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6
votes
1answer
61 views

space of reduced positive definite quadratic forms

What is the highest dimension for which the space of reduced positive definite quadratic forms (or the fundamental domain of $SL_n(\mathbb{R})/SL_n(\mathbb{Z})$) has been explicitly calculated? I know ...
0
votes
0answers
41 views

Complex Phase Problem Relating to Binary Quadratic Forms [on hold]

Let $S$ be a linear subspace of $\mathbb{F}_2^n$, and $Q:\mathbb{F}_2^n\to\mathbb{F}_2$ a quadratic function. Assume that there exist complex phases $\{c_i\}$ such that for every $x\in S$: ...
1
vote
0answers
68 views

What is classified by $H^1(\mathbb{R},SO(p,q))$ and by $H^1(\mathbb{R},SU(p,q))$?

We denote by $F^{\mathbb{R}}_{p,q}$ the quadratic form over the field ${\mathbb{R}}$ $$ F^{\mathbb{R}}_{p,q}(x)=x_1^2+\dots+x_p^2-(x_{p+1}^2+\dots+x_{p+q}^2) $$ on the vector space ...
5
votes
1answer
301 views

Easiest way to distinguish $E_8 \oplus E_8$ from $E_{16}$

For $n$ an integer divisible by $8$, let me denote by $E_n$ the "usual" even non-degenerate positive definite integral symmetric bilinear form over $\mathbf Z^n$. It is well known that in dimension ...
4
votes
3answers
383 views

Class number for binary quadratic forms discriminant $\Delta$ to class number $\mathbb Q(\sqrt \Delta)$

Jyrki Lahtonen has suggested I write a blog post relating binary quadratic forms to quadratic field class numbers, ...
1
vote
1answer
150 views

The Chebotarev Density Theorem and the representation of infinitely many numbers by forms

Let $ax^{2}+bxy+cy^{2}$ be a primitive positive definite quadratic form of discriminant $\Delta<0$. It is well known that $ax^{2}+bxy+cy^{2}$ represents infinitely many prime numbers. One of the ...
5
votes
1answer
152 views

Positive Primes represented by an indefinite binary form, reducing poly degree from 8 to 4

In his lovely answer at Positive primes represented by indefinite binary quadratic form Noam found that a (positive) odd prime $p$ is represented by the indefinite form $x^2 + 13 x y - 9 y^2$ if and ...
7
votes
1answer
307 views

Positive primes represented by indefinite binary quadratic form

Neil Sloane asked me about commands in computer languages to find the (positive) primes represented by indefinite binary quadratic forms. So I wrote something in C++ that works. This is for the OEIS, ...
5
votes
1answer
153 views

Rank four quadratic Form with non trivial discriminant in I(k)

Im sure this is a beginners question. Let $k$ be a field and $I(k)$ the fundamental ideal in the Witt-ring W(k). The Arason-Pfister-Hauptsatz states: "If $\varphi$ is any anisotropic class in ...
1
vote
0answers
122 views

Equation in the Gaussian Integers

Let $a,b \in \mathbb{N}$. Is there a possibility to characterize the solutions of $a N(\alpha) - b N(\beta)=1$ where $\alpha,\beta \in \mathbb{Z}[i]$? In particular I am interested in the case $a=1$ ...
2
votes
3answers
225 views

Solving a quadratic matrix equation with non-squared matrix

I was trying to solve the problem of finding the value of a non-squared matrix $T$ ($n \times m$) which solves $$ T^T T = X$$ where $X$ is a symmetric and positive semidefinite $m \times m$ matrix, ...
0
votes
1answer
81 views

Rost Correspondence and minimal Pfister-Neighbors

In http://www.math.uiuc.edu/K-theory/0357/ Karpenko utters the following, $Conjecture$ $1.6.$ If an anisotropic quadric $X = Q$ possesses a Rost correspondence, then the quadratic form (defining $X$) ...
2
votes
0answers
119 views

n-ary quadratic forms with $S$-integer values

Let $Q(x_1,\ldots,x_n):=x_1^2+\cdots+x_n^2$ be an $n$-ary quadratic form. Given a finite set of (rational) primes $S$ is there an algorithm or theorem that describes all solutions to ...
1
vote
0answers
49 views

