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.
523
questions
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Convergence in probability of quadratic form with positive mean
Let $\boldsymbol{X}_n\in\mathbb{R}^n$ be a sequence of Gaussian random vectors with independent entries, such that $X_{n,i}\sim \mathcal{N}(\mu_i,\sigma^2)$ (that is, all entries of the $n$th vector ...
0
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0
answers
82
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Totally isotropic space for bilinear pairing over ring
A duplicate of this:
Consider the following well-known inequality: Let $b$
be a non-degenerate symmetric bilinear pairing over a (finite-dimensional) $\mathbb F$-vector space $V$ and $W$
a totally ...
2
votes
1
answer
153
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)\...
1
vote
0
answers
133
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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
$$\displaystyle q(...
8
votes
4
answers
836
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A cubic equation, and integers of the form $a^2+32b^2$
I am trying to determine whether there are any integers $x,y,z$ such that
$$
1+2 x+x^2 y+4 y^2+2 z^2 = 0. \quad\quad\quad (1)
$$
It is clear that $x$ is odd. We can consider this equation as quadratic ...
2
votes
0
answers
48
views
Quadratic surjective map between spheres
The quadratic function $f:\mathbb R^4\to\mathbb R^3$
$$f(a,b,c,d)=\begin{bmatrix} 2(ac + bd)&2(ad - bc)&a^2 + b^2 - c^2 - d^2\end{bmatrix}$$
surjectively maps the sphere $S^3$ to the sphere $S^...
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answers
76
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Squares in division ring extensions $\ell/k$ with $[\ell:k] = 2$
Let $k$ and $\ell$ be division rings such that $\ell$ contains $k$, and $[\ell : k] = 2$. When do I know that there is an element $a \in k$ such that $x^2 = a$ has solutions in $\ell$, but not in $k$?
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answers
108
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Riemannian geometry applied to diophantine equations
I am looking for a book or lecture notes on applications of riemannian geometry to diophantine equations. The main motivation is the following : suppose i have a quadratic form $q$ on $\mathbf{Q}^n$ ...
2
votes
1
answer
169
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Integers $8k+3>0$ not represented by $2x^2+4y^2+4yz+9z^2$ over the integers
Oeis A306970 lists positive integers of the form $8k+3$ which are not reprented by
$$f(x,y,z):=2x^2+4y^2+4yz+9z^2$$
over the integers as $3,43,163,907$. It says this list may not be complete and ...
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answers
138
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Positive definite quadratic form algorithm
Let $f(x,y)= ax^2+bxy+cy^2$, or similarly denote it by $(a,b,c)$. This question is about the case $(1,0,p)$ where $p$ is prime. Suppose I have one solution $\bar{x}_1=(x_0,y_0)$ for $f(x,y)=m$ for ...
12
votes
1
answer
884
views
Positive 4-form
Denote by $W$ the space of all symmetric bilinear forms on $\mathbb{R}^n$.
Let $Q$ be a quadratic form on $W$.
Suppose that $Q(b)\geqslant 0$ for any $b\in W$ such that $b(X,Y)=\ell(X)\cdot\ell(Y)$ ...
0
votes
1
answer
169
views
number of representations by sums of three squares (with coefficients)
There are formulas for counting the number of representations of a positive integer $N$ as a sum of three integer squares. What is a reference for
$$
\#\{(x,y,z)\in \mathbf{N}^3: 5^4 x^2+y^2+z^2=N\}
?$...
1
vote
0
answers
78
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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 ...
15
votes
2
answers
1k
views
Positive quadratic polynomial
Let $S$ be solutions of a system of quadratic polynomials on $\mathbb{R}^n$.
Suppose $q$ is another quadratic polynomial such that $q|_S\geqslant 0$.
Is it possible to find a polynomial $\tilde q$ ...
2
votes
1
answer
89
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Sufficient condition for pair of real quadrics to have real intersection
In the following, when I talk about the zero of a homogeneous polynomial I always mean a projective zero.
Let $ q $ be a real quadric. Then $ q $ has a real zero if and only if $ q $ has indefinite ...
0
votes
0
answers
34
views
Intersection of a hyper-ellipsoid with an hyperplane
Let consider an hyper-ellipse $\mathcal{Q}$ in $\mathbb{R}^n$ given by $x^\top Q x = K$ and a hyperplane $\mathcal{H}$ in $\mathbb{R}^n$ given by $a^\top x = b$. We assume that $Q \in \mathbb{R}^{n \...
2
votes
1
answer
254
views
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 ...
4
votes
1
answer
206
views
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 ...
0
votes
0
answers
112
views
Genus of quadratic form
I am trying to understand the genus of a lattice from Conway and Sloane textbook. They said two quadratic forms $Q_1$ and $Q_2$ lie in the same genus if they are equivalent over $\mathbb{R}$ and over ...
4
votes
1
answer
293
views
Fields in which $ -1 $ can't be written as sum of two square elements
We say a field $F$ has the property $*$ if the equation $x^2 + y^2=-1$ has no solution in $F$. For an example if $F$ is a subfield of real numbers then $F$ satisfies $*$. On the other hand if $ F $ is ...
0
votes
0
answers
75
views
Is a Lagrangian subgroup of a metric group isomorphic to its quotient?
A metric group is a finite abelian group $G$ with a quadratic function
$$q:G\rightarrow \mathbb R/\mathbb Z\;,$$
that is,
$$M(a,b):= q(a+b)-q(a)-q(b)$$
is bilinear in $a$ and $b$ [edit: and non-...
1
vote
0
answers
98
views
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 $PGL_3(k(\mathbb{P}^n))$...
2
votes
0
answers
71
views
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=\...
