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
2
votes
0
answers
38
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 ...
- 173
2
votes
0
answers
266
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 ...
- 453
2
votes
0
answers
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, ...
- 313
2
votes
0
answers
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 ...
- 323
2
votes
0
answers
276
views
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 \...
- 739
2
votes
0
answers
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 $...
2
votes
0
answers
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
$$q(x)=\sum_{i =1}^n \alpha_i ...
- 980
2
votes
0
answers
150
views
The number of orbits of a two-point stabilizer of the symplectic group $Sp(2m,2)$
I am trying to figure out the number of orbits of a two-point stabilizer of the action of $Sp(2m,2)$ on its two orbits $\Omega^+$ and $\Omega^-$ as detailed in Dixon and Mortimer's "Permutation ...
- 121
2
votes
0
answers
96
views
Linear forms and the second Voronoi decomposition
This is not my area of expertise, so forgive me if the question is a bit naive. Given a collection of vectors $v_1,\ldots,v_d$ in $\mathbb{R}^n$ (with $d\geq n$), there is a corresponding set of ...
- 147
2
votes
0
answers
114
views
Existence theorem for symmetric nondegenerate forms over a ring
There exists a rich theory for inner product spaces (i.e. vector spaces with a symmetric nondegenerate bilinear form) over fields, and it can be discussed in the context of local rings and free ...
- 327
2
votes
0
answers
192
views
Maximal connected subgroup of orthogonal group
Let $(Q,V)$ be a quadratic space over an algebraically closed field $k$ with $\dim(V) \geq 3$
Define
$$ SO_Q:= \{ \sigma \in GL(V) : Q(\sigma v) = Q(v) \ \text{for all} \ v \in V \ \text{and} \det(\...
- 63
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 $...
- 356
2
votes
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?
$$ 1-\sum_{k=1}^n a_{ik} \lambda_k + \sum_{j=1}^n \sum_{k=1}^n \lambda_j a_{jk} \lambda_k > 0, \...
- 21
2
votes
0
answers
111
views
Inseparable field extensions of degree p and linear independence
Let $F$ be a field of characteristic $p$; let $\alpha \in F$ such that $\alpha \neq \beta^p$ for any $\beta \in F$, and let $K := F(x)$ where $x=\sqrt[p]{\alpha}$.
Is it true that the elements $1,(x-...
- 21
2
votes
0
answers
47
views
Generalization of positive definiteness for a non-smooth quadratic form
I have a "quadratic form" defined as follows:
$$f(\vec x) = \sum_{i<j} c_{ij} x_i x_j + \sum_i a_i(x_i) x_i^2$$
where the coefficients $c_{ij}$ are fixed constants, but the coefficients $a_i(x_i)$...
- 884
2
votes
1
answer
233
views
Superlevel sets of a parametrized quadratic forms
Let $N$ be an odd integer, $n\in\mathbb{N}$, and $-\frac{2T}{NR^2}\leq a\leq0$ for given $R,T\in\mathbb{R}$ with $\frac{T}{NR^2}\leq\frac{\pi}{2}$.
Now consider the quadratic form $\Omega(a)=\sum_{l\...
- 213
2
votes
0
answers
58
views
Number of representations by the norm in a division algebra corresponding to endomorphism rings of elliptic curves
Let $E$ be a supersingular curve over a field of characteristic $p$ with endomorphism ring $\mathcal O_D$ which is a maximal order in a division ring $D$ over $Q$ ramified at $p$ and $\infty$.
The ...
- 7,746
2
votes
1
answer
374
views
A "nice" (but non-definite) quadratic programme
For integers $n\geq k>0$, let $f$ be the following quadratic form:
$$f(x_1,\ldots,x_n)=\sum_{i=1}^n\sum_{j=0}^{k-1}x_ix_{i+j\bmod n}.$$
Is it true that the minimum of $f$ over the unit simplex is ...
- 947
2
votes
0
answers
330
views
Concrete Hanson-Wright inequality?
I'm working on a paper that requires bounding
$$\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 ...
- 21
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 ...
- 452
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 ...
- 1,590
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-...
- 1,813
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 ...
- 2,207
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 (...
- 325
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 ...
- 3,873
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(...
- 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 ...
- 21
2
votes
1
answer
185
views
Real slices of Minkowski space, using a complex quadratic form
Ordinary Minkowski space is $\mathbb{R}^{3,1}:=(\mathbb{R}^4,\phi)$
where $\phi:\mathbb{R}^4\rightarrow\mathbb{R}$
is a quadratic form of signature $(3,1)$.
Lying within this is a hyperboloid model ...
- 2,436
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\...
- 52.3k
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}}:={\...
- 14.2k
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 $a=1$ ...
- 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,...
- 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. ...
- 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 ...
- 2,446
1
vote
2
answers
401
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 ...
