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Questions tagged [quadratic-forms]

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4
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0answers
42 views

On the orthogonal group of a lattice on a quadratic space over dyadic local field

Let $F$ be a local field with valuation ring $R$. $V$ is a n dimensional non-singular quadratic space over $F$ with bilinear form $B$ and quadratic map $Q$. As usual, $O(V)$ denotes the orthogonal ...
1
vote
0answers
43 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 ...
3
votes
0answers
163 views

Can we write each positive integer as $x^2+y^2+\varphi(z^2)$?

As odd squares are congruent to $1$ modulo $8$, any integer of the form $4^k(8m+7)$ with $k,m\in\mathbb N=\{0,1,2,\ldots\}$ cannot be written as the sum of three squares. To avoid such congruence ...
1
vote
1answer
79 views

Existence of symplectic basis

Let $R$ be a PID and $M$ a free, finite rank $R$-module with a perfect billinear form $\omega$ such that $\omega(v,v)=0$ for all $v \in M$. Does anyone know a reference for the fact that a symplectic ...
0
votes
1answer
158 views

Proof of A Positive Definite Covariance Matrix

I would like to prove such a matrix as a positive definite one, $$ (\omega^T\Sigma\omega) \Sigma - \Sigma\omega \omega^T\Sigma $$ where $\Sigma$ is a positive definite symetric covariance matrix ...
0
votes
0answers
29 views

How many points are still contained in a common (hyper-)ellipsoid?

It is known that $${d+2\choose 2}-1$$ points uniquely determine a quadric in $\Bbb R^d$. However, I want my points not on an arbitrary quadric, but on a centered hyperellipsoid in $\Bbb R^d$, or ...
6
votes
4answers
414 views

Solutions to the Diophantine equation $x^2+3y^2+3z^2=n$

For a fixed positive integer $n$, the Diophantine equation $$x^2 + y^2 + z^2 = n$$ was studied by Gauss in Disquisitiones Arithmeticae. As is known, this equation is intimately connected to the ...
1
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0answers
65 views

Reference request: Number of elliptic and hyperbolic quadratic forms of a given rank over a finite field

My question is over the finite field $\mathbf{F}_q$ of $q$ elements. It is well known that a symmetric matrix of odd rank corresponds to a parabolic quadratic form but even rank symmetric matrices ...
2
votes
0answers
42 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 ...
16
votes
0answers
448 views

The number 1680 and Lagrange's four-square theorem

The number $1680$ has the factorization $2^4\times3\times5\times7$. Rather to my surprise, I found that this number has certain mysterious connection with Lagrange's four-square theorem. QUESTION: ...
3
votes
0answers
120 views

Can we write each positive integer as $w^2+x^2(1+2y^2+2z^2)$ with $w,x,y,z\in\mathbb Z$ and $x\not=0$?

Lagrange's four-square theorem states that each nonnegative integer is the sum of four squares. Here I ask the following question concerning a refinement of Lagrange's four-square theorem. QUESTION: ...
5
votes
1answer
236 views

Are lattice points in thin spherical shells uniformly distributed?

Consider the spherical shell (annulus) $$A(R,r) = \{ x \in \mathbb{R}^3 : R \leq | x|\leq R+r \}.$$ Think of the limit $R \to \infty$. Assume that $r$ depends on $R$ as $r(R) = R^{-\delta}$. We are ...
5
votes
0answers
132 views

Is every integer $n>1$ the sum of two triangular numbers and two powers of $5$?

Recall that the triangular numbers are those integers $$T_n=n(n+1)/2\ \ \ (n=0,1,2,\ldots).$$ In 1796 Gauss proved that each $n\in\mathbb N=\{0,1,2,\ldots\}$ is the sum of three triangular numbers, ...
1
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0answers
45 views

Question about Notation for Spaces of $n$-ary $k$-ic Forms

Define an $n$-ary $k$-ic form to be a polynomial over the integers of homogeneous degree $k$ in $n$ variables. In Section 1 of the paper "Higher Composition Laws I" (linked below), Bhargava writes $(\...
0
votes
1answer
118 views

Polyhedral conditions for quadratic inequalities in fixed dimension

Denote $\mathcal T$ be set of $(T_1,T_2,T_3,T_4)\in\mathbb Z^4$ that satisfy $$0<T_1,T_2,T_3,T_4$$ conditions? Define the level set $$M_{\gamma}(Q,\mathcal T)=\{(T_1,T_2,T_3,T_4)\in\mathcal T:Q(...
0
votes
1answer
74 views

Vanishing bilinear forms

For a symmetric or antisymmetric bilinear form $\varphi$ on a vector space $V$, if $\varphi(x,y)=0$ then also $\varphi(y,x)=0$ ($x,y\in V$). I was wondering if this is also a necessary condition for ...
2
votes
0answers
39 views

Clarification on FPTAS optimization in a paper

In the abstract of this paper by Hildebrand, Weismantel & Zemmer it is stated that they provide an FPTAS for $$\min x'Qx$$ over a fixed dimension polyhedron when $Q$ has at most one negative or ...
5
votes
1answer
355 views

higher order analogues of sylvester's law of inertia?

