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.
585 questions
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Spectral gap for a second-variation operator on a jet Hilbert space
Let $J_0$ be a (separable) complex Hilbert space with scalar product $\langle \cdot,\cdot\rangle_{J_0}$ and norm $\|\cdot\|_{J_0}$.
Suppose we are given a real, symmetric, non-negative bilinear form
$$...
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3
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363
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Weakening of the Idoneal Number condition
This is about the following Math.SE Q&A: "What is the largest integer $d$ such there is a congruence relation on primes p so that $x^2 + dy^2 = p^2$ has a non-zero integral solution?".
...
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Can $w^2+bx^2+cy^2+dz^2$ be universal over a sparse subset of $\mathbb N$?
Let $b,c,d\in\mathbb N=\{0,1,2,\ldots\}$ with $1\le b\le c\le d$. For a subset $S$ of $\mathbb N=\{0,1,2,\ldots\}$, if
$$\{w^2+bx^2+cy^2+dz^2:\ w,x,y,z\in S\}=\mathbb N$$
then we say that $w^2+bx^2+cy^...
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Approximations to pairs of algebraic numbers by quadratic elements over a number field
Question / conjecture
Let $K$ be a real number field and consider a pair of real numbers $(x, x')$.
Assume that there are real numbers $\epsilon > 0$, $C > 0$
and infinitely many pairs of real ...
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94
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Conics and Hermitian curves over $\mathbb{F}_{q^2}$
Let $\mathcal{H} \subset \mathbb{P}^2(\mathbb{F}_{q^2})$ be the Hermitian curve, defined (up to projective equivalence) by
$$
X^{q+1} + Y^{q+1} + Z^{q+1} = 0.
$$
Its automorphism group is $\mathrm{PGU}...
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Find integer coefficients of polynomials from approximate irrational roots [duplicate]
Let take quadratic equations
$$x^2+ax+b=0$$
assume here $a,b$ both are integer and the roots of the equation are irrational if I give you one root in irrational form then is there any method to find $...
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The quadratic forms corresponding to canonical (Neron-Tate) height of an elliptic curve
Surely this is well-known in the arithmetic geometry community, which I (unfortunately) do not regard as a member of, so I will include some basic exposition for the laymen (including myself).
We deal ...
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Is there an elementary proof for this identity involving a simplex, quadratic forms and determinants?
In a recent project we found a curious identity for simplices (Theorem 5.6).
Let $\Delta\subset\Bbb R^d$ be a $d$-simplex with facets $F_0,...,F_d$, $v_i\in\Bbb R^d$ the vertex opposite to $F_i$, $u_i\...
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$\left\{\frac{x(ax+b)}2+\frac{y(ay-b)}2:\ x,y=0,1,2,\ldots\right\}$ and asymptotic bases of order 2
A subset $A$ of $\mathbb N=\{0,1,2,\ldots\}$ is called an asymptotic base of order $h$ if any sufficiently large $n\in\mathbb N$ belongs to the set
$$hA=\{a_1+\ldots+a_h:\ a_1,\ldots,a_h\in A\}.$$
...
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3
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Minimum L2 norm of polynomial and the Hilbert matrix
Hilbert introduced his famous matrix when he studied the following problem. How small can the integral
$$\int_{a}^b|p(x)|^2dx $$
become for a non-zero polynomial $p$ with integer coefficients? He ...
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Orthogonalization of quadratic forms over a $p$-adic Banach space
Let $X$ be an arbitrary set. Let $H = c_0(X, \mathbb{Q}_p)$ be the $p$-adic Banach space with sup norm. Let $\langle \cdot, \cdot \rangle$ be a symmetric, nondegenerate $\mathbb{Q}_p$-bilinear form on ...
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Existence of rank 3 lattice of signature (1,2) containing two copies of $U$ intersecting in a positive vector
Let $U = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ denote the standard hyperbolic plane. I am trying to construct, for a given natural number $N$, an explicit example of a rank 3 lattice $...
