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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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The quadratic form $x^2+ny^2$ via prime factors

Elementary algebra shows that the product of two numbers in the form $x^2 + ny^2$ again has the same form, since if $p = (a^2 + nb^2)$ and $q = (c^2 + nd^2)$, $$pq = (a^2 + nb^2)(c^2 + nd^2) = (ac \...
Stemkoski's user avatar
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1 answer
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When are rings of the form $K[x_1,...,x_n]/(Q)$ principal ideal domains when $Q$ is quadratic?

By a result of Klein-Nagata rings of the form $A_Q=K[x_1,...,x_n]/(Q)$ are factorial when $K$ is a field, $n \geq 5$ and $Q$ is a non-degenerate quadratic form. Question 1: When is $A_Q$ a principal ...
Mare's user avatar
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1 answer
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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 ...
Stanley Yao Xiao's user avatar
2 votes
1 answer
747 views

Why is the Fano variety of lines on a smooth three-dimensional quadric isomorphic to $\mathbb{P}^3$?

Let $Q \subset \mathbb{P}^4$ be a smooth three-dimensional quadric over an algebraically closed field $k$ ($\mathrm{char}(k) \neq 2$) and let $F$ be the Fano variety of lines on $Q$. In "Iskovskikh ...
Dimitri Koshelev's user avatar
2 votes
1 answer
850 views

Simultaneous quadratic equations

I have a set of 3 quadratic equtions: $A_1µ_1^2 + B_1µ_1µ_2 + C_1µ_2^2 + D_1µ_1 + E_1µ_2 + F_1 = 0$ $A_2µ_2^2 + B_2µ_2µ_3 + C_2µ_3^2 + D_2µ_2 + E_2µ_3 + F_2 = 0$ $A_3µ_3^2 + B_3µ_3µ_1 + C_3µ_1^2 + ...
Jakube's user avatar
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1 answer
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Primes as the first coefficient of a reduced indefinite quadratic form

Given a discriminant d>0 (make it fundamental if that is easier), when can a prime p be the the $x^2$ coefficient of a reduced indefinite quadratic form? That is, for what p is there a reduced form $...
paarshad's user avatar
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2 answers
506 views

Lower bounds for split primes in Real quadratic fields

Snippet portion: From Iwaniec and Kowalski's Analytic Number Theory: If the class number $h=h(D)$ is small, then there are only few prime ideals $\bf{p}$ of degree one with small norm. Indeed, if $p=...
paarshad's user avatar
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Can each natural number be represented by $2w^2+x^2+y^2+z^2+xyz$ with $x,y,z\in\mathbb N$?

It is well known that each $n\in\mathbb N=\{0,1,2,\ldots\}$ can be written as $2w^2+x^2+y^2+z^2$ with $w,x,y,z\in\mathbb N$. Furthermore, $$\{2w^2+x^2+y^2:\ w,x,y\in\mathbb N\}=\mathbb N\setminus\{4^k(...
Zhi-Wei Sun's user avatar
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The stabiliser group of an isotropic quadratic form over $\mathbb{Q}_p$ is non-compact?

Let $\mathbb{Q}_p$ denote the $p$-adic integers. Let $V$ be a $\mathbb{Q}_p$-vector space and $Q : V \rightarrow \mathbb{Q}_p$ be a non-degenerate integral quadratic form. We say that the pair $(Q,V)$ ...
Andrew Musso's user avatar
2 votes
1 answer
116 views

Sufficient condition for pair of real quadrics to have real intersection

In the following, when I talk about the zero of a homogeneous polynomial I always mean a projective zero. Let $ q $ be a real quadric. Then $ q $ has a real zero if and only if $ q $ has indefinite ...
Ian Gershon Teixeira's user avatar
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1 answer
436 views

how to determine a biquadratic form is positive-definite

A biquadratic form $\sum_{i,j,k,l}b_{i,j,k,l}x_{i}x_{j}y_{k}y_{l}$, how to determine whether it is positive-definite? A necessary and sufficient condition? In fact, I have a matrix $B=\sum_{1\leq i,...
snow's user avatar
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1 answer
486 views

Involution on the components of a group algebra

If $G$ is a finite group and $k$ a field, there is a canonical involution (ie an involutive anti-automorphism) $\sigma$ on $k[G]$ induced by $g\mapsto g^{-1}$. Given that the center of $k[G]$ has $(\...
Captain Lama's user avatar
2 votes
1 answer
107 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 $x, y$ ...
Albertas's user avatar
  • 704
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1 answer
358 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
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1 answer
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Question about Gauss composition law over PID.

Let $m$ be a square free integer, $\mathbb{Q}(\sqrt{m})$ a quadratic field extension of $\mathbb{Q}$, $\Delta$ is its discriminant and $O_{\mathbb{Q}(\sqrt{m})/\mathbb{Q}}$ its ring of integers. We ...
M.Souf's user avatar
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Orthogonal group of the lattice $I_{p,q}$?

