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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System of quadratic equations

Let $B_1,\ldots,B_s$ be $(s\times s)$ symmetric real matrices and $x=\left(x_1,\ldots,x_s\right)^\prime$ a $(s\times 1)$ vector of unknowns. Is there a way or reference theory for studying the ...
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+50

Witt ring of a field with Pythagoras number $2$

I am currently looking at a few simple properties of the Witt ring of a field $K$ (by which I mean the ring of Witt classes of quadratic forms, not the ring of Witt vectors), which are clearly true ...
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59 views

Existence of solutions to quadratic vector equation

Given $$A_{i,j,k} X_j^* X_k + C_i = 0$$ where $ūĚźī_{i,j,k}$ and $ūĚź∂_i$ are arbitrary complex numbers for all $ūĚĎó, ūĚĎė$ which are $ūĚĎĀ$-dimensional indices and $i$ which is an $m$-dimensional index where $...
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40 views

Simultaneous diagonalization of two rational forms

It is known that any two real quadratic forms are equivalent iff they have the same signature. If we consider rational quadratic forms, they are $\mathbb{Q}$-equivalent iff the have the same signature,...
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156 views

Is there an integer-valued quadratic polynomial $P(x,y)$ such that $\{P(x,y)+2^k:\ x,y\in\mathbb Z\ \text{and}\ k\in\mathbb N\}=\mathbb Z^+$?

I seek for very sparse representations of positive integers. Let $$\mathbb N=\{0,1,2,\ldots\}\ \ \ \text{and}\ \ \ \mathbb Z^+=\{1,2,3,\ldots\}.$$ Recall that a polynomial $P(x,y)$ is integer-valued ...
2
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1answer
261 views

When are rings of the form $K[x_1,…,x_n]/(Q)$ principal ideal domains when $Q$ is quadratic?

By a result of Klein-Nagata rings of the form $A_Q=K[x_1,...,x_n]/(Q)$ are factorial when $K$ is a field, $n \geq 5$ and $Q$ is a non-degenerate quadratic form. Question 1: When is $A_Q$ a principal ...
4
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1answer
160 views

Computing the genus of certain ternary indefinite lattices

For $k$ a positive integer, let us consider the rank 3 ternary indefinite lattice $L=L_k$ with quadratic form $$6kx^2-2(y^2+yz+z^2).$$ Its discriminant group has length $2$. Question. Is this lattice ...
4
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1answer
223 views

Classification of vector spaces with a quadratic form and an order n automorphism

Introductory general nonsense (for motivation: feel free to skip): Let $G$ be a finite group and $k$ be a field of characteristic $0$. Consider the set $\mathcal{S}$ of isomorphism classes of finite ...
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2answers
79 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{...
3
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1answer
59 views

The covariance matrix of quadratic form, without normal assumption

Assume $\mathbf{x}$ is a random vector with mean $\mathbf{\mu}$ and covariance matrix $\mathbf{\Sigma}$. Symmetric matrices $\mathbf{A}$ and $\mathbf{B}$ are given. Without assuming normality, how to ...
3
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2answers
113 views

Generators of the orthogonal group of a quadratic form in odd dimension in characteristic 2

In characteristic not $2$, the Theorem of Cartan-Dieudonné states: [Grove, Theorem 6.6]: Let $q$ be a nondegenerate symmetric quadratic form of dimension $n$ in characteristic not $2$. Then every ...
4
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1answer
212 views

Is the map $GL_n(\mathbb{Z})\to GL_n(\mathbb{Z}/2\mathbb{Z})$ surjective?

Suppose $F$ is a field. I want to know whether the map $GL_n(GW(F))\to GL_n(W(F))$ is surjective, where $GW$ means Grothendieck-Witt and $W$ means Witt. In the case $F$ is algebraic closed, it reduces ...
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79 views

finding automorphisms of binary hermitian forms

Set $K$ 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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1answer
310 views

Under what conditions is the polynomial of degree $6$ irreducible?