Sylvester's law of inertia (here I quote wikipedia) If A is the symmetric matrix that defines the quadratic form, and S is any invertible matrix such that D = SAS^{T} is diagonal, then the number ...
4
votes
1answer
658 views

Do $G$-invariant non-degenerate quadratic forms come from $G$-invariant even lattices?

The following is a somewhat well-known fact: Given an even lattice $L$ with the pairing $\langle,\rangle: L\times L\to \mathbb{Z}$, we extend the pairing to $L\otimes \mathbb{Q}$ by tensoring with $\...
5
votes
1answer
357 views

The statement that $A \ge B$ implies $A^{-1} \le B^{-1}$ is still true for matrices? [closed]

Problem: Suppose we have two real, symmetric and positive definite square matrices $A$ and $B$, i.e., $$A_{ij}, B_{ij}\in \mathbb{R}$$ $$A^T=A$$ $$B^T=B$$ $$x^TAx>0 \forall x$$ $$x^TBx>0 \...
3
votes
1answer
193 views

« Generalized simultaneous diagonalization » of a pair of symmetric, non-commuting, positive semi-definite matrices

I hope my question is trivial for some of you but for the time being I’m lost somewhere between the generalized eigenproblem, simultaneous diagonalization of quadratic forms, simultaneous SVD, ...
1
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0answers
36 views

Isotropy of skew-Hermitian forms over division algebras

Assume char(F) $\neq$ 2. Let $D$ be a central division algebra over a field $F$ and $h: V \rightarrow D$ be an anisotropic skew-Hermitian form. We can easily see that $h_{\bar{F}}$ is totally ...
1
vote
0answers
60 views

Showing a modified system of quadratic equations is stable

I have and $n$ dimensional dynamical system, given by $\dot{x} = M D(x) P x - \frac{c}{2}x$ $P$ is a full rank $n \times n$ matrix, with $p_{ij} \in [0,c]$, such that $p_{ij}=c-p_{ji}$ for some ...
0
votes
0answers
54 views

Suggestions to solve an optimization problem that involves quadratic forms

I am in a crucial part of my research, I have arrived at an optimization problem that I can not solve, I need to solve it to be able to perform simulations and thus complete my research, due to this ...
7
votes
0answers
206 views

Which continuous function is optimal for sieving?

In 1968, Barban and Vehov considered [1] the problem of determining for which continuous functions $\rho:\mathbb{R}^+\to [0,1]$ satisfying certain properties ($\rho(t)=1$ for $t\leq U_0$, $\rho(t)=0$ ...
1
vote
1answer
132 views

Maximizing quadratic form subject to inequality constraints [closed]

Given a $n \times n$ symmetric matrix $\rm S$, solve the optimization problem in $n \times k$ (where $n \geq k$) matrix $\rm X$ $$\begin{array}{ll} \text{maximize} & \mbox{tr} \left( \mathrm X^\...
5
votes
1answer
244 views

Realizing root-system roots as polynomial roots without Lie theory

The vectors of a root-system were originally called "roots" because they are the zeros of a characteristic polynomial that comes from the connection of (crystallographic) root-systems to classifying ...
0
votes
0answers
50 views

Lattices of prime level

Let $L$ be an even positive definite lattice of level $2$ (i.e. $2L^*$ is even). How can we prove that the rank of $L$ is divisible by $4$? In general, if $L$ is a lattice of prime level, what can we ...
8
votes
2answers
291 views

When is a bilinear form equivalent to a trace form?

Associated to a finite, separable field extension $L/K$, there is a natural nondegenerate bilinear form, the trace form, defined by $$\langle x,y \rangle := \mathrm{Tr}_{L/K}(xy)$$ Now, given a ...
0
votes
1answer
101 views

Elementary description to count of perfect squares - II

What can we say about growth of smallest gap $g(a)$ which is the smallest $|x-y|$ where $0\leq x,y\leq\Big\lfloor\frac a2\Big\rfloor$ and $\sqrt{x(a-x)},\sqrt{y(a-y)}\in\Bbb Z$? Is $g(a)=1\iff a=b^2+...
0
votes
1answer
117 views

Elementary description to count of perfect squares - I

Is there an elementary description of $$N(a)=\Big|\Big\{x\in\{0,1,\dots,\Big\lfloor\frac a2\Big\rfloor-1,\Big\lfloor\frac a2\Big\rfloor\Big\}:\sqrt{x(a-x)}\in\Bbb Z\}\Big|$$ and though likely non-...
5
votes
1answer
284 views

A curious determinant of quadratic forms

In a work about the Wave Equation, I encountered the following symmetric matrix of size $1+n$, whose entries are quadratic forms. The arguments are a scalar $a$ and a vector $X\in k^n$. $$S(a,X)=\...
3
votes
1answer
247 views

