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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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Epstein zeta function for non-fundamental discriminant to L-series

Let $Q(x,y) = ax^2+b xy + cy^2$ be a primitive integral positive-definite quadratic form, with associated number field $K$. If $D=b^2-4ac$ is a fundamental discriminant, then it's well-known that $$\...
pisco's user avatar
  • 211
4 votes
1 answer
159 views

Schur multiplier of a Chevalley group of type $D_5$

$\DeclareMathOperator\EO{EO}\DeclareMathOperator\SO{SO}\DeclareMathOperator\St{St}\DeclareMathOperator\Sp{Sp}$This is sort of a follow up question to my post here regarding the commutator subgroup of $...
Noah B's user avatar
  • 263
2 votes
0 answers
96 views

Families of quadratic forms over a function field

I have a family of quadratic form in two variables $q_{t}(x,y) = ax^2 + bxy+cy^2$ where $a,b,c\in\mathbb{C}(u,v)$ are rational functions depending on a parameter $t\in\mathbb{P}^1$. I would like to ...
TopGatLu's user avatar
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4 votes
0 answers
123 views

Is an orthogonal direct sum decomposition with respect to two quadratic forms necessarily unique up to isomorphism

Consider two quadratic forms $Q$ and $P$ over a finite dimensional vector space $V$ over a quadratically closed (or perhaps Pythagorean) field $F$. If $V$ can be decomposed as $V = V_1 \oplus V_2 \...
wlad's user avatar
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1 vote
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26 views

Criterion for unicity and existence of pre-image in multivariate cryptography

Repost from math.stackexchange since no one could help me there and it concerns my research. I am reading Ding's Multivariate Public Key Cryptosystems and in the book the author explains the so-called ...
Saegusa's user avatar
  • 153
6 votes
1 answer
422 views

Computing a Commutator Subgroup

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}$I’m studying the group $\O(5,5,\mathbb{Z})$, the indefinite orthogonal matrices with integer entries. In particular, I ...
Noah B's user avatar
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0 votes
0 answers
28 views

How to express the following formula with quadratic form?

A and B are two constants, and $x_1,x_2,x_3$ are all binary vectors with the same length. How to express the following formula with quadratic form? $$AB \vec{x}_1^T \vec{p}_1 \cdot \vec{x}_2^T \vec{p}...
christian_lee's user avatar
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0 answers
46 views

How to express the product of elements of a vector with quadratic form?

$x$ is a binary vector which means the elements in $x$ are 0 or 1, and $p$ is another vector with the same length. How to express the product of elements in $p$ whose corresponding elements are 1 in $...
christian_lee's user avatar
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0 answers
47 views

Is the absolute value of a complex quadratic form a convex real function?

Consider a complex vector space $V = \mathbb{C}^n$ and a quadratic form $Q(x) = x^TAx$ on $V$ where $A$ is a symmetric matrix i.e., $A^T = A$. Is is true that the absolute value $|Q(x)|$, seen as a ...
user493645's user avatar
2 votes
0 answers
91 views

Solutions to the quadratic matrix equation $X A X^T = B$

Let $A, B \in \mathbb{R}^{n \times n}$ be symmetric, positive-semidefinite, full-rank matrices. I would like to understand the set of $X \in \mathbb{R}^{n \times n}$ which are themselves symmetric and ...
dotdashdashdash's user avatar
4 votes
1 answer
206 views

Is there a good notion of kernels of quadratic forms on abelian groups?

Let $G$ be an abelian group and let $q:G \to \mathbb{Q/Z}$ be a quadratic form, i.e. $q(a)=q(-a)$ and $b(x,y)=q(x+y)-q(x)-q(y)$ is a bihomomorphism. On vector spaces, when people speak about the ...
Bipolar Minds's user avatar
1 vote
0 answers
76 views

When does a system of homogeneous quadratic equations have integer solutions?

