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.
543 questions
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Cohomological dimension of transcendental p-adic extensions
Let $q$ denote a quadratic form over a field $k$.
The u-invariant of a field $u(k)$ is defined by $u(k):=\{ max (\mathrm{rank}(q)) $ | $ q $ is anisotropic over $k\}$.
Let $k = \mathbb{Q}_p$ for any ...
- 1,479
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''Local-global-principle'' for certain isometries of lattices
Hi everybody.
I am trying to understand a proof of Kneser. the assertion is on a ''weak version'' of the local-global principle certain isometries: It is Satz (30.9) in kneser book ''Quadratische ...
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if $\Pi_1$ and $\Pi_2$ be elliptic planes then $\Pi_1 \oplus \Pi_2 $ is still elliptic?
Let $\Omega \in\Lambda^{4}\big(V^{ \star }\big)$ be volume form. Define symplectic bilinear form
$q: \Pi \oplus \Pi \rightarrow R $
$\big( \alpha ,\beta \big) \longrightarrow \alpha \...
user21574
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probably Lagrange or Legendre, Pell variant
Evidently Legendre showed that, for positive primes, if $p \equiv 3 \pmod 8$ there is an integral solution to $x^2 - p y^2 = -2.$ Next, if $q \equiv 7 \pmod 8$ there is an integral solution to $x^2 -...
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Rationality of quadric fibrations
Let $Q\to S$ be a quadric fibration over a rational base $S$, over an algebraically closed field of non zero characteristic. Is it true the following?
$Q$ is rational if and only if $Q \to S$ has a ...
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133
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Graceful labeling of the complete bipartite graph and its laplacian quadratic form diagonalized
A graceful labeling of a connected simple undirected graph $G=(V,E)$ is a map $f:V\to\lbrace 1,...,|E|+1\rbrace$ such that for all $t\in\lbrace 1,...,|E|\rbrace$ there is a (trivially unique) $\langle ...
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How to solve for bounds restricting ${\Sigma}$ to symmetric-positive-semi-definiteness?
Scenario
I have a equation for a covariance matrix ${\Sigma}$ where everything but a vector of correlations is known aka $x=(x_{1}, \dots, x_{D})$ for $x_{i}\in [-1, 1]$.
Problem
I know that ${x}$ ...
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A question about the sign of quadratic forms on nonnegative vectors
Let $M$ be a real square matrix of order $n\ge 3$.
Assume that for every nonnegative vector $\textbf{z}\in \mathbb R^n$ which has at lease one zero entry we have $\textbf{z}^T M \textbf{z} \ge 0$.
Can ...
- 13
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Degenerate representation
Let $\rho : \mathbb{R}^n\to \mathfrak{so}(2m)$ be a faithful representation of the commutative Lie algebra $\mathbb{R}^n$ into the Lie algebra of skew-symmetric matrices. There is an orthonormal basis ...
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A question on vectors in $\mathbb{R}^4$
Let $M$ be an invertible $4 \times 4$ real matrix, and let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^4$. Consider the matrix
$$\displaystyle \mathcal{M} = \mathcal{M}(\mathbf{u}, \mathbf{v}) = [\mathbf{u}...
- 26.9k
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Solving a pair of ternary quadratic form equations
Let $Q_1(x_0, x_1, x_2), Q_2(x_0, x_1, x_2) \in \mathbb{Z}[x_0, x_1, x_2]$ be two primitive, non-singular ternary quadratic forms (possibly indefinite). Suppose we want to solve the simultaneous ...
- 26.9k
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Maximizing quadratic form subject to inequality constraints [closed]
Given a $n \times n$ symmetric matrix $\rm S$, solve the optimization problem in $n \times k$ (where $n \geq k$) matrix $\rm X$
$$\begin{array}{ll} \text{maximize} & \mbox{tr} \left( \mathrm X^\...
- 21
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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:
$V$ ...
- 185
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Counting integral points on a diagonal conic
Let $q(x,y,z) = x^2 - by^2 - cz^2$ where $b,c$ are co-prime positive integers. Suppose that the binary quadratic form $f(x,y) = x^2 - by^2$ is irreducible. I am interested in counting integral points ...