- 1,109
1
vote
1
answer
155
views
If $\Vert A-B\Vert_{op}\leq \varepsilon$ then $A^{-1}$ and $B^{-1}$ are uniformly close
Let $A,B$ are two $p\times p$ positive definite matrices such that $0<\delta_0\leq \min\{\lambda_{\min}(A), \lambda_{\min}(B)\}\leq \max\{\lambda_{\max}(A), \lambda_{\max}(B)\}\leq \delta_1$. Also ...
- 399
1
vote
1
answer
152
views
Positive solutions to simultaneous real quadratic equations
I have a system of $n$ quadratic equations with $n$ unknowns. It can be written as
$diag(x)Ax=1$
$x$ is an $n$-vector, $A$ is $n\times n$, real, symmetric and positive definite, the diagonal ...
- 13
1
vote
1
answer
270
views
Local densities of hermitian forms
I think this is an easy question, but I need some time to introduce it. I need to apply Yumiko Hironaka's computations on local densities of hermitian forms (see 1).
I would have liked to create the ...
- 2,446
1
vote
1
answer
155
views
Dimension of binary motives of a quadric
Let $Q$ be a anisotropic quadric of dimension $d$ over $k$.
We work in the category of effective Chow-Motives over $k$.
Let $T$ be the Tate-Motive.
For a motive $M$ we write $M(l)$ for its $l$-th Tate-...
- 1,479
1
vote
2
answers
252
views
Impossible Range for Minkowski-Like Sum of Squares
Given coprime positive integers M,N, and a corresponding integer z outside of the range (for all integers x,a,b,c) of $Mx^2-N(a^2+b^2+c^2)$, is there any such z which is "deceptive", meaning that it ...
- 1,547
1
vote
1
answer
396
views
Tangent space to positive oriented Grassmannians
Let $L$ be a real vector space of dimension 22 and $q$ a quadratic form on $L$ of signature $(3,19)$.
Let $V\subset L$ be a positive oriented subspace of dimension 2 and $G^{po}(2,L)$ be the ...
1
vote
2
answers
850
views
Convexity of inverse quadratic form
Let $\alpha_i\in[0,1]^k$, $x_i\in\mathbb{R}^d$ for all $i\in[k]$, with $k \geq d$. Define $X: \operatorname{col}(X) = \{x_i\}_{i\in[k]}$, $\Lambda(\alpha) = \operatorname{diag}(\alpha)$, $y\in\mathbb{...
- 245
1
vote
1
answer
397
views
How do you solve this quadratic matrix equation?
could you please help me solve this quadratic matrix equation? I look around, seems like there is no general solution for it..
$$-BX^2 + X - C = 0$$ for X, B and C are (3x3) matrices. B and C are ...
- 11
1
vote
1
answer
223
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 ...
- 41
1
vote
1
answer
343
views
Constructing groups of Type E^{66}_{7,1} having non trivial Tits algebra
This can be considered as a continuation of my last useful question:
Constructing groups of Type E7 with certain Tits Index
It is known that a quadratic form $q$ of dimension $12$, having splitting ...
- 1,479
1
vote
1
answer
928
views
Block Covariance Matrix - Positive Definite? (Quadratic Optimization) [closed]
I have a covariance matrix C. I have then formulated an quadratic optimization problem that involves the following matrix in the quadratic form:
[ C C ]
[ C C ]
However, the quadratic solver ...
- 139
1
vote
1
answer
653
views
Existence of non-trivial solution to non linear polynomial system
I need to find conditions for the existence of non-trivial solutions to a multivariable polynomial system in two cases:
The first case:
$f1: a_1x^2+a_2xy+a_3y^2+a_4z^2=0$
$f2: b_1x^2+b_2xy+b_3y^2+...
- 11
1
vote
1
answer
870
views
Diagonalization of quadratic forms over euclidean rings
Let $A$ be a commutative euclidean ring, (probably) with 2 a unit in $A$. I'm trying to compute Witt and Grothendieck-Witt rings and since $A$ is a PID any f.g.p. module over it is free so I only need ...
- 998
1
vote
1
answer
550
views
Number of integers $<\sqrt{d}/2$ represented by an indefinite quadratic form
Given an indefinite integral quadratic form $Q(x,y)=ax^2 +bxy + cy^2$ with $b^2-4ac=d>0$, is there an easy way to count the number of integers in $t \in (-\sqrt{d}/2, \sqrt{d}/2)$ such that there ...
- 809
1
vote
1
answer
349
views
Non-representability by a binary quadratic form
Let $k$ be an arbitrary field, $d\in k$, and $d$ is not a square in $k$.
Consider the binary quadratic form $f(x,y)=x^2-d y^2$
(it is the norm from $k(\sqrt{d})$ to $k$).
I am looking for a reference ...
- 14.2k