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Questions tagged [quadratic-forms]

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.

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What are the Voronoi cones in 4 variables?

Question: What are the top dimensional cones of the 2nd Voronoi decomposition of the space of positive definite forms in $4$ variables? The 2nd Voronoi decomposition of the cone of positive definite ...
Numma1Stunna's user avatar
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pavings and quadratic forms

Hi, let $L$ be a lattice isomorphic to $\mathbb{Z}^r$ for some positive integer $r$ and $E=L\otimes \mathbb{R}$. An integral paving in $E$ is a set $\Sigma$ of integral polytopes (the vertices are ...
melan's user avatar
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Quadratic forms over symmetric $3\times3$ matrices

This question is motivated by a technique called compensated compactness (CC), which was elaborated by L. Tartar and F. Murat for the analysis of PDEs. The foreword of CC is a problem about quadratic ...
Denis Serre's user avatar
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Spinor norm interpretation for a specific orthogonal group?

I hate to admit I don't know the answer to this but a referee has asked me about it in a paper of mine so here goes. Let F be a p-adic field with p odd. Let Q by a quaternary quadratic form over F of ...
mander's user avatar
  • 234
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Sets represented by integral quadratic forms

Let $f(x) = x^\intercal A x$ be a positive definite integral quadratic form on $d$ variables. A positive integer $n$ is said to be represented by $f$ if $f(x) = n$ for some $x \in \mathbb Z^d$. A set ...
Andre Kornell's user avatar
2 votes
1 answer
196 views

Estimate for the operator $A A_D^{-1}$

Let $O\subset\mathbb{R}^d$ be a bounded domain of the class $C^{1,1}$ (or $C^2$ for simplicity). Let the operator $A_D$ be formally given by the differential expression $A=-\operatorname{div}g(x)\...
Yulia Meshkova's user avatar
2 votes
0 answers
55 views

Quadratic surjective map between spheres

The quadratic function $f:\mathbb R^4\to\mathbb R^3$ $$f(a,b,c,d)=\begin{bmatrix} 2(ac + bd)&2(ad - bc)&a^2 + b^2 - c^2 - d^2\end{bmatrix}$$ surjectively maps the sphere $S^3$ to the sphere $S^...
rikhavshah's user avatar
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Are the following two characterisations of symplectic modules, using the language of form rings, the same?

Page 205 of the book Classical Groups and Algebraic K-Theory defines a symplectic module to be an arbitrary quadratic module $(M,h,q)$ over a form ring $(R,\Lambda)$ with $(J,\varepsilon)$ where $J=\...
wlad's user avatar
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Criterion for unicity and existence of pre-image in multivariate cryptography

Repost from math.stackexchange since no one could help me there and it concerns my research. I am reading Ding's Multivariate Public Key Cryptosystems and in the book the author explains the so-called ...
Saegusa's user avatar
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Solutions to the quadratic matrix equation $X A X^T = B$

Let $A, B \in \mathbb{R}^{n \times n}$ be symmetric, positive-semidefinite, full-rank matrices. I would like to understand the set of $X \in \mathbb{R}^{n \times n}$ which are themselves symmetric and ...
dotdashdashdash's user avatar
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59 views

Equidistribution of lattice points on quadratic forms without certain values

I have recently been studying some results about equidistribution of lattice points on positive definite quadratic forms $Q$ with $3$ or more variables. Concretely the article Duke and Schulze-Pillot, ...
MathqA's user avatar
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118 views

Computing coefficients of theta functions associated to quadratic forms

If we take an integral positive definite quadratic form $Q$ and set $\Theta_Q(z) = \sum_{k\geq 0}R_Q(k)e^{2\pi ikz}$, what are the most efficient algorithms to compute the $R_Q(k)$? I am aware e.g. of ...
a196884's user avatar
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Correspondence between class group of binary quadratic forms and the narrow class group via Dirichlet composition: an elementary approach?