Cassels-Birch-Davenport theorem for multiple quadratic forms of certain type

A classical theorem of Cassels states that if a homogenous quadratic form $Q$ has an integer zero, then there is a zero of small height (bounded solely by the coefficients and number of variables). ...
3
votes
1answer
113 views

Sum of Squares Length of a Product

Let $n \geq 2$. Let $g_1, \ldots , g_{n-1} \in \mathbb{R}[x_1,\ldots,x_n]$ such that $q=g_1^2+\ldots +g_{n-1}^2$ is not divisible by $p=x_1^2+\ldots +x_n^2$. Let $m \geq 1$ be the smallest integer ...
9
votes
4answers
357 views

Is any quadric birational to a product of Brauer-Severi varieties?

Let $k$ be a field with algebraic closure $\bar k$. Assume that $k$ is perfect and not of characteristic $2$ for simplicity. Let $$X: \quad Q(x)=0, \quad \subset \mathbb{P}^n_k,$$ be a non-singular ...
1
vote
0answers
56 views

Is this related to a simple property of a lattice?

I am looking for a certain notion of sparseness of lattices. I want to find a vector in $\mathbb{Z}^N$ that the minimal possible inner product with all the vectors of a given lattice. Or at least, I ...
3
votes
2answers
188 views

equivalence of quadratic forms over finitely generated fields

Over number fields, two quadratic forms are equivalent iff they have the same dimension, signature, discriminant and Hasse invariant. How is the situation like over finitely generated fields?
2
votes
1answer
76 views

Principally split primes with factors in arbitrarily small angular sectors

I wonder if the following is known: let $n$ be a (square-free) positive integer. Is there ever/always a sequence of prime numbers $p$ that can be written in the form $$p = x^2 + ny^2,$$ where $x, y$ ...
2
votes
1answer
72 views

Stabilizer of a nonsingular vector in a quadratic space (char (k)=2)

suppose that $k$ is a finite field of characteristic 2 and $(V,q)$ a quadratic space, i.e., $V$ is a $k$-vector space and $q:V\to k$ quadratic form. Suppose that $\dim(V)\geq 4$ and that $q$ is ...
2
votes
1answer
109 views

Block Covariance Matrix - Positive Definite? (Quadratic Optimization) [closed]

I have a covariance matrix C. I have then formulated an quadratic optimization problem that involves the following matrix in the quadratic form: [ C C ] [ C C ] However, the quadratic solver ...
2
votes
2answers
281 views

Connection between quadratic forms and ideal class group

I'm studying the classic results on binary (integer) quadratic forms and I'm looking for a reference on the following result (maybe a book that contains a proof): Let $O_k$ be the ring of algebraic ...
2
votes
1answer
77 views

Can one determining the p-adic lattice just from the values of the quadratic form on a p-group?

Given a finite $p$-group $A$, with a non-degenerate quadratic form $q:A\rightarrow \mathbb Q/2\mathbb Z$ (that is a map satisfying $q(na)=n^2q(a)$ for all $n\in \mathbb Z,a\in A$), an important result ...
2
votes
1answer
135 views

Indefinite orthogonal groups over p-adics

Let $q$ be a rational quadratic form. How can we think of a Cartan decomposition of $O_q(Q_p)$? Is there a notion of Cartan involution for p-adic field, so that we can execute same process as we do ...
-1
votes
1answer
77 views

Uniqueness of the solution to a quadratic problem [closed]

Consider a positive definite matrix $\boldsymbol H$, the known vectors ${\boldsymbol b}$ and ${\boldsymbol a}_i$. Now the minimization problem is casted with respect to the vector ${\boldsymbol x} $ ...
2
votes
1answer
171 views

Representation of rationals by quadratic form

In one paper about namber theory author stated 2 lemmas Lemma 1. If $p$ is a prime $\equiv3(mod $ $4)$ then $x^2+y^2-pz^2$ represents a non-zero rational number $m$ if and only if $m$ is not of the ...
3
votes
3answers
245 views

Can you efficiently solve a system of quadratic multivariate polynomials?