Let $k$ be a perfect field of characteristic $p \neq 2,3$ such that $\omega := \sqrt[3]{1} \in k$, where $\omega \neq 1$. Consider an absolutely irreducible (not necessarily homogenous) quadratic ...
4
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1answer
114 views

Finding Motzkin's original paper on copositive quadratic forms

I am currently in the process of writing my thesis about copositive matrices and would like to write a chronological narrative about the ascent of these matrices to the prominent place they have today ...
2
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1answer
133 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 (...
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1answer
604 views

A list of proofs of the Hasse‚ÄďMinkowski theorem

I am currently doing a project in which I intend to include the most insightful possible proof of the Hasse‚ÄďMinkowski theorem (also known as the Hasse principle for quadratic forms, among other names) ...
5
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1answer
136 views

Modular forms and number of representations by binary quadratic forms

Let $Q(x,y)$ be a positive definite quadratic form of discriminant $d$. Let $r_Q(n)$ be the number of solutions of $Q(x,y)=n$. It is known that the function $f_Q(\tau)=\sum_{n=0}^{\infty}r_Q(n)q^n$ is ...
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60 views

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 $X$ ...
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58 views

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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35 views

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 ...
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1answer
71 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 ...
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2answers
221 views

Correspondence between binary quadratic representations and proper ideals of quadratic number fields

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Aut{Aut}$Fix $d < 0$, a fundamental quadratic discriminant and $n$ a positive integer. Suppose $Q$ is a primitive binary quadratic form of ...
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152 views

Linearly independent quadratic forms vanishing on a finite set of points

The question I am interested in can be summed up as follows: given positive integers $n,m,k$, how do we write down $m$ linearly independent quadratic forms $Q_1, \cdots, Q_m \in \mathbb{C}[x_0, \cdots,...
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138 views

How should the first n natural numbers be arranged in a circle to minimize the sum of the products of adjacent pairs? [closed]

I was able to find (and prove) arrangements that would result in the sum of the products of adjacent pairs attain the maximum. I am able to conjecture that the arrangement that would result in the ...
6
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1answer
114 views

Hasse invariant and the Clifford algbera

Let $$q = a_1 x_1^2 + \cdots + a_n x_n^2$$ be a quadratic form over some $p$-adic field $\mathbb{Q}_p$. We thus have its Hasse invariant $$\mathcal{h}(q) = \prod_{1 \leq i < j \leq n} (a_i,a_j)_p \...
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80 views

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 ...
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16 views

Integral of the product l of the pdf of two univariate gaussian regression models with respect to the parameter

good evening to everyone. I am having trouble finding a way of calculating the integral of the product of the pdfs of two univariate Gaussian regression models with respect to the parameter, and I was ...
7
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125 views

the gaps between values of a positive binary quadratic form at integer points

Suppose that $s$ is a positive irrational number. Consider all possible values of the sum $x^2+sy^2$ where $x$ and $y$ are integers. These values form a set $W=\{0=w_1<w_2<\ldots\}$. Can the ...
8
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1answer
223 views

Standard conjecture on u-invariants?

This is well beyond my expertise, but I just learned some of the history behind $u$-invariants of fields $F$, where ($u(F)+1$)-variable quadratic equations always have a non-trivial solution, but $u(F)...
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1answer
214 views

Questions on $x^2+y^2+z^2$, $x^2+y^2+2z^2$ and $x^2+2y^2+3z^2$

Question 1. My computation in 2018 suggests that a sum of two integer squares is a sum of three nonzero integer squares if and only if it is not of the form $$4^km\ \ (k=0,1,2,\ldots;\ \ m=1,2,5,10,13,...
6
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1answer
114 views

Invariant Lagrangian in the first homology of a cover of a surface

If $\tilde{S}\to S$ is a Galois cover of a surface of genus $g$ and group $G$, a well-know theorem of Chevalley-Weil tells that, as a $\mathbb{Q}[G]$-module, one has $$H_1(\tilde{S},\mathbb{Q})=\...
4
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1answer
309 views

Is there a trigonometric field which is different enough from real numbers?

I found this topic in a book 'Metric Affine Geometry' by Ernst Snapper and Robert J. Troyer. I call a field $k$ trigonometric iff there is a quadratic form $q$ over $k^2$ such that every two lines ...
2
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0answers
54 views

Is there a method to solve a non-linear quadratic matrix equation?

