2
votes
1answer
155 views

Involution on the components of a group algebra

If $G$ is a finite group and $k$ a field, there is a canonical involution (ie an involutive anti-automorphism) $\sigma$ on $k[G]$ induced by $g\mapsto g^{-1}$. Given that the center of $k[G]$ has $(\...