Given a system of 2nd-degree polynomials, $P=\{p_1,\dots,p_m\}$ where $p_i: \mathbb{R}^n \rightarrow \mathbb{R}$, can you efficiently find $x_1,\dots,x_n\in\mathbb{R}$? I know that in the general ...
11
votes
1answer
292 views

Positive ternary quadratic forms in the same genus that represent the same numbers

There are three genera of positive, integral, ternary quadratic forms in which both forms (classes...) are regular, so the paired forms represent the same numbers. These pairs (complete genera) are: ...
4
votes
1answer
123 views

Which power of $2$ kills $W(k)$?

Is the following fact "well-known": if $-1$ is a sum of squares in a field $k$, then the Witt group $W(k)$ of quadratic forms is killed by multiplication by $2^N$ for some $N\ge 0$? What can one say ...
3
votes
1answer
239 views

S genus of quadratic forms

Let $f$ be a non-degenerate quadratic form with integral coefficients. The genus of $f$ is the set of quadratic forms up to integral equivalence which are equivalent to $f$ over the $p$-adic integers ...
4
votes
0answers
100 views

Fully Homomorphic Error Correction?

Consider a field $F$. Suppose we have two vectors $a,b\in F^n$, and an invertible matrix $G\in F^{n\times n}$. Let $c\in F^n$ be the point-wise product of $a$ and $b$, that is, $c_i=a_ib_i$. Let ...
2
votes
0answers
80 views

Terminology and reference question

I am working on a problem involving bilinear forms over complex Hilbert spaces, and in my case it is not natural to make the forms sesquilinear, i.e., $a(u,v)$ is linear in both complex arguments. ...
0
votes
0answers
65 views

Distribution of Quadratic Forms in Rademacher Random Vector

This is a fairly well-known theorem on quadratic form in normal distribution, which is taken from here. If $\mathbf y ∼ N(0, \sigma^2)$ is an $n\times 1$ column vector and $\mathbf M$ is an ...
5
votes
3answers
340 views

Optimization problem on trace of rotated positive definite matrices

Given two $n \times n$ symmetric positive definite matrices $A$ and $B$, I am interested in solving the following optimization problem over $n \times n$ unitary matrices $R$: $$ \mathrm{arg}\max_R ...
2
votes
1answer
166 views

quadratic programming on hypercube

I want to maximize a quadratic form $\mathbf x^T\mathbf Q\mathbf x$ and also want to find out which vector $\mathbf x$ maximizes the quadratic form when $\mathbf Q$ is an $n\times n$ positive ...
10
votes
3answers
399 views

orbits of automorphism group for indefinite lattices

I have a question about indefinite lattices. QUESTION: Let $\Lambda\times\Lambda\rightarrow {\Bbb Z}$ be a lattice, that is, ${\Bbb Z}^n$ with a non-degenerate integer quadratic form, not necessarily ...
3
votes
1answer
174 views

Are these powers of a characteristic 3 power series annihilated by certain Hecke operators?