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Reference request: analogue of Cramér's conjecture for integers represented by binary quadratic form
The Prime Number Theorem asserts that $\pi(N) \sim N/\log N$ as $N \to \infty$ where $\pi(N)$ is the prime counting function. Colloquially, the (average) density of primes $\le N$ is like $1/\log N$. ...
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Prime Inheritance and Prime-Generating Subsequence Trees in Class Number 1 Quadratic Polynomials [closed]
This question is inspired by the classical behavior of Euler’s polynomial
$$
\mathbf{f(x) = x^2 - x + 41},
$$
which is well-known for producing prime values for integer inputs $x = 0$ to $39$, and is ...
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Prime inheritance in class number 1 quadratic polynomials
This conjecture is based on computational exploration of quadratic polynomials associated with imaginary quadratic fields of class number one.
Let us define:
• For a polynomial $f(x) \in \mathbb{Z}[...
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Concentration inequality for quadratic form involving random matrix
I have $N$ i.i.d random vectors $\{X_k\}_{k=1}^N$ in $\mathbb{R}^n$ where each entry is bounded and positive.
I construct a matrix $M_N$ as
\begin{align}
M_N=\frac{1}{N}\sum_{k=1}^NX_kX_k^T
\end{align}...
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Explicit diagonalization of a pair of ternary quadratic forms
The topic of when a pair of $n$-ary quadratic forms can be simultaneously diagonalized is certainly a well-tread topic. However, I do not recall seeing the following result in the literature.
Let $(A,...
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Canonical basis for a subspace associated to a pair of ternary quadratic forms
Let $F$ be an algebraically closed field of characteristic zero, and $(A,B) \in F^2 \otimes \operatorname{Sym}^2 F^3$ be a pair of linearly independent symmetric $3 \times 3$ matrices with ...
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Quadratic equations over division rings of dimension 2 with specified (non)solutions
Let $\ell$ be a division ring of left dimension $2$ (as a vector space) over the sub division ring $k$.
Suppose that all quadratic equations $x^2 + ax + b = 0$ with $a, b \in k$, either have no root ...
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Determining the stabilizer of a pair of ternary quadratic forms
This is a revisit of an old question of mine: Stabilizers of pairs of ternary quadratic forms
The first part of the post is simply an expansion of Noam Elkies' answer. In the answer he gave in the ...
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Symmetric matrices in orthogonal groups of ternary quadratic forms
This is related to my earlier question: Intersection of orthogonal groups
(See also Matrix expression for elements of $SO(3)$)
I am interested in extracting symmetric elements of the orthogonal groups ...
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Intersection of orthogonal groups
Let $f$ be a ternary quadratic form, say with real coefficients. Then there is an associated symmetric matrix to $f$, say $A_f$. The orthogonal group of $f$ is then the group
$$\displaystyle O_f = \{H ...
- 30.7k
3
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A new kind of series for $1/\pi$
As in Question 491655, Question 491762 and Question 491811, we define
$$a_n(x):=\sum_{i,j=0}^n\binom ni^2\binom nj^2\binom{i+j}ix^{i+j}$$
for each nonnegative integer $n$.
Using my own way (mentioned ...
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Group algebra representations that preserve involution
This question is based on my unanswered post here.
Let $K$ be a field equipped with an involution $\bar{}$, which may be the identity. Suppose $(V,b)$ consists of a $K$-vector space $V$ together with ...
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Extending automorphisms of quadratic forms
Sorry for this question. If the statement is true, it should be found in all textbooks; if not, it should be mentioned as such in all textbooks. But I couldn't find a single trace, leave alone ...
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On positive integers not representable as $ax^k+by^l+cz^m$
It is well known that for any $a,b,c\in\mathbb Z^+=\{1,2,3,\ldots\}$ there are infinitely many $n\in \mathbb N=\{0,1,2,\ldots\}$ not representable as $ax^2+by^2+cz^2$ with $x,y,z\in\mathbb N$. See, e....