Here $I_{p,q}$ is the unique-up-to-isometry unimodular lattice of signature $(p,q)$, whose Gram matrix is diagonal with $p$ 1s and $q$ -1s. In his paper "ON GROUPS OF UNIT ELEMENTS OF CERTAIN ...
A. Pascal's user avatar
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2 votes
2 answers
1k views

Rank of a linear combination of quadratic forms

Suppose we have a set of quadratic forms $Q_i (x_1, \dots, x_n)$ for $1 \leq i \leq k$ in $n$ variables, defined over $\mathbb{R}$. We suppose these are 'collectively nondegenerate' in the sense that ...
sobe86's user avatar
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1 answer
148 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
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1 answer
148 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
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1 answer
177 views

Simultaneous diagonalization of two rational forms

It is known that any two real quadratic forms are equivalent iff they have the same signature. If we consider rational quadratic forms, they are $\mathbb{Q}$-equivalent iff the have the same signature,...
Артемий Соколов's user avatar
2 votes
2 answers
281 views

Witt index of the sum of 24 squares

Consider the quadratic forms $q(x)=x_1^2+\dots +x_8^2$ in 8 variables and $p(x)=x_1^2+\dots+x_{24}^2$ in 24 variables over the field of rational numbers $\mathbb{Q}$. Let $E/\mathbb{Q}$ be a field ...
Mikhail Borovoi's user avatar
2 votes
5 answers
1k views

Does the Diophantine equation $(x^2+ay^2)(u^2+bv^2) = p^2+cq^2$ admit a complete solution?

In this MSE question/thread, I have been discussing the equation $$ (x^2+ay^2)(u^2+bv^2) = p^2+cq^2, \tag{$\star$} $$ where $x,a,y,u,b,v,p,c,q$ are integers. I posed a conjecture which turned out to ...
Kieren MacMillan's user avatar
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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
2 votes
1 answer
2k views

Diagonalization of a quadratic form in integers

Hello, Recently I've been studying the problem of quadratic form diagonalization. Suppose that we have a form $F(x,y,z)$ with corresponding symmetric matrix $M$. This form is equivalent to another ...
Anton's user avatar
  • 1,625
2 votes
2 answers
262 views

on the determination of a quadratic form from its isotropy group in char. 2

So this question is a continuation of the following one [1] On the determination of a quadratic form from its isotropy group For some motivations and relevant backgrounds related to this question ...
Hugo Chapdelaine's user avatar
2 votes
1 answer
171 views

Integers $8k+3>0$ not represented by $2x^2+4y^2+4yz+9z^2$ over the integers

Oeis A306970 lists positive integers of the form $8k+3$ which are not reprented by $$f(x,y,z):=2x^2+4y^2+4yz+9z^2$$ over the integers as $3,43,163,907$. It says this list may not be complete and ...
Mastrem's user avatar
  • 458
2 votes
1 answer
258 views

An arithmetic problem involving a system of equations

Fix a positive integer $r$. Describe the solutions to the system of equations given by: $$\begin{equation}\sum_{1\leq i\leq r}X_i^2\equiv0\pmod{X_k}(1\leq k\leq r)\end{equation}$$ Example: In the case ...
semisimpleton's user avatar
2 votes
2 answers
323 views

Determinants of minors occurring 'within' determinant of full matrix

$A= (a_{ij})$ is an $n\times n$ symmetric positive matrix. It induces a quadratic form $f(x):= x^tAx$ on $\mathbb{R}^n$. $D_m$ denotes the determinant of the top left $m\times m$ submatrix of $A$ (or ...
Calamardo's user avatar
  • 675
2 votes
1 answer
69 views

Clarification on FPTAS optimization in a paper

In the abstract of this paper by Hildebrand, Weismantel & Zemmer it is stated that they provide an FPTAS for $$\min x'Qx$$ over a fixed dimension polyhedron when $Q$ has at most one negative or ...
Turbo's user avatar
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2 votes
1 answer
65 views

Limit of $(X' (\Delta + \varepsilon I_n)^{-1} X)^{-1}$ with $\Delta$ an $n \times n$ diagonal matrix

(Note: I have asked this question before on math.stackexchangecom, but it wasn't answered, so I am trying again here). The question is pretty much in the title; $X$ is an $n\times r$ matrix with $n&...
Elvis's user avatar
  • 123
2 votes
1 answer
223 views

Uniformity in the error term of counting ideals up to a certain norm

There is a beautiful formula to count the number of ideals $I$ in the ring of integers $\mathcal{O}_K$ of a number field $K$, given by $$\sum_{n \leq X} a_n \sim C_K X,$$ where $a_n$ is the number ...
Stanley Yao Xiao's user avatar
2 votes
1 answer
210 views

u-Invariants of p-adic function fields

In his Paper "Fields of u-invariant 9" Oleg Izhboldin points out that for a algebraic closed, finitely generated field $k$ we have $u(k)= 2^{cd(k)}$. In particular we have $u(\mathbb{C}((t_1),..(t_n)...
nxir's user avatar
  • 1,479
2 votes
2 answers
187 views