I learned that in general, solving systems of quadratic Diophantine equations is a difficult problem. But I wonder if there are special (and non-trivial) types of systems that are easier to handle. ...
Louie's user avatar
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7 votes
0 answers
204 views

K3 surfaces with no −2 curves

I seem to remember that a K3 surface with big Picard rank always has smooth rational curves. This question is equivalent to the following question about integral quadratic lattices. Let us call a ...
Misha Verbitsky's user avatar
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0 answers
175 views

Points at which a polynomial becomes reducible

Let $n \geq 10$ and set $\mathbf{y} = (y_1,\ldots,y_n)$. Let $Q_1(\mathbf{y}),\ldots,Q_5(\mathbf{y})$ be non-zero quadratic forms with integer coefficients such that the cubic form $x_1Q_1(\mathbf{y})+...
admissiblecycle's user avatar
0 votes
1 answer
147 views

Compatibility conditions for quadratic equations

In the context of physics, I stumbled over the following problem: I have $N$ equations, all are quadratic in a single scalar, real variable $x$: \begin{eqnarray} 0 &= A_1x^2 + B_1x + C_1 \\ &...
Michael Schindler's user avatar
1 vote
1 answer
90 views

Projectivity of the fundamental ideal of Witt groups

Suppose $k$ is a field. I wonder when the Witt ring of the quadratic forms $\textbf{W}(k)$ has a projective fundamental ideal, which is the kernel of the rank modulo 2 morphism. Here I want a ...
Nanjun Yang's user avatar
1 vote
1 answer
61 views

A question about the sign of quadratic forms on nonnegative vectors

Let $M$ be a real square matrix of order $n\ge 3$. Assume that for every nonnegative vector $\textbf{z}\in \mathbb R^n$ which has at lease one zero entry we have $\textbf{z}^T M \textbf{z} \ge 0$. Can ...
user139975's user avatar
5 votes
1 answer
148 views

Equivalence of quadratic forms over $p$-adic integers vs over localisation at $p$

To discern whether two integral quadratic forms are equivalent over the $p$-adic integers, one can compute a Jordan decomposition at $p$ and read off some invariants. Restricting to $p\ne2$ for ...
a196884's user avatar
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1 vote
2 answers
271 views

Integral solutions of quadratic equation $5 X² − 14 X⁢Y + 5 ⁢Y² = n$

Solve for all integers $x$ and $y$ the quadratic form $5 X² − 14 X⁢Y + 5 ⁢Y² = n$ for some integer n. I know that for some cases there are recurrence solutions, but I'm not sure how to solve these ...
Benjamin Warren's user avatar
2 votes
0 answers
47 views

Equidistribution of lattice points on quadratic forms without certain values

I have recently been studying some results about equidistribution of lattice points on positive definite quadratic forms $Q$ with $3$ or more variables. Concretely the article Duke and Schulze-Pillot, ...
MathqA's user avatar
  • 163
11 votes
1 answer
796 views

How to describe all integer solutions to $x^2+y^2=3z^2+1$?

The question is in the title. Here is a short motivation. The general quadratic Diophantine equation is $$ x^TAx+bx+c=0, $$ where $x$ is a vector of $n$ variables, $A$ is $n \times n$ matrix with ...
Bogdan Grechuk's user avatar
1 vote
0 answers
94 views

Can I solve this quadratic program "fast"?

We are given a matrix $D \in \mathbb{Z}^{C \times C}$ of non-negative entries, an integer $k \geq 1$ and we need to maximize the quadratic form $x^T D x$ under some simple constraints. For all ...
reservoir's user avatar
0 votes
0 answers
50 views

A certain property of positive-semidefinite infinite matrices

In this answer I concluded with this: For which arrays $\big(\sigma_{ij}\big)_{(i,j)\in\mathbb N^2}$ [of real numbers] whose every upper-left corner is positive-semidefinite does line $(1)$ above ...
Michael Hardy's user avatar
1 vote
0 answers
35 views

Eigenvalues of orthogonal group element

Let $q$ be a quadratic form over a nonarchimedean local field $F$, and let $\operatorname{O}(q)$ be the corresponding orthogonal group. Let $g\in\operatorname{O}(q)$ be semisimple. Can we know ...
Windi's user avatar
  • 689
-1 votes
1 answer
127 views

Does this quadratic system admit an integral or a rational solution?