I am interested in solving the following quadratic equation: $$x^{\top} A x = \sqrt{x^{\top} B x}$$ Here, $x \in \mathbb{R^q}$ is an unknown vector, and A and B are two q$\times$q-dimensional ...
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1answer
168 views

$7n=x^2+2y^2+4z^2$ with or without $x^2\equiv y^2\equiv z^2\pmod 7$

It is well known that any positive odd integer can be written as $x^2+2y^2+4z^2$ with $x,y,z\in\mathbb Z$. Question 1. Whether for any odd integer $n>93$ there are $x,y,z\in\mathbb Z$ such that $7n=...
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169 views

Leech lattice and rational varieties

Question: Is there a smooth rational variety $X$ of complex dimension $4n$, $n \in \mathbb{N}$; such that the intersection form on $H^{4n}(X,\mathbb{Z})$ is the Leech lattice? My motivation is mainly ...
4
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1answer
157 views

Conics and triples of binary quadratic forms

Let $C \subset \mathbb{P}^2$ be a planar conic curve, defined by a ternary quadratic form $Q(x_1, x_2, x_3)$ say. Suppose that $C(\mathbb{Q}) \ne \emptyset$, or equivalently, that $C$ is everywhere ...
5
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1answer
375 views

Proof that $x^2 + y^2 - z^2$ is universal

The (ternary) quadratic form $x^2 + y^2 - z^2$ is universal, meaning that any integer $n$ can be represented as $n = x^2 + y^2 - z^2$ for some integers $x, y, z$. My question is this: who proved this ...
5
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1answer
828 views

Set of quadratic forms that represents all primes

A SPECIFIC CASE: Any prime number can be classified as either $p \equiv 1 \pmod 3$ or $p \equiv 2 \pmod 3$. If $p = 3$ or $p = 1 \pmod 3$, then the prime $p$ can be represented by the quadratic form $ ...
3
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1answer
170 views

Reference request: relationship between discriminant and smoothness of a conic over arbitrary fields

I'm looking for an (ideally modern) reference of the relationship between the discriminant and smoothness of projective conics over arbitrary fields (including those of characteristic 2). Let $k$ be a ...
2
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0answers
71 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 ...
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2answers
376 views

Determining if a quadratic form is non-negative if variables are non-negative

Let $f(x_1,\dots,x_n) = \sum_{1 \le i \le j \le n} c_{i,j}x_ix_j$ be a homogeneous quadratic form. Is there a quick-ish way to determine whether $f(x_1,\dots,x_n) \ge 0$ for all $x_1,\dots,x_n \ge 0$? ...
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151 views

Characterization of certain 4-dimensional lattices

Let $\Lambda \subset {\bf Q}^4$ be a lattice, i.e., $\Lambda$ is a free abelian group and $\Lambda \otimes {\bf Q} = {\bf Q^4}$. The determinants of those dilation-rotations (i.e. linear maps of ${\bf ...
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45 views

Can you efficiently solve a linear system with a quadratic multivariate polynomial of degree 2?

Given a system of linear equations, $P=\{p_1,\dots,p_m-1\}$ and a $2$nd degree polynomial $p_m$ where $p_i: \mathbb{R}^n \rightarrow \mathbb{R},$ can you efficiently find a common zero of all of these ...
0
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1answer
80 views

The stabiliser group of an isotropic quadratic form over $\mathbb{Q}_p$ is non-compact?

Let $\mathbb{Q}_p$ denote the $p$-adic integers. Let $V$ be a $\mathbb{Q}_p$-vector space and $Q : V \rightarrow \mathbb{Q}_p$ be a non-degenerate integral quadratic form. We say that the pair $(Q,V)$ ...
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1answer
123 views

Doubt about lemma for polynomial equivalence [closed]

Multivariate polynomials $f,g$ are equivalent if there exists invertible linear transformation $A$ such that $f(X)=g(A\cdot X)$ From paper p.1: Lemma 1.1. (Structure of quadratic polynomials). Let $F$...
6
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3answers
238 views

Representation theorem for quadratic form on Hilbert space

I think my question is more suitable for Mathematics Stack Exchange than to MathOverflow but I've already posted two related questions there and I got even more confused, so maybe I can clarify things ...
2
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0answers
92 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(\...
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33 views

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 \...
3
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1answer
276 views

Close integer solutions to $ab-cd=1$?

I am looking for infinite set of Diophantine solutions. Suppose we require $$0<\min(a,d)<\max(a,d)<\min(b,c)<\max(b,c)\leq\sqrt 2\min(a,d)$$ $$a,b,c,d\in\mathbb Z$$ then can we still find ...

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