- 26.9k
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Weyl asymptotics vs. form perturbations
Consider Hilbert spaces $V$,$H$; a closed quadratic form $a$ with domain $V$; and its associated operator $A$ on $H$. (If necessary, the form can be assumed to be coercive.) For the sake of simplicity,...
- 3,388
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Lorentz quotient and orientation
$$ U \; = \;
\left( \begin{array}{cc}
0 & 1 \\\
1 & 0
\end{array}
\right) ,
$$
Given a real oriented vector space $V$ with inner product, form Lorentzian $L = V \oplus U....
- 25.7k
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About lattice $\pmod q$
For any matrix $A \in Z^{n\times m}$, Let $$\wedge_q(A)=\{ y\in Z^m\mathpunct{:}\exists s\in Z^n,\text{ s.t. }y=A^ts \pmod q \},$$ $$\wedge_q^\bot(A)=\{x\in Z^m: Ax=0 \pmod q\}.$$ There is a result ...
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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 ...
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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\| \|...
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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 ...
- 287
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40
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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 ...
- 4,547
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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 ...
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85
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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 ...
- 4,547
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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 ...
- 297
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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(...
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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 ...
- 420
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Automorphism group of conic bundle fixing the base
Let $\pi: X \to \mathbb{P}^n$ be a conic bundle over an (algebraically closed) field $k$. Let $g \in Aut(X)$ so that $g$ preserves the fibres of $\pi$. Clearly $g$ lives inside $PGL_3(k(\mathbb{P}^n))$...
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Is there any point in considering Form Rings when 2 admits an inverse?
In the study of quadratic spaces over general rings, there is a type of scalar which people consider called a
Form ring $(R,\Lambda)$ relative to some anti-automorphism denoted $(-)^J:R\to R$ and ...
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When does a system of homogeneous quadratic equations have integer solutions?
I learned that in general, solving systems of quadratic Diophantine equations is a difficult problem. But I wonder if there are special (and non-trivial) types of systems that are easier to handle.
...
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Projectivity of the fundamental ideal of Witt groups
Suppose $k$ is a field. I wonder when the Witt ring of the quadratic forms $\textbf{W}(k)$ has a projective fundamental ideal, which is the kernel of the rank modulo 2 morphism. Here I want a ...
- 918
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Can I solve this quadratic program "fast"?
We are given a matrix $D \in \mathbb{Z}^{C \times C}$ of non-negative entries, an integer $k \geq 1$ and we need to maximize the quadratic form $x^T D x$ under some simple constraints. For all ...
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Eigenvalues of orthogonal group element
Let $q$ be a quadratic form over a nonarchimedean local field $F$, and let $\operatorname{O}(q)$ be the corresponding orthogonal group. Let $g\in\operatorname{O}(q)$ be semisimple.
Can we know ...
- 833
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Solution that minimizes the sum of squared errors, with quadratic constraints
Given symmetric and positive definite $n \times n$ (real) matrices $A_1, \dots, A_m$ and $b_1, \dots, b_m \in {\Bbb R}^{n}$, I am trying to find the solution with the least sum of squared errors of ...
- 11
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Continuous choice of null directions for a family of bilinear forms
Let $E$ and $F$ be (real) Hilbert spaces, where $\dim E = \infty$ and $1 \leq \dim F < \infty$. Let $T : E \to \operatorname{Sym}(F \times F,\mathbb{R})$ be a continuous linear map, where $\...
- 2,095
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Generalize spinor bundles over orthogonal Grassmannians
We will work over $\mathbb C$ and the notation will be coherent with the paper of Ottaviani (see [Ott]).
Consider a $n$-dimensional quadric hypersurface $Q_n \subset \mathbb P^{n+1}$. We have ...
- 381
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Maximal $l^2$ restriction of psd quadratic form on $\mathbb{R}^n$
I've been stuck on this problem for a few days as part of a bigger research problem and am looking for some help.
Let $S : \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$ be a positive ...
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Obstruction in construction of some lattices, related with $K3$ surfaces
I am considering a certain $K3$ surface that is lattice-polarized in two ways.
This leads to the following simple problem in lattice theory:
(Let me borrow notations for lattice from Ch.14 of this ...