Let D in Z/3[[x]] be sum ((a_n)(x^n)) where the sum runs over all n prime to 6 and a_n is the mod 3 reduction of the number of ideals of norm n in the ring of integers of Q(root(-3)). (So ...
9
votes
1answer
440 views

The Dissertation of F. J. van der Linden

Does anyone have access to the 1984 dissertation of Franciscus Jozef van der Linden under Hendrik Lenstra? It is called Euclidean Rings with two infinite primes. The theory is that this has the ...
2
votes
1answer
126 views

maximizing convex quadratic form over the intersection of unit sphere and positive orthant

For a positive semi-definite matrix $C$, I want to find the solution to the following problem: $\arg\max_{h\geq 0} h^T C h\quad$ s.t. $\quad h^T h\leq 1$ Any pointers are welcome.
12
votes
1answer
418 views

primes represented by an indefinite binary quadratic form

Suppose I have a form $$ f(x,y) = a x^2 + b x y + c y^2, $$ with $a,b,c$ integers, $\gcd(a,b,c)=1$ and $\Delta = b^2 - 4 a c > 0,$ but $\Delta \neq n^2$ for any integer $n.$ Do there exist ...
0
votes
0answers
152 views

A quadratic form pair

Let $Q_s(x)\in\Bbb Z[x_1,x_2,\dots,x_s],\hat{Q}_{\hat s}(y)\in \Bbb Z[y_1,y_2,\dots,y_\hat s]$ be pair of homogeneous purely non-diagonal (every term of form $x_ix_j$ or $y_iy_j$) quadratic forms and ...
3
votes
1answer
71 views

Set of isomorphisms of Pfister forms corresponding to first cohomology of algebraic group

Assume $k_0$ is a field with char($k_0$) not $2$. Let us define functors from $\rm Field_{/k_0}\to \rm Sets$ as $\rm Pfister_n(k):=\{\text{isomorphism classes of n-fold Pfister forms over k}\}$; ...
4
votes
0answers
209 views

Hermitian forms over quaternion algebra

Notations: Let $Q=(a,b)$ be a quaternion algebra over a field of characteristic $\neq 2$, i.e. $i^2=a, j^2=b, k=ij, ij=-ji$. Consider $K=k(t)(\alpha)$, where $\alpha=\sqrt{at^2+b}$. Let ...
5
votes
2answers
477 views

Integral orthogonal group for indefinite ternary quadratic form

I have the indefinite quadratic form $q(x,y,z) = 19 x^2 + 5 y^2 - z^2.$ It's not my fault. I find, on reflection, that I have no idea how to describe the orthogonal group of this over the integers. ...
6
votes
2answers
392 views

Does this quadratic form over a large field represent 1?

I have a field $K$ of transcendence degree two over $\mathbb{R}$, and elements $a_1,a_2,a_3\in K$. I would like to understand the set $$ Q = \{ u\in K^3 : \sum_i a_iu_i^2 = 1\} $$ In particular, I ...
2
votes
4answers
1k views

Solving a System of Quadratic Equations

I have many polynomial equations in many variables which I want to jointly minimize (in a mean square sense, but you could pick a different reasonable measure which favors anything where all ...
4
votes
2answers
242 views

Proving the existence of an integral quadratic form

Theorem 11 (Conway & Sloane, Sphere Packings, Lattices and Groups, 3rd Edition, pp 383, Ch 15). If a system of putative $p$-adic symbols for each $p$ satisfies the determinant, oddity, and ...
3
votes
1answer
185 views

Intuition on a certain class of quadratic optimization problems

Let $\mathcal{X} = \{\mathbf{X}\in\mathbb{C}^{d\times d}:\|\mathbf{X}\|\leq 1\}$, where $\|\cdot\|$ is the Frobenius norm. Let $\mathbf{y}\in\mathbb{C}^{d\times 1}$. We are familiar with the following ...
0
votes
1answer
72 views

A description of the isometry group $O(U\oplus E_8)$?

Are there any good description of the isometry group $O(U\oplus E_8)$? Here $U$ denotes the hyperbolic lattice and $E_8$ the root lattice of type $E_8$.
2
votes
1answer
256 views

How to determine $O(L)$ is finite or not?

Let $L$ be an indefinite {\it non-unimodular} integral lattice. I am particularly interested in unimodular cases, such as $U(2)\oplus A_4, U\oplus D_4$. Are there any general method to determine ...