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 $A=-\mathrm{div}g(x)\nabla$ ...
Yulia Meshkova's user avatar
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(...
Stanley Yao Xiao's user avatar
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 ...
Bogdan Grechuk's user avatar
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^...
rikhavshah's user avatar
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0 answers
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: ...
zeta's user avatar
  • 327
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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 $k$?
THC's user avatar
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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 $\mathbf{Q}^n$ ...
Tuvasbien's user avatar
  • 156
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 ...
Mastrem's user avatar
  • 458
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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 ...
ReverseFlowControl's user avatar
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 $b(X,Y)=\ell(X)\cdot\ell(Y)$ ...
Anton Petrunin's user avatar
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\} ?$...
Dr. Pi's user avatar
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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 ...
Drew Brady's user avatar
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 $\tilde q$ ...
Anton Petrunin's user avatar
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 ...
Ian Gershon Teixeira's user avatar
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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 \...
Mathieu le provost's user avatar
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 ...
skd's user avatar
  • 5,520
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 ...
Andrea Antinucci's user avatar
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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 ...
vent's user avatar
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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 ...
Sky's user avatar
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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-...
Andi Bauer's user avatar
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1 vote
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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 $PGL_3(k(\mathbb{P}^n))$...
TCiur's user avatar
  • 409
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=\...
wlad's user avatar
  • 4,752
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 ...
semisimpleton's user avatar
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 ...
wlad's user avatar
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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 $$\...
pisco's user avatar
  • 331
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 $...
Noah B's user avatar
  • 397
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 \...
wlad's user avatar
  • 4,752
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 ...
Saegusa's user avatar
  • 173
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 ...
Noah B's user avatar
  • 397
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}...
christian_lee's user avatar
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 $...
christian_lee's user avatar
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 ...
user493645's user avatar
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 ...
dotdashdashdash's user avatar
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 ...
Bipolar Minds's user avatar
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. ...
Louie's user avatar
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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 ...
Misha Verbitsky's user avatar
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})+...
admissiblecycle's user avatar
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 \\ &...
Michael Schindler's user avatar
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 ...
Nanjun Yang's user avatar
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 ...
user139975's user avatar
5 votes
1 answer
197 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 ...
a196884's user avatar
  • 323
1 vote
2 answers
303 views

Integral solutions of quadratic equation $5 X² − 14 X⁢Y + 5 ⁢Y² = n$

Solve for all integers $x$ and $y$ the quadratic form $5 X² − 14 X⁢Y + 5 ⁢Y² = n$ for some integer n. I know that for some cases there are recurrence solutions, but I'm not sure how to solve these ...
Benjamin L. Warren's user avatar
2 votes
0 answers
52 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, ...
MathqA's user avatar
  • 313
11 votes
1 answer
848 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 ...
Bogdan Grechuk's user avatar
1 vote
0 answers
113 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 ...
reservoir's user avatar
1 vote
0 answers
39 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 ...
Windi's user avatar
  • 813
-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-\...
Turbo's user avatar
  • 13.5k
2 votes
1 answer
98 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 ...
MathqA's user avatar
  • 313
4 votes
1 answer
183 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 ...
a196884's user avatar
  • 323
1 vote
0 answers
166 views

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 ...
gtg's user avatar
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