I have been trying to explore and learn about connections between the form class group and the ideal class group. To be on the same page, we define the form class group of a negative discriminant $D \...
asrxiiviii's user avatar
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112 views

Possible values of hyperbolic quadratic forms

$\newcommand\Z{\mathbf{Z}}$Given an integral quadratic form $q$ of signature $(n-1,1)$ and a value $\lambda \in \Z$ is there an algorithm that can determine whether there exist $x\in \Z^n$ such that $...
Mathieu Dutour Sikiric's user avatar
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100 views

Sum of binary quadratic forms over inputs of equal Hamming weight

$\DeclareMathOperator{\field}{\mathbb{F}}$ Let $n$ be a positive integer, and let $q : \field_2^{n} \rightarrow \field_2$ be a quadratic form, specified in coordinates as $$q(x)=\sum_{i =1}^n \alpha_i ...
Ben's user avatar
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The number of orbits of a two-point stabilizer of the symplectic group $Sp(2m,2)$

I am trying to figure out the number of orbits of a two-point stabilizer of the action of $Sp(2m,2)$ on its two orbits $\Omega^+$ and $\Omega^-$ as detailed in Dixon and Mortimer's "Permutation ...
F.Tomas's user avatar
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Linear forms and the second Voronoi decomposition

This is not my area of expertise, so forgive me if the question is a bit naive. Given a collection of vectors $v_1,\ldots,v_d$ in $\mathbb{R}^n$ (with $d\geq n$), there is a corresponding set of ...
Yoav Len's user avatar
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Existence theorem for symmetric nondegenerate forms over a ring

There exists a rich theory for inner product spaces (i.e. vector spaces with a symmetric nondegenerate bilinear form) over fields, and it can be discussed in the context of local rings and free ...
Plank's user avatar
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Maximal connected subgroup of orthogonal group

Let $(Q,V)$ be a quadratic space over an algebraically closed field $k$ with $\dim(V) \geq 3$ Define $$ SO_Q:= \{ \sigma \in GL(V) : Q(\sigma v) = Q(v) \ \text{for all} \ v \in V \ \text{and} \det(\...
Andrew Musso's user avatar
2 votes
0 answers
114 views

Projective group of Plucker quadric over the reals

A somewhat elementary question but seemingly difficult to find a suitable reference: Consider the six-dimensional real space $\wedge^2(\mathbb R^4)$ with basis $e_i \wedge e_j \ (i < j)$ where $...
A. Gupta's user avatar
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1 answer
307 views

Positivity of quadratic form minus linear form on the simplex

Let $a_{ij}$ be the elements of a $n$-dimensional covariance matrix. Can we prove the following? $$ 1-\sum_{k=1}^n a_{ik} \lambda_k + \sum_{j=1}^n \sum_{k=1}^n \lambda_j a_{jk} \lambda_k > 0, \...
J M Grandola's user avatar
2 votes
0 answers
111 views

Inseparable field extensions of degree p and linear independence

Let $F$ be a field of characteristic $p$; let $\alpha \in F$ such that $\alpha \neq \beta^p$ for any $\beta \in F$, and let $K := F(x)$ where $x=\sqrt[p]{\alpha}$. Is it true that the elements $1,(x-...
adam chapman's user avatar
2 votes
0 answers
47 views

Generalization of positive definiteness for a non-smooth quadratic form

I have a "quadratic form" defined as follows: $$f(\vec x) = \sum_{i<j} c_{ij} x_i x_j + \sum_i a_i(x_i) x_i^2$$ where the coefficients $c_{ij}$ are fixed constants, but the coefficients $a_i(x_i)$...
valle's user avatar
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2 votes
1 answer
233 views

Superlevel sets of a parametrized quadratic forms

Let $N$ be an odd integer, $n\in\mathbb{N}$, and $-\frac{2T}{NR^2}\leq a\leq0$ for given $R,T\in\mathbb{R}$ with $\frac{T}{NR^2}\leq\frac{\pi}{2}$. Now consider the quadratic form $\Omega(a)=\sum_{l\...
SoYu's user avatar
  • 213
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Number of representations by the norm in a division algebra corresponding to endomorphism rings of elliptic curves

Let $E$ be a supersingular curve over a field of characteristic $p$ with endomorphism ring $\mathcal O_D$ which is a maximal order in a division ring $D$ over $Q$ ramified at $p$ and $\infty$. The ...
Asvin's user avatar
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2 votes
1 answer
374 views

A "nice" (but non-definite) quadratic programme

For integers $n\geq k>0$, let $f$ be the following quadratic form: $$f(x_1,\ldots,x_n)=\sum_{i=1}^n\sum_{j=0}^{k-1}x_ix_{i+j\bmod n}.$$ Is it true that the minimum of $f$ over the unit simplex is ...
Ron P's user avatar
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2 votes
0 answers
330 views

Concrete Hanson-Wright inequality?

