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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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
$$|\{(n_1,n_2,\ldots,n_k)\in\mathbb{Z}^k: \ n_1^2+n_2^2+\cdots+n_k^2=N\}|\leq C_\varepsilon N^{\frac{...
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Positivity of quadratic form minus linear form on the simplex
Let $a_{ij}$ be the elements of a $n$-dimensional covariance matrix. Can we prove the following?
$$ 1-\sum_{k=1}^n a_{ik} \lambda_k + \sum_{j=1}^n \sum_{k=1}^n \lambda_j a_{jk} \lambda_k > 0, \...
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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 ...
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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-...
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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 $a_i(x_i)$...
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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 $$N(F,w,m)...
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On quadratic forms in four variables
Let $F$ be a non-singular integral quadratic form in four variables. Then a result of Heath-Brown from the 90's states for $m \to \infty$, $$|\{ x \in \mathbb{Z}^4 \,:\, F(x) = m \}| = C_F\sigma(F,m)m ...
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Computing local residues of traces of bilinear forms on algebraic number fields
This is a question about computing the local residues of a general symmetric bilinear form over $\mathbb{Q}$. I've been using Lam's Introduction to Quadratic Forms as a reference, but I'm stuck on the ...
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Superlevel sets of a parametrized quadratic forms
Let $N$ be an odd integer, $n\in\mathbb{N}$, and $-\frac{2T}{NR^2}\leq a\leq0$ for given $R,T\in\mathbb{R}$ with $\frac{T}{NR^2}\leq\frac{\pi}{2}$.
Now consider the quadratic form $\Omega(a)=\sum_{l\...
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Number of representations by the norm in a division algebra corresponding to endomorphism rings of elliptic curves
Let $E$ be a supersingular curve over a field of characteristic $p$ with endomorphism ring $\mathcal O_D$ which is a maximal order in a division ring $D$ over $Q$ ramified at $p$ and $\infty$.
The ...
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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 ...
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When does one quadratic form divide another?
Let $Q_1, Q_2$ be two quadratic forms with integer coefficients in 4 variables $x_1, x_2, x_3, x_4$, both non-singular and not proportional. For a positive number $X$, which we may assume to be large ...
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how to prove an equation involving sums of Kronecker symbol
Let $p\equiv 8 \mod 9$ be a prime, I find the following equation:
$$2\sum_{\substack{0<x<p\\ 2|x}}\sum_{r|p^2-x^2}\left(\frac{-3}{r}\right)=p+1.$$
where $\left(\frac{-3}{r}\right)$ is the ...
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What is $\mathbb{E} [\max_{\sigma \in \{ \pm 1\}^n} \sigma^T Z \sigma]$ for a random Gaussian matrix $Z$?
Given an $n \times n$ random matrix $\mathbf{Z}$ with each entry i.i.d. $\mathcal{N} (0,1)$, what is $\mathbb{E} [\max_{\sigma \in \{ \pm 1\}^n} \sigma^T Z \sigma]$ as $n \to \infty$? If this is too ...
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Tensor product of preordered rings
All rings in this post are commutative, unital, and contain $\frac{1}{2}$.
To study "real" properties of a ring $R$, one is often interested in the orderings which exist on fraction fields of ...
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some problems on sum of two squares
During my experiments with "Mathematica" I arrived to the following observations. My question is that are they interesting, known, solved or not. If they are known could you please give me a reference....
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Matrices whose pairwise products form a basis
Over the vector space of 2x2 matrices, the Pauli matrices $I, X, Y, Z$ form a complete basis. Each of these matrices square to $I$, and with the additional relation that $Z = iXY$, we see that every ...
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When does $axy+byz+czx$ represent all integers?
For which $a,b,c$ does $axy+byz+czx$ represent all integers?
In a recent answer, I conjectured that this holds whenever $\gcd(a,b,c)=1$, and I hope someone will know. I also conjectured that $axy+byz+...
user44143
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The number of quadratic forms attaining Hermite's constant
$\require{AMScd}$
I'm considering minimum values (at non-zero integer points) of real, positive-definite, quadratic forms of determinant $1$. These are functions $f:\mathbb{R}^n\to \mathbb{R}$ which ...
0
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1
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Difference of two sums of two squares [closed]
Is there a way to prove that if n = (a^2 + b^2) - (c^2 + d^2), then for every natural n, there are infinitely many a, b, c and d?
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applications of finding least quadratic nonresidue mod $p$?