Let $a,b$ be coprime and say $0<a<b<2a$. Consider the quadratic system: $$\alpha\delta-\beta\gamma=1$$ $$(\alpha^2-(\alpha\delta+\beta\gamma))a^2b+\beta^2b^3+(2\alpha\beta-\beta\delta)ab^2-\...
Turbo's user avatar
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2 votes
1 answer
82 views

Primitive representation of integers by some form on the genus of a quadratic form

Some time ago, I asked a question about equidistribution on a paper of Duke and Schulze-Pillot that was usefully answered. However, on the answer there was a statement that was unimportant for me back ...
MathqA's user avatar
  • 163
0 votes
0 answers
226 views

Representations with noticeable property

Let $\rho : \mathbb{R}^n\to \mathfrak{so}(2m)$ be a faithful representation of the commutative Lie algebra $\mathbb{R}^n$ into the Lie algebra of skew-symmetric matrices (a Lie algebra morphism). ...
user56980's user avatar
  • 282
0 votes
0 answers
42 views

Properties of the quadratic form for the sum of the Jacobian and its transpose

Consider the Jacobian matrix of $$ \begin{bmatrix} f_{1}(x_1(t), x_2(t),\;\; \ldots \;\;x_{N_1}(t))\\ \vdots \\ f_{N_2}(x_1(t), x_2(t),\;\; \ldots \;\;x_{N_1}(t)) \end{bmatrix} $$ Let the jacobian ...
Atharva's user avatar
4 votes
1 answer
172 views

Computing spinor equivalence for positive definite forms

Given an integral positive-definite rank $n$ quadratic form $f$, one can use the algorithm in Conway and Sloane (Chapter 15, SPLaG) to efficiently determine if the genus of $f$ contains more than one ...
a196884's user avatar
  • 323
1 vote
0 answers
130 views

Solution that minimizes the sum of squared errors, with quadratic constraints

Given symmetric and positive definite $n \times n$ (real) matrices $A_1, \dots, A_m$ and $b_1, \dots, b_m \in {\Bbb R}^{n}$, I am trying to find the solution with the least sum of squared errors of ...
gtj's user avatar
  • 11
3 votes
1 answer
168 views

Relation between positive roots of $E_8$ and $\mathbb{F}_2^8 /{0}$

There exists an explicit bijection (due to Cayley, that has built up a very nice table to describe this) between the positive roots of the Lattice $E_7$ and $\mathbb{F}_2^6/{0}$ (where $\mathbb{F}_2$ ...
IMeasy's user avatar
  • 3,747
3 votes
2 answers
229 views

Inverse quadratic norms

The following problem seems easy at a first glance but I can't see the way to prove it. Actually I don't even know if it's true but it is assumed implicitly in a research paper. Help highly ...
Felix's user avatar
  • 33
1 vote
0 answers
78 views

Continuous choice of null directions for a family of bilinear forms

Let $E$ and $F$ be (real) Hilbert spaces, where $\dim E = \infty$ and $1 \leq \dim F < \infty$. Let $T : E \to \operatorname{Sym}(F \times F,\mathbb{R})$ be a continuous linear map, where $\...
Eduardo Longa's user avatar
2 votes
1 answer
129 views

Does the F-unitary group isomorphism arises from a conformal isometry?

Let $K$ be a CM-field with totally real subfield $F$. Let $(V_1, h_1)$ and $(V_2, h_2)$ be two $n$-dimensional $K$-vector spaces with nondegenerate Hermitian forms, where $n\geq 3$. Question 1 Does ...
Zhiwei Zheng's user avatar
1 vote
1 answer
249 views

Degenerate representation

Let $\rho : \mathbb{R}^n\to \mathfrak{so}(2m)$ be a faithful representation of the commutative Lie algebra $\mathbb{R}^n$ into the Lie algebra of skew-symmetric matrices. There is an orthonormal basis ...
user56980's user avatar
  • 282
1 vote
0 answers
78 views

Generalize spinor bundles over orthogonal Grassmannians

We will work over $\mathbb C$ and the notation will be coherent with the paper of Ottaviani (see [Ott]). Consider a $n$-dimensional quadric hypersurface $Q_n \subset \mathbb P^{n+1}$. We have ...
Bobech's user avatar
  • 371
3 votes
2 answers
190 views

Generalization of the geometric series representation of the Kronecker delta for arbitrary lattices

In On construction of holomorphic cusp forms of half integral weight by Shintani, in the last equation of page 96, an identity for the Kronecker delta for elements of $L^*/L$ is defined. Here, $L$ is ...
user avatar
1 vote
0 answers
107 views

Polynomial parametrization for solutions of quadratic Diophantine equations

A previous Mathoverflow question asks if there is an algorithm that would determine all integer solutions to a given quadratic Diophantine equation. To make this question more formal, we need to agree ...
Bogdan Grechuk's user avatar
4 votes
1 answer
341 views