Asymptotic property of a quadratic form

suppose $x=\Delta$, $y=M \Phi \Delta$, where $\Delta\in N\times 1$, $M^T=M \in N \times N$ and $\Phi^T=\Phi \in N \times N$. Define $Z=xy^T+yx^T$. It is known from the answer to my previous question ...
Michael Fan Zhang's user avatar
2 votes
1 answer
522 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
  • 7,377
2 votes
2 answers
293 views

Convex optimization problem to QPP

Briefly, have the following problem: \begin{equation} \sum_{i = 0}^n a_i \ (max [ F_i( \bar x ), 0 ] )^2 \rightarrow min, \\\\ s.t.\\\\ A \bar x \leq b \end{equation} where $ F( \bar x ) $ is a ...
Artem Pianykh's user avatar
2 votes
2 answers
468 views

Orthogonal transformations fixing a subspace (setwise)

Let $(V,Q)$ be a non-degenerate quadratic space of dimension $n$ over an algebraically closed field of characteristic zero. Let $W$ be a subspace of $V$ of dimension $m < \frac12 n$ which is ...
Wanderer's user avatar
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2 votes
1 answer
186 views

$k[X_1,\ldots,X_n]/Q$ is UFD for non-singular quadratic form $Q$ and $n\ge 5$

I am looking for a reference for the following result. Thanks in advance. Let $k$ be a field of any characteristic other than $2$. Klein and Nagata showed that the ring $R:=k[X_1,\ldots,X_n]/Q$ is a ...
reggie's user avatar
  • 21
2 votes
1 answer
242 views

Shifted lattices and the discriminant group

I'm studying a geometrical problem where an (even, negative-definite) lattice $L$ arises. Roughly, as an intersection pairing for curves on a surface. In fact, the problem naturally leads me to ...
Benighted's user avatar
  • 1,701
2 votes
2 answers
119 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 ...
Ron P's user avatar
  • 947
2 votes
1 answer
1k views

Irreducible variety

I asked a similar question at MSE, as the question seemed quite basic to me, but did not get any hint in 24 hours, except for one upvote for the question itself. I still think I am stuck with some ...
Vadim's user avatar
  • 187
2 votes
1 answer
364 views

Bai and Silverstein's "Lemma on Quadratic Forms" - question about the constant $C_p$

In the book "Spectral Analysis of Large Dimensional Random Matrices" by Bai and Silverstein, there is the following lemma: Lemma B.26 (pg. 530) Let $A=(a_{ij})$ be an $n\times n$ nonrandom matrix ...
Lepidopterist's user avatar
2 votes
1 answer
186 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 non-...
César Galindo's user avatar
2 votes
1 answer
376 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
  • 2,108
2 votes
1 answer
210 views

Gram matrix modulo 4

Suppose we have a full rank, integer sublattice $L$ of the integer lattice $\mathbb Z^d$, where we fix the dimension $d$. Consider the Gram matrix $M$ of $L$, relative to some basis for $L$, and ...
Sinai Robins's user avatar
2 votes
2 answers
299 views

Involution of $E_{8}$ lattice

Let $L$ be a lattice associate to the Dykin matrix of type $E_{8}$. I would like to understand involutions of $L$ and their invariant $L^{+}$ and coinvariant lattice $L^-$ (I think they are isomorphic)...
M Pandhari's user avatar
2 votes
1 answer
398 views

finitely many orbits of integer solutions module unimodular groups

Let $Q$ be the Lorentzian matrix, that is $Q=\mathrm{diag}(I_n,-1)$, where $I_n$ denotes de $n\times n$ identity matrix. Let $\mathcal R$ the set of integer solutions $x\in\mathbb Z^{n+1}$ of $$ Q[x]:...
emiliocba's user avatar
  • 2,446
2 votes
0 answers
156 views

Sets represented by integral quadratic forms

Let $f(x) = x^\intercal A x$ be a positive definite integral quadratic form on $d$ variables. A positive integer $n$ is said to be represented by $f$ if $f(x) = n$ for some $x \in \mathbb Z^d$. A set ...
Andre Kornell's user avatar
2 votes
1 answer
196 views

Estimate for the operator $A A_D^{-1}$

Let $O\subset\mathbb{R}^d$ be a bounded domain of the class $C^{1,1}$ (or $C^2$ for simplicity). Let the operator $A_D$ be formally given by the differential expression $A=-\operatorname{div}g(x)\...
Yulia Meshkova's user avatar
2 votes
0 answers
55 views

Quadratic surjective map between spheres

The quadratic function $f:\mathbb R^4\to\mathbb R^3$ $$f(a,b,c,d)=\begin{bmatrix} 2(ac + bd)&2(ad - bc)&a^2 + b^2 - c^2 - d^2\end{bmatrix}$$ surjectively maps the sphere $S^3$ to the sphere $S^...
rikhavshah's user avatar
2 votes
0 answers
76 views

Are the following two characterisations of symplectic modules, using the language of form rings, the same?

Page 205 of the book Classical Groups and Algebraic K-Theory defines a symplectic module to be an arbitrary quadratic module $(M,h,q)$ over a form ring $(R,\Lambda)$ with $(J,\varepsilon)$ where $J=\...
wlad's user avatar
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