- 1,841
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99
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Integral zeros of quadratic forms
Suppose I have a definite quadratic form over $\mathbb{Z}$ in $n$ variables
$$Q(\mathbf{x}) = \sum_{i,j} a_{i,j}x_ix_j$$
I'm looking to find whether an integer $m$ is representable by $Q$. Everyone ...
- 901
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Can $12n+5$ be written as $2x^2+5y^2+9z^2+xyz$ with $x,y,z$ nonnegative integers?
Let $\mathbb N=\{0,1,2,\ldots\}$. By the Gauss-Legendre theorem on sums of three squares, for any $n\in\mathbb N$ we can write $4n+1$ as $x^2+y^2+z^2$ with $x,y,z\in\mathbb N$.
Motivated by this, here ...
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When the summands of a positive definite matrix are positive definite
Let $A,B$ be two real symmetric matrices. Let $C = A+B$ be a positive-definite matrix. Write $C>0$ for $C$ being positive-definite. Suppose that $A>0 \implies C>0$ and $B > 0 \implies C>...
- 71
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Is there an effective genus theory for indefinite quadratic forms?
For positive definite quadratic forms, there is a way to check if two forms are isomorphic by arithmetic equivalence over $GL_n(Z)$ by computing configurations of vector up to some norm and then ...
1
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348
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Tail bounds for random Gaussian chaos?
Let $g = (g_1, \dots, g_d)$ be a sequence of independent standard Normal random variables, and suppose $\Sigma$ is a $d \times d$ (deterministic), real, symmetric, positive definite matrix. The Hanson-...
- 420
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93
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Uniqueness of decomposition for positive-definite integral bilinear forms?
Let $\Lambda$ be a lattice, that is a free finitely generated abelian group with a symmetric bilinear form.
In general, decomposition of lattices into indecomposable orthogonal sublattices is not ...
- 2,300
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111
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Roots of the system of quadratic equations
For a set of $m$ positive semi-definite $d\times d$ matrices $Q_j$ I have the following system of equations over column-vector $\vec{x}$:
$$
q_j = \vec{x}^T Q_j \vec{x}, \quad j=1,\dots,m
$$
with ...
- 171
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0
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36
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Values on minkowski square of rational ellipsoid
Let $q(x)$ be positive-definite rational quadratic form. Let $C_q$ be the following set:
$$C_q = \{q(x + y) \quad | \quad q(x) = q(y) = 1\}$$
Is it true or not that set $C_q$ defines the equivalence ...
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0
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109
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Existence of solutions to quadratic vector equation
Given
$$A_{i,j,k} X_j^* X_k + C_i = 0$$
where $𝐴_{i,j,k}$ and $𝐶_i$ are arbitrary complex numbers for all $𝑗, 𝑘$ which are $𝑁$-dimensional indices and $i$ which is an $m$-dimensional index where $...
- 11
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195
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finding automorphisms of binary hermitian forms
Let $K$ be an imaginary quadratic field and $\mathcal{O}_K$ be its ring of integers. Write $\mathcal{H}(\mathcal{O}_K)$ for the set of hermitian 2 x 2 matrices with entries in $\mathcal{O}_K$. Now, we ...
- 33
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90
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The splitting pattern of the Killing form of an algebraic group and the Tits index
Let us assume that $G$ is an anisotropic semisimple, connected algebraic group over a field $k$ of characteristic zero.
Let $K_G$ denote the class of its Killing form in the Witt ring of $k$.
Let $X$ ...
- 1,479
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0
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100
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Rational quadratic form with given determinant and Hasse-Witt invariant
Let $Q$ be the diagonal quadratic form denoted by
$$Q(x_1, \ldots, x_k) = \sum_{i = 1}^{k} d_i x_i^2, \quad x_i \in \mathbb{Q}, \quad d_i > 0$$
Also let ${\mathbb{Q}^{*}}^2$ be the set of nonzero ...
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0
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43
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About weight generator of quadratic lattices
I get stuck on Example 93:5 in O'Meara's book "Introduction to quadratic forms", where it is explained a method of find a weight generator of a quadratic lattice.
In this question I assume ...
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