I saw some papers from famous mathematicians (assuming GRH or without it) which are devoted to finding bound for least quadratic nonresidues modulo prime number $p$.
My question is that why it is so ...
- 841
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Representation of two related integers by the same binary quadratic form
Let $f(x,y) = ax^2 + bxy - cy^2$ be an indefinite, irreducible, and primitive binary quadratic form. That is, we have $\gcd(a,b,c) = 1$ and $\Delta(f) = b^2 - 4ac > 0$ and not equal to a square ...
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Evaluating a binary quadratic form at convergents
We use the notation
$$\displaystyle [a_0; a_1, \cdots, a_n] = a_0 + \cfrac{1}{a_1 + \cfrac{1}{\ddots + \cfrac{1}{a_n}}}$$
to denote a finite continued fraction, and for a given real number $\alpha$, ...
- 25.5k
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2
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Quadratic diophantine equations and geometry of numbers
Let (for concreteness) $a = 2$, $b = \sqrt{5}$ and $\varphi = (\sqrt{5}+1)/2$. I am interested in solutions $(w,x,y,z) \in \mathbb{Z}[\varphi]^4$ of the system
$$
w^2 - ax^2 -by^2 + abz^2 = 1
$$
$$
\...
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Topological vs algebraic intersection forms
Let $X$ be a simply connected complex projective surface (hence a real $4$-manifold). Let $(H^2(X,\mathbb Z)/\mathrm{tors}, q_X)$, $(A^1(X)/\mathrm{tors},q_X')$ be the corresponding lattices in ...
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Proper ideals are invertible
I am reading through Cox's book Primes of the form $x^2+ny^2$ and I am stuck with some proofs in Chapter 7 (I have the 2nd edition). There, the author presents the following Lemma:
Lemma 7.5: Let $...
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Strong Approximation for solutions to quadratic Diophantine equations
Can anyone either direct me to an relatively elementary proof in the literature--or show me why this (Conjecture 1 stated below) is true--or if I am mistaken and it is not true:
For any 4-tuple $\xi =...
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Integer positive definite quadratic form as a sum of squares
Let $A$ be a symmetric $d\times d$ matrix with integer entries such that the quadratic form $Q(x)=\langle Ax,x\rangle, x\in \mathbb{R}^d$, is non-negative definite. For which $d$ does it imply that $Q$...
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Solving diagonal simultaneous quadratic equations
A problem I am trying to solve has led to me to the following system of equations:
$$A(x^2) + Bx + c = 0$$
Where $A$ and $B$ are known matrices, $c$ is a known vector, $x$ is the vector of unknowns ...
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Norm quadrics and their motives
Let $k$ be a field of characteristic $\neq 2$ and $\langle\!\langle a_{1},\cdots,a_{n}\rangle\!\rangle$ a Pfister form over $k$. Denote by $Q_{\underline{a}}=Q_{a_{1},\cdots,a_{n}}$ the projective ...
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Is it possible for the Witt group of a scheme to have non-trivial odd torsion?
Let $F$ be a field of characteristic not $2$. A well-known theorem of Pfister asserts that the torsion of $W(F)$, the Witt group of $F$, is $2$-primary.
Baeza [B, V.6.3] extended this result to Witt ...
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On sums of three squares
Let $\mathbb N=\{0,1,2,\ldots\}$. By the Gauss-Legendre theorem on sums of three squares, any $n\in\mathbb N$ with $n\equiv1,2\pmod4$ can be written as $x^2+y^2+z^2$ with $x,y,z\in\mathbb N$. Clearly, ...
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Sums of two integer squares in arithmetic progressions
Is there an explicit formula in the literature for the number of representations of a positive integer $n$ as a sum of two integer squares, the second of which is divisible by $5$? So this means to ...
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1
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A "nice" (but non-definite) quadratic programme
For integers $n\geq k>0$, let $f$ be the following quadratic form:
$$f(x_1,\ldots,x_n)=\sum_{i=1}^n\sum_{j=0}^{k-1}x_ix_{i+j\bmod n}.$$
Is it true that the minimum of $f$ over the unit simplex is ...
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On universal sums $x(ax+b)/2+y(cy+d)/2+z(ez+f)/2$ over $\mathbb N$
Let $a,b,c,d,e,f$ be integers with $a\ge c\ge e>0$, $b>-a$ and $a\equiv b\pmod2$, $d>-c$ and $c\equiv d\pmod 2$, $f>-e$ and $e\equiv f\pmod2$. If each $n\in\mathbb N=\{0,1,2,\ldots\}$ can ...