I'm working on a paper that requires bounding $$\Pr\left[|\vec x^\top Q \vec y| >= t\right]$$ where $Q$ is a matrix (happens to be symmetric) and $\vec x,\vec y$ are iid real mean-zero subgaussian ...
user3538175's user avatar
2 votes
0 answers
108 views

Angle between subspaces in indefinite signature

I know there is a notion of principal angles between subpaces of a Euclidean space, but what about subspaces of a finite dimensional vector space $V$ equipped with a non-degenerate quadratic form of ...
Fabien Besnard's user avatar
2 votes
0 answers
98 views

Classification of quadratic submodules in $\mathbb C[[t]]$

Let $\mathbb C[[t]]$ be the ring of formal series with complex coefficients. Let $M$ be a finite rank free module over this ring. Let $Q$ be a regular quadratic form on $M$. (E.g., the standard ...
Roman Fedorov's user avatar
2 votes
0 answers
100 views

Alternating bihomomorphism is skew of 2-cocycle - relative situation

Let $G$ be a finite abelian group. It is well-known that every alternating bihomomorphism $\Omega:G\times G \to \mathbb{C}^\times$ (i.e. $\Omega(g,g)=1$) arises as the skew $\kappa/\kappa^T$ of a 2-...
Bipolar Minds's user avatar
2 votes
0 answers
156 views

Optimization of quadratic form with band matrices

Let $A_1$ be the $N \times N$-matrix for which $a_{i,j} = 1$ for $i=j$ and 0 otherwise. Let $A_2$ be the matrix for which $a_{i,j}=1$ for $|i-j| \leq 1$ and 0 otherwise. Similarly define $A_3$ (which ...
Kurisuto Asutora's user avatar
2 votes
0 answers
120 views

Transversality of quadrics containing a projective curve

Let $C$ be a curve of genus $g$ and $L$ a $g^r_d$ on it and assume that we are in the range ${r+2\choose 2}>2d-g+1$. If $C$ and $L$ are chosen to be general then by the maximal rank conjecture (...
Irfan Kadikoylu's user avatar
2 votes
0 answers
150 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 ...
Arnold Neumaier's user avatar
2 votes
0 answers
110 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(...
Keivan Karai's user avatar
  • 6,224
2 votes
0 answers
236 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 ...
nicosiekirka's user avatar
2 votes
0 answers
113 views

Hlawka inequality for Lorentz quadratic form

Let $K$ be a convex cone in ${\mathbb R}^n$. A continuous function $f:K\rightarrow\mathbb R$ satisfies a Hlawka inequality if $$f(0)+f(x+y)+f(y+z)+f(z+x)\le f(x)+f(y)+f(z)+f(x+y+z),\qquad\forall x,y,z\...
Denis Serre's user avatar
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2 votes
0 answers
123 views

What is classified by $H^1(\mathbb{R},SO(p,q))$ and by $H^1(\mathbb{R},SU(p,q))$?

We denote by $F^{\mathbb{R}}_{p,q}$ the quadratic form over the field ${\mathbb{R}}$ $$ F^{\mathbb{R}}_{p,q}(x)=x_1^2+\dots+x_p^2-(x_{p+1}^2+\dots+x_{p+q}^2) $$ on the vector space $V^{\mathbb{R}}:={\...
Mikhail Borovoi's user avatar
2 votes
0 answers
192 views

Equation in the Gaussian Integers

Let $a,b \in \mathbb{N}$. Is there a possibility to characterize the solutions of $a N(\alpha) - b N(\beta)=1$ where $\alpha,\beta \in \mathbb{Z}[i]$? In particular I am interested in the case $a=1$ ...
Martin's user avatar
  • 1,101
2 votes
0 answers
207 views

n-ary quadratic forms with $S$-integer values

Let $Q(x_1,\ldots,x_n):=x_1^2+\cdots+x_n^2$ be an $n$-ary quadratic form. Given a finite set of (rational) primes $S$ is there an algorithm or theorem that describes all solutions to $Q(x_1,\ldots,...
Eric Rowell's user avatar
  • 1,639
2 votes
0 answers
86 views