2
votes
1
answer
250
views
An arithmetic problem involving a system of equations
Fix a positive integer $r$. Describe the solutions to the system of equations given by:
$$\begin{equation}\sum_{1\leq i\leq r}X_i^2\equiv0\pmod{X_k}(1\leq k\leq r)\end{equation}$$
Example: In the case ...
1
vote
0
answers
58
views
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 ...
4
votes
0
answers
139
views
Epstein zeta function for non-fundamental discriminant to L-series
Let $Q(x,y) = ax^2+b xy + cy^2$ be a primitive integral positive-definite quadratic form, with associated number field $K$. If $D=b^2-4ac$ is a fundamental discriminant, then it's well-known that $$\...
4
votes
1
answer
190
views
Schur multiplier of a Chevalley group of type $D_5$
$\DeclareMathOperator\EO{EO}\DeclareMathOperator\SO{SO}\DeclareMathOperator\St{St}\DeclareMathOperator\Sp{Sp}$This is sort of a follow up question to my post here regarding the commutator subgroup of $...
4
votes
0
answers
169
views
Is an orthogonal direct sum decomposition with respect to two quadratic forms necessarily unique up to isomorphism
Consider two quadratic forms $Q$ and $P$ over a finite dimensional vector space $V$ over a quadratically closed (or perhaps Pythagorean) field $F$. If $V$ can be decomposed as $V = V_1 \oplus V_2 \...
2
votes
0
answers
33
views
Criterion for unicity and existence of pre-image in multivariate cryptography
Repost from math.stackexchange since no one could help me there and it concerns my research.
I am reading Ding's Multivariate Public Key Cryptosystems and in the book the author explains the so-called ...
6
votes
1
answer
488
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Computing a Commutator Subgroup
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}$I’m studying the group $\O(5,5,\mathbb{Z})$, the indefinite orthogonal matrices with integer entries. In particular, I ...
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votes
0
answers
32
views
How to express the following formula with quadratic form?
A and B are two constants, and $x_1,x_2,x_3$ are all binary vectors with the same length. How to express the following formula with quadratic form?
$$AB \vec{x}_1^T \vec{p}_1 \cdot \vec{x}_2^T \vec{p}...
0
votes
0
answers
51
views
How to express the product of elements of a vector with quadratic form?
$x$ is a binary vector which means the elements in $x$ are 0 or 1, and $p$ is another vector with the same length. How to express the product of elements in $p$ whose corresponding elements are 1 in $...
0
votes
0
answers
78
views
Is the absolute value of a complex quadratic form a convex real function?
Consider a complex vector space $V = \mathbb{C}^n$ and a quadratic form $Q(x) = x^TAx$ on $V$ where $A$ is a symmetric matrix i.e., $A^T = A$.
Is is true that the absolute value $|Q(x)|$, seen as a ...
2
votes
0
answers
189
views
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 ...
4
votes
1
answer
237
views
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 ...
1
vote
0
answers
120
views
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.
...
7
votes
0
answers
236
views
K3 surfaces with no −2 curves
I seem to remember that a K3 surface with big Picard rank always
has smooth rational curves.
This question is equivalent to the following question about integral quadratic lattices. Let us call a ...
0
votes
0
answers
177
views
Points at which a polynomial becomes reducible
Let $n \geq 10$ and set $\mathbf{y} = (y_1,\ldots,y_n)$. Let $Q_1(\mathbf{y}),\ldots,Q_5(\mathbf{y})$ be non-zero quadratic forms with integer coefficients such that the cubic form $x_1Q_1(\mathbf{y})+...
0
votes
1
answer
178
views
Compatibility conditions for quadratic equations
In the context of physics, I stumbled over the following problem: I have $N$ equations, all are quadratic in a single scalar, real variable $x$:
\begin{eqnarray}
0 &= A_1x^2 + B_1x + C_1 \\
&...
1
vote
1
answer
91
views
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 ...
1
vote
1
answer
68
views
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 ...
5
votes
1
answer
213
views
Equivalence of quadratic forms over $p$-adic integers vs over localisation at $p$
To discern whether two integral quadratic forms are equivalent over the $p$-adic integers, one can compute a Jordan decomposition at $p$ and read off some invariants. Restricting to $p\ne2$ for ...
1
vote
2
answers
330
views
Integral solutions of quadratic equation $5 X² − 14 XY + 5 Y² = n$
Solve for all integers $x$ and $y$ the quadratic form $5 X² − 14 XY + 5 Y² = n$ for some integer n. I know that for some cases there are recurrence solutions, but I'm not sure how to solve these ...
2
votes
0
answers
53
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, ...
10
votes
1
answer
871
views
How to describe all integer solutions to $x^2+y^2=3z^2+1$?
The question is in the title. Here is a short motivation. The general quadratic Diophantine equation is
$$
x^TAx+bx+c=0,
$$
where $x$ is a vector of $n$ variables, $A$ is $n \times n$ matrix with ...
1
vote
0
answers
121
views
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 ...
1
vote
0
answers
40
views
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 ...
-1
votes
1
answer
128
views
Does this quadratic system admit an integral or a rational solution?
Let $a,b$ be coprime and say $0<a<b<2a$.
Consider the quadratic system:
$$\alpha\delta-\beta\gamma=1$$
$$(\alpha^2-(\alpha\delta+\beta\gamma))a^2b+\beta^2b^3+(2\alpha\beta-\beta\delta)ab^2-\...
2
votes
1
answer
105
views
Primitive representation of integers by some form on the genus of a quadratic form
Some time ago, I asked a question about equidistribution on a paper of Duke and Schulze-Pillot that was usefully answered.
However, on the answer there was a statement that was unimportant for me back ...
4
votes
1
answer
184
views
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 ...