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.
543 questions
3
votes
1
answer
481
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 ...
- 661
4
votes
1
answer
174
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 ...
- 34.3k
6
votes
1
answer
331
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 ...
- 25.7k
1
vote
0
answers
90
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 $X$ ...
- 12.9k
1
vote
0
answers
100
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 ...
- 50.6k
1
vote
0
answers
43
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 ...
- 4,729
8
votes
1
answer
720
views
Are lattice points in thin spherical shells uniformly distributed?
Consider the spherical shell (annulus)
$$A(R,r) = \{ x \in \mathbb{R}^3 : R \leq |
x|\leq R+r \}.$$ Think of the limit $R \to \infty$.
Assume that $r$ depends on $R$ as $r(R) = R^{-\delta}$. We are ...
- 12.9k
1
vote
1
answer
397
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 ...
- 20.2k
12
votes
2
answers
3k
views
On the positive definiteness of a linear combination of matrices
In my work in PDE, the following problem in linear algebra came up. Any help in this direction is appreciated.
QUESTION:
Let $m,n\in\mathbb{N}$ and let $A_1,\ldots, A_m\in M_n(\mathbb{R})$ be real, ...
- 63.9k
3
votes
2
answers
635
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 ...
- 316
5
votes
0
answers
237
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,...
- 26.9k
6
votes
1
answer
263
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
$$\mathcal{h}(q) = \prod_{1 \leq i < j \leq n} (a_i,a_j)_p \...
- 476
1
vote
0
answers
150
views
How should the first n natural numbers be arranged in a circle to minimize the sum of the products of adjacent pairs? [closed]
I was able to find (and prove) arrangements that would result in the sum of the products of adjacent pairs attain the maximum.
I am able to conjecture that the arrangement that would result in the ...
CommunityBot
- 1
1
vote
0
answers
129
views
Siegel's formula for generalized theta series with characteristics?
Siegel's formula(Siegel-Weil) directly relates the weighted sum of theta functions to Eisenstein series. (Or equivalently, the weighted sum of the cusp form is zero). I wonder if there is a ...
- 11
7
votes
0
answers
224
views
the gaps between values of a positive binary quadratic form at integer points
Suppose that $s$ is a positive irrational number. Consider all possible values of the sum $x^2+sy^2$ where $x$ and $y$ are integers. These values form a set $W=\{0=w_1<w_2<\ldots\}$. Can the ...
- 109k
-1
votes
1
answer
252
views
Questions on $x^2+y^2+z^2$, $x^2+y^2+2z^2$ and $x^2+2y^2+3z^2$
Question 1. My computation in 2018 suggests that a sum of two integer squares is a sum of three nonzero integer squares if and only if it is not of the form
$$4^km\ \ (k=0,1,2,\ldots;\ \ m=1,2,5,10,13,...
- 43.7k
4
votes
1
answer
377
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 ...
- 63.9k
3
votes
0
answers
138
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 ...
- 31
-1
votes
1
answer
182
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=...
CommunityBot
- 1
4
votes
1
answer
197
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 ...
- 1,766
9
votes
0
answers
195
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 ...
- 6,995
5
votes
1
answer
503
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 ...
- 15.6k
6
votes
1
answer
1k
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 $ ...
- 105k
3
votes
1
answer
220
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 ...
- 7,527
11
votes
0
answers
158
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 ...
- 11.9k
2
votes
0
answers
114
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 ...
- 327
7
votes
2
answers
632
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 $x_1,\dots,x_n \ge 0$?
...
- 3,979
-3
votes
1
answer
140
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 $F$...
- 12.9k
6
votes
3
answers
830
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 ...
- 1,305
2
votes
0
answers
192
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
$$ SO_Q:= \{ \sigma \in GL(V) : Q(\sigma v) = Q(v) \ \text{for all} \ v \in V \ \text{and} \det(\...
- 63
1
vote
0
answers
53
views
Stabilizer group uniquely determines subspace
Let $(Q,V)$ be a quadratic space over an algebraically closed field $k$.
Let
$$ SO_Q(k):= \{ \sigma \in GL(V) : Q(\sigma v) = Q(v) \ \text{for all} \ v \in V \ \text{and} \det(\sigma) = 1 \}$$
Let $L \...
- 63
3
votes
1
answer
370
views
Close integer solutions to $ab-cd=1$?
I am looking for infinite set of Diophantine solutions.
Suppose we require
$$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$$
then can we still find ...
- 1,826
6
votes
1
answer
1k
views
Explicit computation of spinor norm
I've asked this on math.stackexchange, unsuccessfully. I hope this question is appropriate for mathoverflow.
Let $V$ be a finite-dimensional vector space over a field $K$ with $\operatorname{char}K\...