Completing the square of a matrix expression

Let $A,C\in\mathbb{R}^{m\times n}$, $n\ge m$, $B\in\mathbb{R}^{n\times m}$, and $P$ be a real positive definite $m\times m$ matrix. Denote by $\mathcal{S}^n$ the space of $n\times n$ real symmetric ...
20
votes
3answers
1k views

Simultaneous “orthonormalization” in $\mathbb{C}^4$

Let $A$ be a positive, invertible $4 \times 4$ hermitian complex matrix. So we have a positive sesquilinear form $\langle Av,w\rangle$. Say that a pair $(v,w)$ of vectors in $\mathbb{C}^4$ is good ...
8
votes
2answers
255 views

Proof of Witt's result about quaternion extensions

I'm searching for a proof of Witt's result that a biquadratic extension $K(\sqrt{a},\sqrt{b})/K$ extends to a Galois extension $L/K$ with quaternion group $Q_8$ iff the quadratic forms $<a,b,\frac{...
12
votes
2answers
1k views

On Siegel mass formula

I have asked this question exactly here. The question is as follows: I am interested deeply in the following problem: Let $f$ be a (fixed) positive definite quadratic form; and let $n$ be an ...
11
votes
1answer
1k views

Do almost all systems of quadratic equations have solutions?

If I have a system of linear equations, $A x = c$, with $A$ an $n\times n$ complex matrix, it is relatively easy to see that the set of matrices $A$ for which there is no (complex) solution has ...
0
votes
0answers
153 views

On sum of squares?

If $(a,b)=1$ and $2|b$ then $p$ is prime and $p|a^2+b^2\implies p\not\equiv3\bmod4$. For any other $k\in\Bbb N_{>2}$, is there a polynomial that represents an odd prime $p$ if and only if $p\not\...
2
votes
0answers
89 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 ...
0
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0answers
37 views

A symmetric matrix with nonzero principal minors is cogredient to a diagonal matrix via an upper triangular

A paper I'm reading in representation theory states the following result: Let $F$ be a field of characteristic zero, and $x$ a symmetric matrix in $M_n(F)$ all of whose principal minors are not zero. ...
5
votes
1answer
225 views

Hasse principle for quadratic forms over finitely generated fields

Does the Hasse principle hold for quadratic forms over finitely generated fields (e.g. for the Henselisations/completions at height-$1$-primes or all places)?
5
votes
0answers
151 views

Selmer Group of number fields and Ideal lattices

Let $K$ be a totally real number field of degree $n$ and dicriminant $d$, in this article of F.Lemmermeyer the selmer group of $K$ is defined as $$\text{Sel}(K):=\{\alpha \in K^{\times}: (\alpha)=...
12
votes
2answers
399 views

A diophantine equation in $\mathbb{N}$

While I was working on a paper on graph theory, I encountered a problem which I think is a number-theory-problem. I don't know if there are any tools to answer the question. Find all natural numbers $...
3
votes
2answers
384 views

Roots of quadratic vector equation

Given $$A_{i j k}X_j X_k + B_{ij} X_j + C_i = 0$$ where $A_{ijk}$, $B_{ij}$, and $C_i$ are arbitrary real numbers for all $i$, $j$, $k$ which are $N$-dimensional indices, such that $A_{ijk}=A_{ikj}$ ...
-1
votes
1answer
138 views

Solving quaternary quadratic forms modulo $q$ efficiently

Given a quaternary quadratic equation of form $$Q(a,b,c,d)=m$$ in $\Bbb Z[a,b,c,d]$ with coefficient sizes and $|m|$ bounded in magnitude by $B\in\Bbb N$ where $m\neq0$ if we are looking for solutions ...
9
votes
1answer
365 views

Quadrics in the Grothendieck ring

Let $\mathcal{Q}$ be an irreducible quadric in $\mathbb{P}^n(k)$, with $n \geq 2$ and $k$ a finite field. Let $K_0(V_k)$ be the Grothendieck ring of $k$-varieties. It is well known (it appears) that ...
1
vote
1answer
76 views

Redistribute diagonal entries of a matrix

Let $d = (d_1,...,d_k)^t$ with positive entries. Denote $D:=diag(d)$ and let $m > k$. What are sufficient conditions on $d$ and $m$ so that there exists $V \in \mathbb{R}^{m \times k}$ with: $V$ ...
5
votes
1answer
125 views

Bounded version of linear and quadratic Hasse--Minkowski theorem

The Hasse-Minkowski theorem states that if $$Q(x_1,\ldots,x_n) = \sum_{i,j=1}^n a_{ij} x_ix_j$$ is a quadratic form with $a_{ij} \in \mathbb Z$ and $\det (a_{ij}) \neq 0$, then the equation $$Q(x_1,\...
5
votes
1answer
218 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 ...
4
votes
0answers
101 views

On the quadratic equivalence of fields

I have spent the past two years studying abstract Witt rings. These objects are a generalization of "The Witt ring of a field," an algebraic invariant of fields of characteristic not equal to 2. ...