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Counting representations by a quadratic form subject to a constraint
I am working over the integers. Suppose I have a quadratic form, such as
$$q(x_1,x_2,y_1,y_2) = x_1^2 + 2x_2^2 + y_1^2 + 2y_2^2,$$
as well as some quadratic constraints, such as
$$x_1 x_2=y_1 y_2.$$
...
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All maximal tori of a special orthogonal group
Let $K$ be a field of characteristic different from 2,
and let $q$ be a non-degenerate quadratic form on $V=K^n$ for $n\ge 5$,
say,
$$q(x_1,\dots, x_n)=a_1x_1^2+\dots+a_nx_n^2.$$
Question. How can ...
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Sum of representation function in arithmetic progression
Given a positive definite integral binary quadratic form $f$, denote by $r_f(n)=\#\{(x,y)\in \mathbb{Z}^2~|~f(x,y)=n\}$, how can we obtain estimates for $$\sum_{~~~~~n\leq x\\ n\equiv k~\mathrm{mod}~q}...
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Sum of representations of primes by quaternary quadratic form
So lets say we have a positive definite integral quaternary form $Q$ of determinant $p^2$ for some prime $p$. It can be shown that every integer $n\gg p^{4+\epsilon}$ with $(n,p)=1$ is represented by $...
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Number of representations of $m = QF(a,b,c)= aw^2 + bx^2 + cy^2,$ where $QF(a,b,c)$ is regular and $w, x, y$ have congruence conditions
Are there any papers, theory or formulas on the number of representations of $m = QF(a, b ,c) = aw^2 + bx^2 + cy^2,$ where $QF(a, b, c)$ is regular and $w, x, y$ have some congruence conditions ...
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Existence of solutions for system of quadratic equations
I found an interesting problem in Post 1, Post 2. Let us suppose to have $ M $ quadratic equations
$$
\underline{x}^T A_i \underline{x} + \underline{b}^T \underline{x} = c \quad i = 1,...,M
$$
with $ \...
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Least-square distance between an array of quadratic forms and a given positive vector
Suppose we are given a list of $N$ positive definite quadratic forms $X^TQ_k X$ (where $k\in[1,N]$ and
$Q_k\in\mathbb{R}^{p\times p}$ $\forall k$), and a positive vector $V$ of same length $N$ i.e. $V=...
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How to prove a theorem of Gauss on automorphisms of the ternary quadratic form $x^2+y^2-z^2$?
[This question was posted before on Math StackExchange, and received two useful comments by @Will Jagy, but his comments were not sufficient for me to reconstruct a concise and direct proof in "...
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integers represented as $x^2+ny^2$ with prime factors condition?
Given positive square-free integers $m$ and $d$. Lets denote the prime factorization of $m$ by $m=\prod p_i$.
I know that if each $p_i$ can be written as $x^2+dy^2$ then $m$ can be written in that ...
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Cornacchia's algorithm with too many prime factors
Given quadratic diophantine equation $x^2+dy^2=m$ where $d,m> 0$ and $d$ is square-free, Cornacchia's algorithm: https://en.wikipedia.org/wiki/Cornacchia%27s_algorithm, solves the problem in ...
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Does a quadratic form depend locally Lipschitz-continuously on the related ellipse?
Let $Q_+$ be the space of positive definite quadratic forms on $\mathbb R^2$, equipped with the metric arising from thinking of it as an open subset of $\mathbb R^3$.
Let $E$ be the set of ellipses in ...
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Multiplicative closure of $ax^2+bxy+cy^2$ with discriminant $d$ and class number $h(d)=3m?$
I. Condition
If there are integers $(r_1, r_2, r_3, r_4)$ such that,
$$ar_1^2+br_1r_2+cr_2^2=ac\\
r_3=(ar_1+br_2)/c\\
r_4=(br_1+cr_2)/a$$
then $(a u^2+buv+cv^2)(a x^2+bxy+cz^2)= (a z_1^2+bz_1z_2+cz_2^...
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8
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2
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630
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Testing if a positive definite quadratic form over $\mathbb Z$ represents 1
I am looking for algorithms that can be used to test if a given positive definite $n$-ary ($n\geq 3$) quadratic form over $\mathbb Z$, whose factorization of the discriminant is known, represents 1.
