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On the automorphism group of binary quadratic forms

This question is a continuation of the following two questions: Discriminants of indefinite integral binary quadratic forms admitting 3 or 6 torsion. On certain solutions of a quadratic form ...
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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-...
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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 $$x^{T}Q_{i}x=a_{i},... 1answer 68 views Positive solutions to simultaneous real quadratic equations I have a system of n quadratic equations with n unknowns. It can be written as diag(x)Ax=1 x is an n-vector, A is n\times n, real, symmetric and positive definite, the diagonal ... 2answers 355 views What's in the genus of the cubic lattice? I'll write \mathbf{Z}^n for the integral quadratic form x_1^2 + \cdots + x_n^2. For which values of n is \mathbf{Z}^n unique in its genus, i.e. isolated in Kneser's graph? In particular can ... 1answer 154 views Pythagorean number in Artin's theorem on nonnegative rational fractions Emil Artin's theorem on nonnegative rational fractions says that a rational fraction Q with n variables with real coefficients which is non-negative on \mathbb R^n is a sum of squares of ... 2answers 721 views Upper bound on answer for Pell equation A user on MSE, @martin , asked http://math.stackexchange.com/questions/1611411/pell-equations-upper-bound about an upper bound for x in x^2 - p y^2 = 1, when p is prime. I checked, it appears ... 1answer 149 views Why is the Fano variety of lines on a smooth three-dimensional quadric isomorphic to \mathbb{P}^3? Let Q \subset \mathbb{P}^4 be a smooth three-dimensional quadric over an algebraically closed field k (\mathrm{char}(k) \neq 2) and let F be the Fano variety of lines on Q. In "Iskovskikh ... 0answers 42 views Is this QCQP convex or nonconvex $$\begin{split} \min_{x\in \mathbb{R}^n}\:f(x)=(1/2)x^{T}Q_0x+c_0^T x \end{split}$$ s.t.$$ g_i(x)=\frac{1}{2}x^T Q_ix-lmax_i\leq0,i\in\{1,...,m/2\}  g_i(x)=\frac{...
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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 $p>5$.
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$p$-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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $$q \leq 1 + \frac{324}{25} t^2.$$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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)...
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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 ...
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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 ...
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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 ...
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 ...