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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Siegel's Mass Formula for ternary indefinite quadratic forms
In his paper "On the theory of indefnite quadratic forms", Siegel gives the formula (Thm. 1)
$$
\mu(S,T)=\prod_p \alpha_p(S,T),
$$
where
$S$ is an $m\times m$ non singular integral symmetric matrix ...
- 2,446
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Relation between positive roots of $E_8$ and $\mathbb{F}_2^8 \setminus \{0\}$
There exists an explicit bijection (due to Cayley, that has built up a very nice table to describe this) between the positive roots of the lattice $E_7$ and $\mathbb{F}_2^6 \setminus \{0\}$ (where $\...
- 3,779
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Calculating the explicit constant – Siegel zeros and class numbers
Let $\chi$ denote the Legendre symbol of conductor $q$. A Siegel zero for the $ L $ series associated to $ \chi $, which we denote by $ L(s,\chi) $ is a real zero $ \sigma $ satisfying $ 1-\frac{c}{\...
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If a lattice can be embedded into $\mathbb Q^n,\langle-1\rangle^n$, then can it be embedded into $\mathbb Z^n,\langle -1 \rangle^n$?
Given a graph with negative integers on each vertex $\Gamma$ there is a corresponding intersection lattice denoted $Q_\Gamma$, a free $\mathbb Z$ module generated by the vertices, endowed with a ...
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Quadratic refinements of a bilinear form on finite abelian groups
$\DeclareMathOperator\Hom{Hom}$Let $A$ be a finite abelian group and $\text{Sym}(A)$ the (abelian) group of symmetric bilinear forms over $A$ valued in $\mathbb{R}/\mathbb{Z}$.
A quadratic function on ...
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Computing spinor equivalence for positive definite forms
Given an integral positive-definite rank $n$ quadratic form $f$, one can use the algorithm in Conway and Sloane (Chapter 15, SPLaG) to efficiently determine if the genus of $f$ contains more than one ...
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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?
...
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Simple comparison of positive ternary quadratic form representation counts
Something came up yesterday in a referee request and I was surprised to find that I did not know the facts in full generality. This is about positive quadratic forms in three variables with integer ...
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Are constant connection coefficients uniquely determined by the (1,3) curvature coefficients?
Suppose that on a certain coordinate system the coefficients $\Gamma^i_{jk}$, $i,j,k=1,\cdots, n$, of a linear connection are constant. We do not require compatibility with a metric, however I am ...
- 1,371
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$p$-adic analogues of $\mathrm{SO}(3)$
I read in the paper " From Laplace to Langlands via representations of orthogonal groups" by Benedict Gross and Mark Reeder that there are, up to isomorphism, two orthogonal groups of the (...
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Can an ellipsoid be moved freely inside another ellipsoid?
An origin centric ellipsoid is defined by any positive semi-definite $n$ by $n$ matrix $X$, by taking all vectors $v$ such that $v^tXv\leq1$. Call two origin centric ellipsoid equivalent if one can be ...
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Classification of vector spaces with a quadratic form and an order n automorphism
Introductory general nonsense (for motivation: feel free to skip): Let $G$ be a finite group and $k$ be a field of characteristic $0$. Consider the set $\mathcal{S}$ of isomorphism classes of finite ...
- 32.5k
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Automorphisms of SO_n(k,f)
Let $k$ be a field, $n\in\mathbb{N}$ and $f:k^n\times k^n\to k$ a non-degenerate symmetric bilinear form. Let
$$O_n(k,f):=\{ g\in GL_n(k) \mid \forall x,y\in k^n : f(x,y)=f(g.x,g.y) \}$$
and
$$SO_n(k,...
- 5,654
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Fundamental domain for group generated by reflections in -2 curves
Given a proper algebraic surface $S$, the Picard group $Pic(S)$ is endowed with the (symmetric) intersection form. We can therefore talk about reflections in the classes of $-2$-curves. These will ...
- 1,329
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Existence of solution for a system of quadratic diophantine equations / symmetric quadratic froms
I am interested in solving, or even just deciding the existence of a solution, for a system of quadratic diophantine equations.
Let $p$ be a prime congruent to 1 modulo 8, so $ p =17$ is the first ...