I ...
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75
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Mapping the magnitude of a complex quadratic form to a Hermitian quadratic form
I have a function $f(X) = |a^T X^{-1} a|$ that maps a complex symmetric matrix $X = X^T \neq X^H$ to a real number. I would like to perform optimization involving this function, and I try to convert $...
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Number of symmetric matrices in a box of bounded determinant
Let $B$ and $T$ be positive real numbers. I'm interested in the following problem, which is about counting $2\times2$ symmetric matrices with bounded determinant and entries lying in a box:
Problem: ...
3
votes
2
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677
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Lowest eigenvalue of Toeplitz matrices: strategies?
Let $\{a_n\}_{n\in \mathbb{Z}}$, $a_n\in \mathbb{R}$, be such that $a_n = O(1/n^2)$ and $a_{-n}=a_n$. The Toeplitz matrix $A_N$ is the $N$-by-$N$ matrix defined by
$$A_{N,i,j} = a_{|i-j|}$$
for $1\leq ...
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Example of group algebra with canonical involution under certain condition
Let $G$ be a finite group and $K$ be a field with involution $\overline\cdot:K \to K$. Suppose the fixed field of $\overline\cdot$ is $K_0$ and $K/K_0$ is a definite extension (and $K_0$ is formally ...
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0
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109
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Quadratic forms with the same roots over GF(2) for low rank problems
Let $Q_1(x)=x^TA_1x$ and $Q_2(x)=x^TA_2x$ with $x\in GF(2)^n$, $A_i\in GF(2)^{n\times n}, i \in \{1, 2\}$. If $rank(A_1)=rank(A_2)=2$, is it possible that $Q_1(x)$ and $Q_2(x)$ can have the same roots ...
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645
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Number of roots of a quadratic form over GF(2)
If $Q(x) = x^T A x$ with $x \in GF(2)^n$ and $A \in GF(2)^{n \times n}$, is there a way to find how many roots $Q(x)$ has based on any properties of $A$ (e.g., rank, etc.)?
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115
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Notion of length in projective space over function field
Given a projective space $\mathbb{P}^n(\mathbb{R})$ and two points $x, y \in \mathbb{P}^n(\mathbb{R})$, the distance between $x$ and $y$ is defined as
$$
d(x, y) = \frac{\|v_x \wedge v_y\|}{\|v_x\| \|...
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3
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Reflections on affine quadric hypersurfaces
Let $f\colon\mathbb{Z}^n\otimes\mathbb{Z}^n\to\mathbb{Z}$ be a non-degenerate symmetric bilinear form and consider the affine quadric hypersurface $$
X:=\{f(x,x)+2=0\}\subseteq\mathbb{Z}^n.
$$
For ...
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1
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If a lattice can be embedded into $\mathbb Q^n,\langle-1\rangle^n$, then can it be embedded into $\mathbb Z^n,\langle -1 \rangle^n$?
Given a graph with negative integers on each vertex $\Gamma$ there is a corresponding intersection lattice denoted $Q_\Gamma$, a free $\mathbb Z$ module generated by the vertices, endowed with a ...
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How to prove this problem about ternary quadratic form?
Is this right? And how to prove it ?
For $n \equiv 1,2 \bmod 4$
$$ \Bigg|\ \mathbb Z^3\cap\Big\{(a_1,a_2,a_3)\ \Big|\
a_1^2+a_2^2+a_3^2=n \Big\}\Bigg| \\
= \frac12\Bigg|\mathbb Z^3\cap\Big\{(a_1,...
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Genus of binary quadratic forms: $f(x,y), g(x,y)$ in same genus if and only if represent same values in $(\mathbb Z/m\mathbb Z)^\ast$ for all $m$
In David Cox's book: Primes of the form $x^2+ny^2$, second edition, there is a theorem(Theorem 3.21, page 52) characterize whether two binary quadratic forms in the same genus. The contents of the ...
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