Calculating the explicit constant – Siegel zeros and class numbers

Let $\chi$ denote the Legendre symbol of conductor $q$. A Siegel zero for the $ L $ series associated to $ \chi $, which we denote by $ L(s,\chi) $ is a real zero $ \sigma $ satisfying $ 1-\frac{c}{\...
Krishnarjun's user avatar
3 votes
1 answer
188 views

Duke and Schulze-Pillot condition for equidistribution

When regarding a ternary quadratic form $Q(x,y,z)$, is a classic question to consider which integers $n$ can be represented by $Q$. It is also classic to wonder how "well distributed" are ...
MathqA's user avatar
  • 163
3 votes
1 answer
386 views

Image of a quadratic form is a closed cone

Let $Q : E \to F$ be a quadratic form induced by a symmetric bilinear form $B : E \times E \to F$ defined in a finite dimensional real normed vector space $E$, with values in the normed vector space $...
Eduardo Longa's user avatar
1 vote
0 answers
65 views

Maximal $l^2$ restriction of psd quadratic form on $\mathbb{R}^n$

I've been stuck on this problem for a few days as part of a bigger research problem and am looking for some help. Let $S : \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$ be a positive ...
confused_guest's user avatar
2 votes
0 answers
132 views

Obstruction in construction of some lattices, related with $K3$ surfaces

I am considering a certain $K3$ surface that is lattice-polarized in two ways. This leads to the following simple problem in lattice theory: (Let me borrow notations for lattice from Ch.14 of this ...
user69559's user avatar
  • 1,483
0 votes
0 answers
88 views

Positive definite matrix and Hörmander theory

Let $\varphi \in C_{0}^{\infty}, \varphi\neq 0$. We'll consider the inner product in $L^{2}.$ Let $\alpha,\beta$ multi-index, $m\in \mathbb{N}$ such that $|\alpha|,|\beta|\leq m$ and set $$ \varphi_{\...
Lucas De Souza's user avatar
1 vote
0 answers
93 views

Integral zeros of quadratic forms

Suppose I have a definite quadratic form over $\mathbb{Z}$ in $n$ variables $$Q(\mathbf{x}) = \sum_{i,j} a_{i,j}x_ix_j$$ I'm looking to find whether an integer $m$ is representable by $Q$. Everyone ...
Rdrr's user avatar
  • 831
2 votes
0 answers
95 views

Computing coefficients of theta functions associated to quadratic forms

If we take an integral positive definite quadratic form $Q$ and set $\Theta_Q(z) = \sum_{k\geq 0}R_Q(k)e^{2\pi ikz}$, what are the most efficient algorithms to compute the $R_Q(k)$? I am aware e.g. of ...
a196884's user avatar
  • 323
4 votes
2 answers
474 views

Existence of solution for a system of quadratic diophantine equations / symmetric quadratic froms

I am interested in solving, or even just deciding the existence of a solution, for a system of quadratic diophantine equations. Let $p$ be a prime congruent to 1 modulo 8, so $ p =17$ is the first ...
margollo's user avatar
1 vote
1 answer
250 views

Integers representable as binary quadratic forms

It is known that odd prime $p$ can be represented as $p=x^2+y^2$ if and only if $p \equiv 1$ mod $4$, represented as $p=x^2+2y^2$ if and only if $p \equiv 1$ or $3$ mod $8$, represented as $p=x^2+3y^2$...
Bogdan Grechuk's user avatar
1 vote
0 answers
134 views

Can $12n+5$ be written as $2x^2+5y^2+9z^2+xyz$ with $x,y,z$ nonnegative integers?

Let $\mathbb N=\{0,1,2,\ldots\}$. By the Gauss-Legendre theorem on sums of three squares, for any $n\in\mathbb N$ we can write $4n+1$ as $x^2+y^2+z^2$ with $x,y,z\in\mathbb N$. Motivated by this, here ...
Zhi-Wei Sun's user avatar
  • 13.7k
3 votes
0 answers
115 views

Bott periodicity in characteristic p via Clifford algebras

I am currently reading Husemoller's wonderful book on fibre bundles, specifically the section on Clifford algebras. He defines these groups $L_k$ as follows. Let $M_k$ denote the free abelian groups ...
Noah Ankney's user avatar

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