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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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Squares in skew fields of dimension 2 over a sub skew field

Let $\ell$ be a skew field (i.e., a division ring), and let $k$ be a sub skew field, such that the dimension of $\ell$ as a left vector space over $k$ is $2$. Then if $a \in \ell \setminus k$, we can ...
THC's user avatar
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Sets represented by integral quadratic forms

Let $f(x) = x^\intercal A x$ be a positive definite integral quadratic form on $d$ variables. A positive integer $n$ is said to be represented by $f$ if $f(x) = n$ for some $x \in \mathbb Z^d$. A set ...
Andre Kornell's user avatar
-4 votes
1 answer
188 views

How to express a quadratic polynomial exactly as a power series [closed]

I claim, for $\operatorname{artanh}(\rho) = \frac{1}{2} \ln\left(\frac{1+\rho}{1-\rho}\right)$, i.e., the inverse hyperbolic tangent function, the following holds approximately under assumptions given ...
virtuolie's user avatar
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1 answer
122 views

Automorphism groups in class sets of ternary lattices

Let $\Lambda$ be an integral lattice in some definite ternary quadratic space $(V,Q)$ over $\mathbb{Q}$. Consider the usual class set $\text{Cl}(\Lambda) = O(V)\backslash\text{Gen}(\Lambda)$, i.e. the ...
fretty's user avatar
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1 vote
1 answer
54 views

Low rank matrices which preserve maximizers of quadratic form

Suppose $x, y \in \{0,1\}^d$ are binary vectors. For a matrix $M$ consider the quadratic form, \begin{align} x^T M y + (\mathbb{1} - x)^T M (\mathbb{1} - y) \end{align} Does there exist an $M$ such ...
Television's user avatar
6 votes
0 answers
350 views

A possible variant of Zagier's one-sentence proof for Fermat's sum of two squares theorem?

Is it possible to modify Zagier's one-sentence proof of Fermat's sum of two squares theorem (see here) to prove certain non-trivial cases of Jacobi's four-square theorem (see here)? Let $p$ be a prime ...
Mathew's user avatar
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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 ...
Student88's user avatar
  • 503
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0 answers
89 views

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 ...
JBuck's user avatar
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2 votes
1 answer
169 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)\...
Yulia Meshkova's user avatar
1 vote
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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(...
Stanley Yao Xiao's user avatar
8 votes
4 answers
848 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
50 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
78 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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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
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2 votes
1 answer
171 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
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0 answers
139 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
894 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
186 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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1 vote
0 answers
83 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 ...
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
1 answer
104 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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0 answers
51 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 \...
Mathieu le provost's user avatar
3 votes
1 answer
288 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
  • 5,590
4 votes
1 answer
222 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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0 answers
153 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
304 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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0 votes
0 answers
81 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-...
Andi Bauer's user avatar
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1 vote
0 answers
107 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
  • 557
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=\...
wlad's user avatar
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2 votes
1 answer
252 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
  • 4,873
5 votes
0 answers
160 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
  • 381
4 votes
1 answer
193 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
  • 403
4 votes
0 answers
182 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
  • 4,873
2 votes
0 answers
34 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
510 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
  • 403
0 votes
0 answers
52 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
2 votes
0 answers
214 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
252 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
136 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
  • 11
7 votes
0 answers
243 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
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})+...
admissiblecycle's user avatar
0 votes
1 answer
182 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
94 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
69 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
240 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
348 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
56 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
10 votes
1 answer
888 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
134 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

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