- 12.9k
1
vote
0
answers
76
views
On the image of a polynomial map modulo two distinct primes
Let $Q_0, Q_1, Q_2 \in \mathbb{Z}[x_0, x_1, x_2]$ be three non-singular ternary quadratic forms with integer coefficients. Let $T$ be a large real number, and let $p, q$ be two distinct primes having ...
- 26.9k
1
vote
0
answers
360
views
How to solve a system of quadratic equations?
Suppose we have a system of $p$ quadratic equations about $\mathbf{x} \in \mathbb{R}^3$ and $\mathbf{x} > 0$
$$ \left\{
\begin{array}{lr}
\mathbf{x}^\top \mathbf{C}_1 \mathbf{x} = 1, ...
- 19
2
votes
0
answers
114
views
Projective group of Plucker quadric over the reals
A somewhat elementary question but seemingly difficult to find a suitable reference:
Consider the six-dimensional real space $\wedge^2(\mathbb R^4)$ with basis $e_i \wedge e_j \ (i < j)$ where $...
- 356
1
vote
0
answers
2k
views
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=...
CommunityBot
- 1
1
vote
0
answers
28
views
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 ...
CommunityBot
- 1
17
votes
2
answers
4k
views
On Siegel mass formula
I have asked this question exactly here. The question is as follows:
I am interested deeply in the following problem:
Let $f$ be a (fixed) positive definite quadratic form; and let $n$ be an ...
9
votes
3
answers
4k
views
quadratic forms over fields of characteristic 2
I was wondering if anyone knows any good sources for the theory of quadratic forms over fields of characteristic 2 which are written in English?
7
votes
1
answer
276
views
Is a 8-dimensional quadratic form recognized by its Lie algebra, modulo equivalence and scalar multiplication?
Question. Let $K$ be a field of characteristic zero (large characteristic should be fine too). Let $q,q'$ be two non-degenerate quadratic forms on $K^n$ with $n=8$. Suppose that the Lie algebras $\...
- 2,178
0
votes
0
answers
221
views
Number of integer solutions to quadratic polynomial with integer coefficients
It is known from for example Representations of Integers as Sums of Squares by Grosswald, E. that
$$|\{(n_1,n_2,\ldots,n_k)\in\mathbb{Z}^k: \ n_1^2+n_2^2+\cdots+n_k^2=N\}|\leq C_\varepsilon N^{\frac{...
- 193
3
votes
3
answers
590
views
Solving diagonal simultaneous quadratic equations
A problem I am trying to solve has led to me to the following system of equations:
$$A(x^2) + Bx + c = 0$$
Where $A$ and $B$ are known matrices, $c$ is a known vector, $x$ is the vector of unknowns ...
- 1,216
3
votes
1
answer
250
views
Subring of quaternion algebra
I am following the book Introduction to Quadratic Forms over Fields by T. Y. Lam. In section VI.2, the author proves that, over an arbitrary local field $F$, there is a unique quaternion division ...
- 5,382
1
vote
0
answers
146
views
Maximizing quadratic forms
Consider the maximization problem
$$\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,$$
and let $M$ be maximum value ...
2
votes
0
answers
111
views
Inseparable field extensions of degree p and linear independence
Let $F$ be a field of characteristic $p$; let $\alpha \in F$ such that $\alpha \neq \beta^p$ for any $\beta \in F$, and let $K := F(x)$ where $x=\sqrt[p]{\alpha}$.
Is it true that the elements $1,(x-...
- 2,051
2
votes
0
answers
47
views
Generalization of positive definiteness for a non-smooth quadratic form
I have a "quadratic form" defined as follows:
$$f(\vec x) = \sum_{i<j} c_{ij} x_i x_j + \sum_i a_i(x_i) x_i^2$$
where the coefficients $c_{ij}$ are fixed constants, but the coefficients $a_i(x_i)$...
- 884
24
votes
2
answers
889
views
Simple conjecture about rational orthogonal matrices and lattices
The following conjecture grew out of thinking about topological phases of matter. Despite being very elementary to state, it has evaded proof both by me and by everyone I've asked so far. The ...
0
votes
0
answers
113
views
On question on quadratic forms in four variables
Let $F$ be a non-singular quadratic form in four variables and let $w: \mathbb{R}^4 \to \mathbb{R}$ be a non-negative compactly supported function satisfying certain suitable conditions. Set $$N(F,w,m)...
- 429
3
votes
1
answer
378
views
On quadratic forms in four variables
Let $F$ be a non-singular integral quadratic form in four variables. Then a result of Heath-Brown from the 90's states for $m \to \infty$, $$|\{ x \in \mathbb{Z}^4 \,:\, F(x) = m \}| = C_F\sigma(F,m)m ...
- 429