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Computing the genus of certain ternary indefinite lattices
For $k$ a positive integer, let us consider the rank 3 ternary indefinite lattice $L=L_k$ with quadratic form
$$6kx^2-2(y^2+yz+z^2).$$
Its discriminant group has length $2$.
Question. Is this lattice ...
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Is there a trigonometric field which is different enough from real numbers?
I found this topic in a book 'Metric Affine Geometry' by Ernst Snapper and Robert J. Troyer.
I call a field $k$ trigonometric iff there is a quadratic form $q$ over $k^2$ such that every two lines ...
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Which power of $2$ kills $W(k)$?
Is the following fact "well-known": if $-1$ is a sum of squares in a field $k$, then the Witt group $W(k)$ of quadratic forms is killed by multiplication by $2^N$ for some $N\ge 0$? What can one say ...
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Invariant for group actions
Hello everybody!
Define the action of $SL_4({\mathbb{Z}})$ on alternating 2-forms or simply skew-symmetric matrices of degree 4 according to the following:
For $B \in SL_4({\mathbb{Z}})$ and an ...
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Generators of the orthogonal group of a quadratic form in odd dimension in characteristic 2
In characteristic not $2$, the Theorem of Cartan-Dieudonné states:
[Grove, Theorem 6.6]: Let $q$ be a nondegenerate symmetric quadratic form of dimension $n$ in characteristic not $2$. Then every ...
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Finding Motzkin's original paper on copositive quadratic forms
I am currently in the process of writing my thesis about copositive matrices and would like to write a chronological narrative about the ascent of these matrices to the prominent place they have today ...
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optimal bound in diophantine representation question
Given that, with integers $t \geq 1$ and $q \geq 3,$ there are solutions to $$ 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. $$
...
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Solving a quadratic matrix equation with fat matrix
I am trying to find an $n \times m$ fat (i.e., $m > n$) matrix $T$ that solves
$$T^T T = X$$
where $X$ is a given $m \times m$ symmetric, positive semidefinite matrix.
I saw this post, but ...
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Is there a good notion of kernels of quadratic forms on abelian groups?
Let $G$ be an abelian group and let $q:G \to \mathbb{Q/Z}$ be a quadratic form, i.e. $q(a)=q(-a)$ and $b(x,y)=q(x+y)-q(x)-q(y)$ is a bihomomorphism. On vector spaces, when people speak about the ...
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Quadratic diophantine equations and geometry of numbers
Let (for concreteness) $a = 2$, $b = \sqrt{5}$ and $\varphi = (\sqrt{5}+1)/2$. I am interested in solutions $(w,x,y,z) \in \mathbb{Z}[\varphi]^4$ of the system
$$
w^2 - ax^2 -by^2 + abz^2 = 1
$$
$$
\...
- 2,050
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Syzygy between covariants of pairs of ternary quadratic forms
In the book Nonlinear Computational Geometry, Page 208 (or page 15 of the online version on the author's website: http://www.loria.fr/~petitjea/papers/imaconics.pdf), Remark 5.1, Petitjean states that
...
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Witt ring of a field with Pythagoras number $2$
I am currently looking at a few simple properties of the Witt ring of a field $K$ (by which I mean the ring of Witt classes of quadratic forms, not the ring of Witt vectors), which are clearly true ...
- 272
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Conics and triples of binary quadratic forms
Let $C \subset \mathbb{P}^2$ be a planar conic curve, defined by a ternary quadratic form $Q(x_1, x_2, x_3)$ say. Suppose that $C(\mathbb{Q}) \ne \emptyset$, or equivalently, that $C$ is everywhere ...
- 26.9k
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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 ...
user125345
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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 ...
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Is an orthogonal direct sum decomposition with respect to two quadratic forms necessarily unique up to isomorphism
Consider two quadratic forms $Q$ and $P$ over a finite dimensional vector space $V$ over a quadratically closed (or perhaps Pythagorean) field $F$. If $V$ can be decomposed as $V = V_1 \oplus V_2 \...
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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. ...
- 141
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Quadrics contained in the (complex) Cayley plane
In the paper
Ilev, Manivel - The Chow ring of the Cayley plane
we can learn, that $CH^8(X)$, with $X := E_6/P_1$, denoting the Cayley plane, has three generators with one of them being the class of ...
- 1,479
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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 ...
