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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Solving a quadratic matrix equation with fat matrix

I am trying to find an $n \times m$ fat (i.e., $m > n$) matrix $T$ that solves $$T^T T = X$$ where $X$ is a given $m \times m$ symmetric, positive semidefinite matrix. I saw this post, but ...
jaraujo's user avatar
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1 answer
143 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 $X$) ...
nxir's user avatar
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2 votes
0 answers
204 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 $Q(x_1,\ldots,...
Eric Rowell's user avatar
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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). ...
DmitryZ's user avatar
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3 votes
1 answer
156 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 ...
Hans's user avatar
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9 votes
4 answers
716 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 ...
Daniel Loughran's user avatar
1 vote
0 answers
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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 ...
Eugeniu Plamadeala's user avatar
3 votes
2 answers
301 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?
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2 votes
1 answer
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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 $x, y$ ...
Albertas's user avatar
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2 votes
1 answer
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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 non-...
César Galindo's user avatar
1 vote
1 answer
915 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 ...
akuz's user avatar
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8 votes
2 answers
3k 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 ...
u1571372's user avatar
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2 votes
1 answer
357 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 ...
HNuer's user avatar
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3 votes
1 answer
334 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 ...
Subhajit Jana's user avatar
-1 votes
1 answer
940 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} $ ...
user98883's user avatar
2 votes
1 answer
499 views

Representation of rationals by quadratic form

In one paper about number 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 ...
SashaP's user avatar
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14 votes
3 answers
9k 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 a common zero of all of these polynomials? In other words, ...
Arc's user avatar
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11 votes
1 answer
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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: $...
Will Jagy's user avatar
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4 votes
1 answer
141 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 ...
Mikhail Bondarko's user avatar
3 votes
1 answer
539 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 $...
Menny's user avatar
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383 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 $x=...
Bill Bradley's user avatar
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2 votes
0 answers
86 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. ...
Nemis L.'s user avatar
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6 votes
3 answers
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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 \,\...
Norouzi's user avatar
  • 352
5 votes
1 answer
2k views

Maximizing quadratic form on the 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 ...
Federico Magallanez's user avatar
14 votes
3 answers
1k 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 ...
Misha Verbitsky's user avatar
3 votes
1 answer
209 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 D=x+2(x^7)+2(...
paul Monsky's user avatar
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9 votes
1 answer
564 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 ...
Will Jagy's user avatar
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3 votes
1 answer
737 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.
Abhishek Kumar's user avatar
13 votes
1 answer
1k 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 (...
Will Jagy's user avatar
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0 votes
0 answers
178 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
1 answer
122 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 $\rm Pfister_n(k):=\{\text{isomorphism classes of n-fold Pfister forms over k}\}$; $\...
Yahoo's user avatar
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4 votes
0 answers
813 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 $\sigma=Int(i)\...
user40597's user avatar
8 votes
3 answers
2k 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. ...
Will Jagy's user avatar
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6 votes
2 answers
441 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 ...
Neil Strickland's user avatar
8 votes
4 answers
6k 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 ...
bobuhito's user avatar
  • 1,537
5 votes
3 answers
462 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 $p$-...
cdubey's user avatar
  • 85
3 votes
1 answer
251 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 ...
Lord Soth's user avatar
0 votes
1 answer
124 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 $E_8$.
Andrew's user avatar
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2 votes
1 answer
352 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 ...
Andrew's user avatar
  • 21
1 vote
2 answers
251 views

Impossible Range for Minkowski-Like Sum of Squares

Given coprime positive integers M,N, and a corresponding integer z outside of the range (for all integers x,a,b,c) of $Mx^2-N(a^2+b^2+c^2)$, is there any such z which is "deceptive", meaning that it ...
bobuhito's user avatar
  • 1,537
4 votes
3 answers
3k views

A nice necessary and sufficient condition on positive semi-definiteness of a matrix with a special structure

Let $$ A = \begin{pmatrix} \sum_{j\ne 1}a_{1j} & -a_{12} & \cdots & -a_{1n}\\ -a_{21} & \sum_{j\ne 2}a_{2j} & \cdots & -a_{2n}\\ \vdots & \vdots & \ddots & \...
Mohammad Khosravi's user avatar
0 votes
1 answer
255 views

Name for a Specific Type of Non-Symmetric Bilinear Form

Let $V$ be a finite dimensional vector space, with some choice of basis $\{e_i\}_{i \in I}$. With respect to an idempotent bijection $B:I \to I$, define a bilinear form by $$ g = \sum_{i=1}^N \lambda_{...
Milan Bernolak's user avatar
0 votes
3 answers
1k views

rank of outer product

I would like to ask if you may know how to prove this claim or any theorem related: Given 9 points (x,y,z) lie on unit sphere in 3 dimensional space such that any 4 points are not on the same plane. ...
thomeou's user avatar
3 votes
1 answer
573 views

Automorphism groups of indefinite non-unimodular integer lattices

Does anyone know of any papers in which structural aspects of the orthogonal group of some indefinite non-unimodular integral lattice are calculated? The exact lattice isn't so important and they don'...
user36896's user avatar
5 votes
0 answers
359 views

primes represented by indefinite quadratic forms

Let $Q$ be an indefinite binary quadratic form with discriminant $D$ and one class per genus (keep the example $x^2 - 2y^2$ in mind). If one asks about the set $P = \{ p : p \text{ prime and } p = Q(...
Victor Miller's user avatar
3 votes
0 answers
136 views

What are the Voronoi cones in 4 variables?

Question: What are the top dimensional cones of the 2nd Voronoi decomposition of the space of positive definite forms in $4$ variables? The 2nd Voronoi decomposition of the cone of positive definite ...
Numma1Stunna's user avatar
2 votes
1 answer
2k views

Maximum dimension of an isotropic subspace in a quadratic space

i hope my question is not too trivial. Let's suppose we have a vector space $V$ with a unimodular quadratic form $q$ of signature $(m,n)$. My question is: which is the maximum dimension of an ...
michael waltz's user avatar
4 votes
0 answers
297 views

Algorithm for solutions to quadratic forms over number fields

Are there any know (preferably implemented) algorithms to find solutions to quadratic forms over number fields (or global fields)? I am especially interested in the quaternary case. There exist some ...
Casaubon's user avatar
0 votes
1 answer
294 views

Subspace generated by positive vectors

Hi everyone, first of all i must admit i'm very familiar with quadratic forms and positive subspaces, so i'm sorry if my question is too trivial. So, here's my problem: Let $L$ be a real vector space ...
Filippo Amaducci's user avatar
1 vote
1 answer
370 views

Tangent space to positive oriented Grassmannians

Let $L$ be a real vector space of dimension 22 and $q$ a quadratic form on $L$ of signature $(3,19)$. Let $V\subset L$ be a positive oriented subspace of dimension 2 and $G^{po}(2,L)$ be the ...
Filippo Amaducci's user avatar