Terminology and reference question

I am working on a problem involving bilinear forms over complex Hilbert spaces, and in my case it is not natural to make the forms sesquilinear, i.e., $a(u,v)$ is linear in both complex arguments. ...
Nemis L.'s user avatar
  • 143
2 votes
0 answers
243 views

Hurwitz integers and $F_4$

The Hurwitz integers are $$ \mathcal H= \{a+bi+cj+dk:a,b,c,d\in\mathbb Z\;\text{ or } \;a,b,c,d\in \tfrac12+\mathbb Z\}. $$ I want to know if there is a formula, for $m\in\mathbb Z$, for the number ...
emiliocba's user avatar
  • 2,446
1 vote
0 answers
70 views

Quadratic forms with the same roots over GF(2) for low rank problems

Let $Q_1(x)=x^TA_1x$ and $Q_2(x)=x^TA_2x$ with $x\in GF(2)^n$, $A_i\in GF(2)^{n\times n}, i \in \{1, 2\}$. If $rank(A_1)=rank(A_2)=2$, is it possible that $Q_1(x)$ and $Q_2(x)$ can have the same roots ...
Fabio Dias's user avatar
1 vote
0 answers
70 views

Notion of length in projective space over function field

Given a projective space $\mathbb{P}^n(\mathbb{R})$ and two points $x, y \in \mathbb{P}^n(\mathbb{R})$, the distance between $x$ and $y$ is defined as $$ d(x, y) = \frac{\|v_x \wedge v_y\|}{\|v_x\| \|...
Sarthak's user avatar
  • 87
1 vote
0 answers
52 views

Genus of binary quadratic forms: $f(x,y), g(x,y)$ in same genus if and only if represent same values in $(\mathbb Z/m\mathbb Z)^\ast$ for all $m$

In David Cox's book: Primes of the form $x^2+ny^2$, second edition, there is a theorem(Theorem 3.21, page 52) characterize whether two binary quadratic forms in the same genus. The contents of the ...
HGF's user avatar
  • 287
1 vote
0 answers
40 views

Quadratic equations over division rings of dimension 2

Let $\ell$ be a division ring, and let $k$ be a sub division ring. I know that a quadratic equation $x^2 + ax + b = 0$, with $a, b \in k$ can have more than two solutions in $\ell$, but what if the ...
THC's user avatar
  • 4,547
1 vote
0 answers
62 views

Isotropic vectors of quadratic forms over number fields

By Meyer's theorem, an indefinite quadratic form $Q$ over $Z$ has an integral isotropic vector is the dimension is at least $5$ and this bound is tight. Indeed, in dimension $4$ there are indefinite ...
Mathieu Dutour Sikiric's user avatar
1 vote
0 answers
85 views

Squares in skew fields of dimension 2 over a sub skew field

Let $\ell$ be a skew field (i.e., a division ring), and let $k$ be a sub skew field, such that the dimension of $\ell$ as a left vector space over $k$ is $2$. Then if $a \in \ell \setminus k$, we can ...
THC's user avatar
  • 4,547
1 vote
1 answer
67 views

Low rank matrices which preserve maximizers of quadratic form

Suppose $x, y \in \{0,1\}^d$ are binary vectors. For a matrix $M$ consider the quadratic form, \begin{align} x^T M y + (\mathbb{1} - x)^T M (\mathbb{1} - y) \end{align} Does there exist an $M$ such ...
Television's user avatar
1 vote
0 answers
136 views

On the equation $q(\mathbf{x}) = 1$ for $q$ a quadratic form

Let $q(\mathbf{x}) = q(x_1, \cdots, x_n)$ be a quadratic form with integer coefficients. For $n \geq 3$, is there a reasonable theory for the set of integer solutions to the equation $$\displaystyle q(...
Stanley Yao Xiao's user avatar
1 vote
0 answers
94 views

Is there a way to linearize matrix quadratic forms?

Say $x$ is a random vector in $\mathbb{R}^n$. Then, given a (deterministic) symmetric real positive definite matrix $A$, if we want to calculate the expectation of the quadratic form, we can use the ...
Drew Brady's user avatar