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.
182 questions with no upvoted or accepted answers
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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 $PGL_3(k(\mathbb{P}^n))$...
- 679
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59
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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 ...
- 4,943
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146
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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.
...
- 11
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143
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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 ...
- 85
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44
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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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226
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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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87
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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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130
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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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94
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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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140
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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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143
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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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115
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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 ...
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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-...
- 410
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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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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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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 $...
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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 ...
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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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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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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 ...
- 620
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129
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Siegel's formula for generalized theta series with characteristics?
Siegel's formula(Siegel-Weil) directly relates the weighted sum of theta functions to Eisenstein series. (Or equivalently, the weighted sum of the cusp form is zero). I wonder if there is a ...
- 11
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53
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Stabilizer group uniquely determines subspace
Let $(Q,V)$ be a quadratic space over an algebraically closed field $k$.
Let
$$ SO_Q(k):= \{ \sigma \in GL(V) : Q(\sigma v) = Q(v) \ \text{for all} \ v \in V \ \text{and} \det(\sigma) = 1 \}$$
Let $L \...
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On the image of a polynomial map modulo two distinct primes
Let $Q_0, Q_1, Q_2 \in \mathbb{Z}[x_0, x_1, x_2]$ be three non-singular ternary quadratic forms with integer coefficients. Let $T$ be a large real number, and let $p, q$ be two distinct primes having ...
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360
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How to solve a system of quadratic equations?
Suppose we have a system of $p$ quadratic equations about $\mathbf{x} \in \mathbb{R}^3$ and $\mathbf{x} > 0$
$$ \left\{
\begin{array}{lr}
\mathbf{x}^\top \mathbf{C}_1 \mathbf{x} = 1, ...
- 19
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Modified straightline complexity of almost square of sums
Assume every linear operation (such as inner product with constant vector) can be performed in one step and multiplication by variables (quadratic operation) can be performed in one step.
We know the ...
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Necessary and sufficient condition for a prime to be represented by an arbitrary positive definite binary quadratic form $ax^2+bxy+cy^2$
Given an arbitrary (but fixed) positive definite primitive integral binary quadratic form $g(x, y)=ax^2+bxy+cy^2$, and let $m$ be an arbitrary integer. We will denote the discriminant of $g$ by $D=D_g=...
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146
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Maximizing quadratic forms
Consider the maximization problem
$$\text{maximize} \quad Q(x)= \sum_{i<j} \Big(\sum_{k} a_{ik}a_{jk}\Big) x_i x_j \quad \text{subject to} \quad \sum_{i}x_i^2=1,$$
and let $M$ be maximum value ...
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Computing local residues of traces of bilinear forms on algebraic number fields
This is a question about computing the local residues of a general symmetric bilinear form over $\mathbb{Q}$. I've been using Lam's Introduction to Quadratic Forms as a reference, but I'm stuck on the ...
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150
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When does one quadratic form divide another?
Let $Q_1, Q_2$ be two quadratic forms with integer coefficients in 4 variables $x_1, x_2, x_3, x_4$, both non-singular and not proportional. For a positive number $X$, which we may assume to be large ...
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Proper ideals are invertible
I am reading through Cox's book Primes of the form $x^2+ny^2$ and I am stuck with some proofs in Chapter 7 (I have the 2nd edition). There, the author presents the following Lemma:
Lemma 7.5: Let $...
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Norm quadrics and their motives
Let $k$ be a field of characteristic $\neq 2$ and $\langle\!\langle a_{1},\cdots,a_{n}\rangle\!\rangle$ a Pfister form over $k$. Denote by $Q_{\underline{a}}=Q_{a_{1},\cdots,a_{n}}$ the projective ...
- 141
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87
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On universal sums $x(ax+b)/2+y(cy+d)/2+z(ez+f)/2$ over $\mathbb N$
Let $a,b,c,d,e,f$ be integers with $a\ge c\ge e>0$, $b>-a$ and $a\equiv b\pmod2$, $d>-c$ and $c\equiv d\pmod 2$, $f>-e$ and $e\equiv f\pmod2$. If each $n\in\mathbb N=\{0,1,2,\ldots\}$ can ...
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How explicitly write a projective transformation between the conics over the univariate function field?
Consider the quadratic forms
$$
Q_1 = x^2 + y^2 - (t^2+1)z^2,\qquad Q_2 = x^2 + y^2 - z^2
$$
over the rational function field $\mathbb{F}_p(t)$, where $p > 2$ is a prime such that $t^2 + 1$ is ...
- 1,833
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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 ...
- 179
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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 $(\...
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85
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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 ...
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79
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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 ...
- 111
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Are there any good references to modular form and its application to quadratic forms
I am a beginner of this interesting branch of mathematics，I have read N.Koblitz's Introduction to Ellipitic Curves and Modular Forms.I am familiar with the arithmetic theory of quadratic forms.Thanks ...
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101
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Counting 'admissible' binary quadratic forms
Let $f(x,y) = f_2 x^2 + f_1 xy + f_0 y^2$ be a primitive, positive definite, and reduced binary quadratic form. Put $k_f$ for the fundamental discriminant associated to $f$. That is, $k_f$ is square-...
- 26.9k
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286
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Generalized eigenvalue problem with nonnegative eigenvector constraint
Consider the following problem that is known to be non-convex but can be solved as a generalized eigenvalue problem (i.e. has a global optimum solution):
$\underset{w}{\text{maximize}}\quad w^{\top}...
- 11
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234
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Is the genus symbol implemented?
Conway and Sloane, as well as Cassels, and also O'Meara, all have their own idiosyncratic way of expressing the following result (for good primes): Every quadratic form with coefficients in $\mathbb{...
- 2,429
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421
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Is this QCQP convex or nonconvex?
\begin{equation}
\begin{split}
\min_{x\in \mathbb{R}^n}\:f(x)=(1/2)x^{T}Q_0x+c_0^T x
\end{split}
\end{equation}
s.t.
$$
g_i(x)=\frac{1}{2}x^T Q_ix-lmax_i\leq0,i\in\{1,...,m/2\}
$$
$$
g_i(x)=\frac{...
- 11
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122
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Statements generalizing representability of integers by binary quadratic forms to $n$-variable higher homogeneous forms?
Representing integers through the theory of binary quadratic forms is a well studied topic. We know that given $a,b,c\in\Bbb N$, based on discrimant $b^2-4ac$, we can study the representability of ...
user76479
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396
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Bound of spectral radius of polynomial of a complex matrix
I am trying to prove or disprove the following inequality.
$$ ||P(A)||_2\leq 2 \max_{\alpha\in W(A)}| P(\alpha)|,$$
where $P(\cdot)$ is a complex polynomial, $A\in \mathbb{C}^{n\times n}$ and $W(A)...
- 163
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158
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Minimal Length of Quadratic Forms
Let $Q(x_1,\dots,x_n)=X'PX$ be a quadratic form with all non-negative and integral coefficients given by $$Q(x_1,\dots,x_n)=\sum_{i=1}^cf_i^+(x_1,\dots,x_n)g_i^+(x_1,\dots,x_n)$$ where $c$ is the ...
- 13.9k
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190
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Cassels-Birch-Davenport theorem for multiple quadratic forms of certain type
A classical theorem of Cassels states that if a homogenous quadratic form $Q$ has an integer zero, then there is a zero of small height (bounded solely by the coefficients and number of variables). ...
- 960
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102
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Is this related to a simple property of a lattice?
I am looking for a certain notion of sparseness of lattices.
I want to find a vector in $\mathbb{Z}^N$ that the minimal possible inner product with all the vectors of a given lattice. Or at least, I ...
