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.

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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 $A=-\mathrm{div}g(x)\nabla$ ...
6 votes
3 answers
661 views

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 ...
1 vote
0 answers
123 views

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(...
5 votes
1 answer
345 views

Binary quadratic forms order four in the form class group not having desired coefficients

I have been looking at binary quadratic forms for a question on MSE, If a binary quadratic form primitively represents $n$ and $n^3$, must it be the identity form?, about forms representing a prime (...
2 votes
0 answers
45 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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90 views

Rank-24 Leech matrix cannot have simultaneous integer entries with unit determinant or integer determinant? [migrated]

A typical $E_8$ lattice is of the form of $E_8$ Cartan matrix: ...
0 votes
0 answers
73 views

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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106 views

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
163 views

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 ...
2 votes
1 answer
246 views

Positivity of quadratic form minus linear form on the simplex

Let us $a_{ij}$ 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, \...
2 votes
1 answer
183 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\...
0 votes
0 answers
129 views

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 ...
2 votes
1 answer
332 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 ...
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 $\tilde q$ ...
12 votes
1 answer
854 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
162 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
1 answer
337 views

Integers representable as binary quadratic forms

It is known that odd prime $p$ can be represented as $p=x^2+y^2$ if and only if $p \equiv 1$ mod $4$, represented as $p=x^2+2y^2$ if and only if $p \equiv 1$ or $3$ mod $8$, represented as $p=x^2+3y^2$...
1 vote
0 answers
72 views

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 ...
2 votes
0 answers
57 views

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 ...
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22 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
211 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 ...
2 votes
1 answer
228 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
195 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
87 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
274 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 ...
1 vote
0 answers
94 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 $PGL_3(k(\mathbb{P}^n))$...
3 votes
1 answer
470 views

$p$-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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65 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-...
2 votes
0 answers
68 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=\...
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
1 answer
241 views

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 $\...
4 votes
0 answers
131 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
180 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 $...
6 votes
1 answer
464 views

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 ...
4 votes
0 answers
158 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 ...
8 votes
2 answers
3k views

Connection between quadratic forms and ideal class group

I'm studying the classic results on binary (integer) quadratic forms and I'm looking for a reference on the following result (maybe a book that contains a proof): Let $O_k$ be the ring of algebraic ...
0 votes
0 answers
31 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
50 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
67 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
146 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
226 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 ...
0 votes
1 answer
174 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 \\ &...
3 votes
1 answer
320 views

Strong Approximation for solutions to quadratic Diophantine equations

Can anyone either direct me to an relatively elementary proof in the literature--or show me why this (Conjecture 1 stated below) is true--or if I am mistaken and it is not true: For any 4-tuple $\xi =...
2 votes
1 answer
133 views

Does the F-unitary group isomorphism arises from a conformal isometry?

Let $K$ be a CM-field with totally real subfield $F$. Let $(V_1, h_1)$ and $(V_2, h_2)$ be two $n$-dimensional $K$-vector spaces with nondegenerate Hermitian forms, where $n\geq 3$. Question 1 Does ...
1 vote
0 answers
99 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
228 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
176 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})+...
2 votes
0 answers
189 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 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 ...

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