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.
543 questions
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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 \...
2
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1
answer
296
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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 ...
2
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1
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391
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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 ...
2
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1
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747
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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 ...
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1
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850
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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 + ...
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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 $...
2
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2
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506
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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=...
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1
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269
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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(...
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2
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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 $(Q,V)$ ...
2
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1
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116
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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 ...
2
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1
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436
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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,...
2
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1
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486
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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 $(\...
2
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1
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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
$x, y$ ...
2
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1
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358
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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 ...
2
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1
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339
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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 ...
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914
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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 ...
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2
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1k
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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 ...
2
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1
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148
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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 ...
2
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1
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148
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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 ...
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177
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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,...
2
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2
answers
281
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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 ...
2
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5
answers
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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 ...
2
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1
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2k
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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 ...
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2k
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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 ...
2
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2
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262
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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 ...
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1
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171
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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 ...
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258
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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 ...
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2
answers
323
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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 ...
2
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1
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69
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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 ...
2
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1
answer
65
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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&...
2
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1
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223
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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 ...
2
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1
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210
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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)...
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2
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187
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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 ...
2
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1
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522
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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 ...
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293
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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 ...
2
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2
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468
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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 ...
2
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1
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186
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$k[X_1,\ldots,X_n]/Q$ 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 ...
2
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1
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242
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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 ...
2
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2
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119
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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 ...
2
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1
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1k
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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 ...
2
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1
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364
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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 ...
2
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1
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186
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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-...
2
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1
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376
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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
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1
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210
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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 ...
2
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2
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299
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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)...
2
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1
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398
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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
$$
Q[x]:...
2
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0
answers
156
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Sets represented by integral quadratic forms
Let $f(x) = x^\intercal A x$ be a positive definite integral quadratic form on $d$ variables. A positive integer $n$ is said to be represented by $f$ if $f(x) = n$ for some $x \in \mathbb Z^d$. A set ...
2
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1
answer
196
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Estimate for the operator $A A_D^{-1}$
Let $O\subset\mathbb{R}^d$
be a bounded domain of the class $C^{1,1}$
(or $C^2$
for simplicity). Let the operator $A_D$
be formally given by the differential expression $A=-\operatorname{div}g(x)\...
2
votes
0
answers
55
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Quadratic surjective map between spheres
The quadratic function $f:\mathbb R^4\to\mathbb R^3$
$$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^...
2
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0
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76
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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=\...