- 1,479
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Two quadratic programming problems always same answer? [closed]
Was exploring quadratic programming optimization and for two types of problems the answers seemed to always equal.
Problem 1:
Minimize $\tfrac{1}{2} \mathbf{x}^T Q\mathbf{x}$
Subject to $ A \mathbf{...
- 171
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Fully Homomorphic Error Correction?
Consider a field $F$. Suppose we have two vectors $a,b\in F^n$, and an invertible matrix $G\in F^{n\times n}$. Let $c\in F^n$ be the point-wise product of $a$ and $b$, that is, $c_i=a_ib_i$. Let $x=...
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Hermitian forms over quaternion algebra
Notations: Let $Q=(a,b)$ be a quaternion algebra over a field of characteristic $\neq 2$, i.e. $i^2=a, j^2=b, k=ij, ij=-ji$. Consider $K=k(t)(\alpha)$, where $\alpha=\sqrt{at^2+b}$. Let $\sigma=Int(i)\...
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Is the class number of the quadratic field $x^2=3\cdot2^n+1$ $O(n)$?
Numerical evidence suggest that the class number of the quadratic field $x^2=3\cdot2^n+1$ is $O(n)$ while the discriminant is $O(2^n)$.
Here are the class numbers for $n=6 \dots 110$ computed with ...
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equivalence of quadratic forms over finitely generated fields
Over number fields, two quadratic forms are equivalent iff they have the same dimension, signature, discriminant and Hasse invariant.
How is the situation like over finitely generated fields?
user19475
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Equivalence of binary quadratic forms over $\operatorname{GL}_2(\mathbb{Q}_p)$ or $\operatorname{GL}_2(\mathbb{Z}_p)$
Let $f(x,y) = ax^2 + bxy + cy^2$ be a binary quadratic form with integer coefficients and non-zero discriminant $\Delta(f)$. It is well-known that $f$ is $\operatorname{GL}_2(\mathbb{R})$-equivalent (...
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Inverse quadratic norms
The following problem seems easy at a first glance but I can't see the way to prove it. Actually I don't even know if it's true but it is assumed implicitly in a research paper.
Help highly ...
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The Chebotarev Density Theorem and the representation of infinitely many numbers by forms
Let $ax^{2}+bxy+cy^{2}$ be a primitive positive definite quadratic form of discriminant $\Delta<0$. It is well known that $ax^{2}+bxy+cy^{2}$ represents infinitely many prime numbers. One of the ...
- 81
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Pairs of quadratic forms and $\mathbf{A}^8/\mathrm{SL}_2^{\times 3}$
$\newcommand{\std}{\mathrm{std}}\newcommand{\SL}{\mathrm{SL}}\newcommand{\mmod}{/\!\!/}$Fix the base field to be the complex numbers $\mathbf{C}$. Let $\std = \mathbf{A}^2$ denote the standard ...
- 5,760
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Image of a quadratic form is a closed cone
Let $Q : E \to F$ be a quadratic form induced by a symmetric bilinear form $B : E \times E \to F$ defined in a finite dimensional real normed vector space $E$, with values in the normed vector space $...
- 2,095
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Close integer solutions to $ab-cd=1$?
I am looking for infinite set of Diophantine solutions.
Suppose we require
$$0<\min(a,d)<\max(a,d)<\min(b,c)<\max(b,c)\leq\sqrt 2\min(a,d)$$
$$a,b,c,d\in\mathbb Z$$
then can we still find ...
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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 $A_{ijk}=A_{ikj}$ ...
- 151
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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 ...
- 800
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Duke and Schulze-Pillot condition for equidistribution
When regarding a ternary quadratic form $Q(x,y,z)$, is a classic question to consider which integers $n$ can be represented by $Q$. It is also classic to wonder how "well distributed" are ...
- 313
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Correspondence between binary quadratic representations and proper ideals of quadratic number fields
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Aut{Aut}$Fix $d < 0$, a fundamental quadratic discriminant and $n$ a positive integer. Suppose $Q$ is a primitive binary quadratic form of ...
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The density of diagonal isotropic ternary quadratic forms with respect to discriminant
Let $q(x,y,z) = ax^2 + by^2 + cz^2$ be a non-singular diagonal ternary quadratic form with integer coefficients. The discriminant $\Delta(q)$ of $q$ is then equal to $abc$, and for any positive number ...
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