Questions tagged [quadratic-forms]

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44 views

Relation between subspaces of diagonal matrix and its “sign” matrix

Let $D$ be a $n \times n$ diagonal matrix with both positive and negative (but all non-zero) entries. Let $J = sign(D)$ be the matrix of $1$s and $-1$s representing the signs of the entries of $D$. ...
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364 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=...
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37 views

trace expressions for matrix quadratic forms [migrated]

Let $A$ be a real symmetric $n \times n$ matrix. Which quadratic forms in $A$ can be written in trace form? Such an expression would naturally generalise some invariant random matrix ensembles. ...
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1answer
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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 $\...
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1answer
85 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 ...
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135 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{...
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1answer
67 views

positivity of quadratic form minus linear form on the simplex

Let us $a_{ij}$ be the elements of a n dimensional covariance matrix. Can we prove that: $ 1-\sum_{k=1}^n a_{ik} \lambda_k + \sum_{j=1}^n \sum_{k=1}^n \lambda_j a_{jk} \lambda_k >0$ for $i=1 \...
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57 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 ...
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81 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-...
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39 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 $a_i(x_i)$...
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54 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)...
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1answer
160 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 ...
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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 ...
2
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1answer
62 views

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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54 views

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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663 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 ...
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142 views

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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1answer
199 views

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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247 views

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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45 views

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+...
5
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1answer
227 views

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 ...
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1answer
69 views

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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22 views

Why would a minimizer be of this form?

I'm trying to find $$\Pi^{*} = \arg \min_{\Pi} \left[ \lambda \left( (\Pi 1 - s)^{\intercal} K (\Pi 1- s) + (\Pi^{\intercal}1-t)^{\intercal}K(\Pi^{\intercal}1-t)\right) + x^{\intercal}\Pi y \right].$$...
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1answer
169 views

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 ...
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1answer
146 views

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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1answer
102 views

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 $\alpha$, ...
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2answers
221 views

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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165 views

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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0answers
96 views

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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1answer
233 views

Strong Approximation for sol'ns 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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3answers
328 views

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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3answers
252 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 ...
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0answers
88 views

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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1answer
230 views

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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0answers
158 views

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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224 views

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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1answer
150 views

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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82 views

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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2answers
93 views

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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0answers
39 views

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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54 views

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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1answer
114 views

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 ...
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0answers
172 views

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 ...
5
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0answers
214 views

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 ...
1
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1answer
140 views

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 ...
0
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1answer
528 views

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 ...
5
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4answers
500 views

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 ...

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