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Correlation between the first and a random position of an ergodic bit sequence
Edit: Since the geometric approach did not work, I try now another approach: phrasing the problem as a quadratic programme.
Probabilistic version.
Let $x=(x_1,x_2, \ldots) $ be an ergodic random ...
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How explicitly write a projective transformation between the conics over the univariate function field?
Consider the quadratic forms
$$
Q_1 = x^2 + y^2 - (t^2+1)z^2,\qquad Q_2 = x^2 + y^2 - z^2
$$
over the rational function field $\mathbb{F}_p(t)$, where $p > 2$ is a prime such that $t^2 + 1$ is ...
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Quadrics over the univariate function field with discriminant of minimal degree
Consider a non-degenerate quadric $Q(x,y,z) \subset \mathrm{P}^2$ over the univariate function field $\mathbb{F}_p(t)$, where $\mathbb{F}_p$ is a prime finite field, $p > 2$. For simplicity assume ...
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On the orthogonal group of a lattice on a quadratic space over dyadic local field
Let $F$ be a local field with valuation ring $R$. $V$ is a n dimensional non-singular quadratic space over $F$ with bilinear form $B$ and quadratic map $Q$.
As usual, $O(V)$ denotes the orthogonal ...
user125345
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Concrete Hanson-Wright inequality?
I'm working on a paper that requires bounding
$$\Pr\left[|\vec x^\top Q \vec y| >= t\right]$$ where $Q$ is a matrix (happens to be symmetric) and $\vec x,\vec y$ are iid real mean-zero subgaussian ...
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Can we write each positive integer as $x^2+y^2+\varphi(z^2)$?
As odd squares are congruent to $1$ modulo $8$, any integer of the form $4^k(8m+7)$ with $k,m\in\mathbb N=\{0,1,2,\ldots\}$ cannot be written as the sum of three squares.
To avoid such congruence ...
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Existence of symplectic basis
Let $R$ be a PID and $M$ a free, finite rank $R$-module with a perfect billinear form $\omega$ such that $\omega(v,v)=0$ for all $v \in M$. Does anyone know a reference for the fact that a symplectic ...
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Proof of A Positive Definite Covariance Matrix
I would like to prove such a matrix as a positive definite one,
$$
(\omega^T\Sigma\omega) \Sigma - \Sigma\omega \omega^T\Sigma
$$
where $\Sigma$ is a positive definite symetric covariance matrix ...
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4
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Solutions to the Diophantine equation $x^2+3y^2+3z^2=n$
For a fixed positive integer $n$, the Diophantine equation
$$x^2 + y^2 + z^2 = n$$
was studied by Gauss in Disquisitiones Arithmeticae. As is known, this equation is intimately connected to the ...
- 1,573
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Reference request: Number of elliptic and hyperbolic quadratic forms of a given rank over a finite field
My question is over the finite field $\mathbf{F}_q$ of $q$ elements. It is well known that a symmetric matrix of odd rank corresponds to a parabolic quadratic form but even rank symmetric matrices ...
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Angle between subspaces in indefinite signature
I know there is a notion of principal angles between subpaces of a Euclidean space, but what about subspaces of a finite dimensional vector space $V$ equipped with a non-degenerate quadratic form of ...
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The number 1680 and Lagrange's four-square theorem
The number $1680$ has the factorization $2^4\times3\times5\times7$. Rather to my surprise, I found that this number has certain mysterious connection with Lagrange's four-square theorem.
QUESTION: ...
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Can we write each positive integer as $w^2+x^2(1+2y^2+2z^2)$ with $w,x,y,z\in\mathbb Z$ and $x\not=0$?
Lagrange's four-square theorem states that each nonnegative integer is the sum of four squares. Here I ask the following question concerning a refinement of Lagrange's four-square theorem.
QUESTION: ...
- 14.5k
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Are lattice points in thin spherical shells uniformly distributed?
Consider the spherical shell (annulus)
$$A(R,r) = \{ x \in \mathbb{R}^3 : R \leq |
x|\leq R+r \}.$$ Think of the limit $R \to \infty$.
Assume that $r$ depends on $R$ as $r(R) = R^{-\delta}$. We are ...
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Is every integer $n>1$ the sum of two triangular numbers and two powers of $5$?
Recall that the triangular numbers are those integers
$$T_n=n(n+1)/2\ \ \ (n=0,1,2,\ldots).$$
In 1796 Gauss proved that each $n\in\mathbb N=\{0,1,2,\ldots\}$ is the sum of three triangular numbers, ...
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