Questions about quadratic forms in many variables, for example $4x_1^2 + 3x_1x_2 + 5x_1x_3 - 8x_3^2$.
25
votes
2answers
793 views
Elementary proof that $2x^2+xy+3y^2$ represents infinitely many primes?
We did in class $x^2+y^2$, which was easy, and we had for homework $2x^2+2xy+3y^2$, which I did (its values (if not square) must be divisible by form primes, or of the form $x^2+5y^2$, and clearly ...
12
votes
0answers
767 views
Proof of Legendre's theorem on the ternary quadratic form
Theorem (Legendre): Let $a,b,c$ coprime positive integers, then $ax^2 + by^2 = cz^2$ has a nontrivial solution in rationals $x,y,z$ iff ...
11
votes
1answer
358 views
Polarization: etymology question
The polarization identity expresses a symmetric bilinear form on a vector space in terms of its associated quadratic form:
$$
\langle v,w\rangle = \frac{1}{2}(Q(v+w) - Q(v) - Q(w)),
$$
where $Q(v) ...
11
votes
3answers
127 views
Are matrices best understood as linear maps?
Any linear map between finite-dimensional vector spaces may be represented by a matrix, and conversely. Matrix-matrix multiplication corresponds to map composition, and matrix-vector multiplication ...
10
votes
5answers
315 views
Every integer vector in $\mathbb R^n$ with integer length is part of an orthogonal basis of $\mathbb R^n$
Suppose $\vec x$ is a (non-zero) vector with integer coordinates in $\mathbb R^n$ such that $\|\vec x\| \in \mathbb Z$. Is it true that there is an orthogonal basis of $\mathbb R^n$ containing $\vec ...
8
votes
2answers
356 views
Proving an inequality with Cauchy-Schwarz
In the "User's guide to viscosity solutions" by Crandall, Ishii and Lions (link), they make the following claim (inequality (A.4) p. 58) :
Given $x$, $\xi$ $\in \mathbb{R}^n$, $A \in \cal{S}(n)$ ...
8
votes
1answer
144 views
construction of the Witt group
I've seen a couple of constructions of the so-called Witt group: it seems that most authors start with the commutative monoid of isometry classes of quadratic spaces under direct sum, pass to the ...
8
votes
4answers
168 views
Representing a number as a sum of at most $k$ squares
Fix an integer $k >0 $ and would like to know the maximum number of different ways that a number $n$ can be expressed as a sum of $k$ squares, i.e. the number of integer solutions to
$$ n = x_1^2 + ...
8
votes
0answers
114 views
On the set of integer solutions of $x^2+y^2-z^2=-1$.
Let
$$
\mathcal R=\{x=(x_1,x_2,x_3)\in\mathbb Z^3:x_1^2+x_2^2-x_3^2=-1\}.
$$
The group $\Gamma= M_3(\mathbb Z)\cap O(2,1)$ acts on $\mathcal R$ by left multiplication.
It's known that there is ...
7
votes
4answers
886 views
Determining if a quadratic is always positive
Is there a quick and systematic method to find out if a quadratic equation is always positive or may have positive and negative or always negative for all values to its variables?
Say for a quadratic ...
7
votes
2answers
161 views
How does the theory of the quadratic number fields relate to the quadratic forms?
As every one knows, the quadratic number fields shares a deep connection with the binary quadratic forms; I have been told this relation when I was a senior high, and now I, learning some difficult ...
6
votes
2answers
299 views
Show $15x^{2} - 7y^{2} = 9$ has no integer solutions
I'm trying to show the quadratic binary has no integer solution. I've used the following process to transform it into a Pell's equation of the form $x^{2} - Dy^{2} = M$
If there is a solution, then ...
6
votes
1answer
94 views
Finiteness of the Witt ring
Is there some slick proof of the fact that for a field $F$, the Witt ring $W(F)$ is finite if and only if $-1$ is a sum of squares and $F^\times/F^{\times 2}$ is finite?
6
votes
1answer
81 views
Completing squares by symplectic transformations
A quadratic polynomial of $2n$ variables is given as
$$
H = \sum_{i,j=1}^{2n} A_{ij} x_i x_j = x^T A x,
$$
where $A$ is a symmetric matrix. I am looking for a symplectic transformation of these ...
6
votes
2answers
65 views
Coercive bilinear form on Hilbert space
I need to show the two following results. If true, it must be a simple proof but I do not seem to be able to make it work. Thank you in advance.
Consider a continuous symmetric bilinear form $B$ on a ...
6
votes
0answers
62 views
Proving the multiplicativity of a quaternary quadratic form
Consider the set $S$ of all integers of the form $f(x,y) + f(z,w)$, where $x,y,z,w$ are integers,
$$ f(x,y) = a x^2 + b x y + a^2 y^2,$$
and $a,b$ are integers with $$a > 1, \; \; 0 < b < ...
5
votes
4answers
232 views
Applications of quadratic forms
It seems that a lot of great mathematicians spent quite a while of their time studying quadratic forms over $\mathbb{Z},\mathbb{Q},\mathbb{Q_p}$ etc. and there is indeed a vast and detailed theory of ...
5
votes
3answers
191 views
Factoring numbers “of the (binary quadratic) form” in two different ways
For some fixed $n$ define the quadratic form $$Q(x,y) = x^2 + n y^2.$$
I think that if $Q$ represents $m$ in two different ways then $m$ is composite.
I can prove this for $n$ prime. I was hoping ...
5
votes
1answer
101 views
Is there a nice way to define the “maximum” of two quadratic forms?
Suppose I have two quadratic forms on $\mathbb R^n$, represented as symmetric matrices $A$ and $B$ on the usual basis. I am interested in approximating the function $x \mapsto \max(x^TAx, x^TBx)$ ...
5
votes
1answer
122 views
Generalizing the 290 theorem.
I have only just come across the remarkable theorem of Conway about universal quadratic forms over $\mathbb{Z}$; namely that in determining whether a integer coefficient, positive definite quadratic ...
5
votes
1answer
117 views
Follow up on intersection forms
For which topological spaces $X$ can I define an intersection form $b(\cdot, \cdot)$?
I know at least one example: If $X$ is a closed orientable $2n$-manifold then one can define an intersection ...
5
votes
1answer
86 views
A good reference for Quadratic Forms
Can anyone recommend a good reference for brushing up on quadratic forms?
They keep coming up (quite naturally of course) in the context of differential geometry and I find I am rustier than I ...
5
votes
0answers
93 views
counterexample for: integral forms that are equivalent as rational forms are also equivalent as integral forms
I have the feeling that I'm missing something very obvious: I'm looking for a counterexmple for the following statement for some $n>1$ (it is trivially true for $n=1$):
Let $A,B\in\mathbb ...
5
votes
0answers
87 views
What numbers are integrally represented by $4 x^2 + 2 x y + 7 y^2 - z^3$
EDIT: a selection of material relevant to this problem is available at INHOMOGENEOUS
In December 2010 my question appeared in the M.A.A. Monthly, show that $4 x^2 + 2 x y + 7 y^2 - z^3 \neq \pm 2 ...
5
votes
0answers
141 views
Solutions to equation in matrix form
Suppose $x=[x_1, x_2, \cdots, x_n]^t$, $b=[b_1,b_2,\cdots,b_n]^t$ with $b_i\in K$ and $A\in M_n(K)$, where $K$ is a field. There are well known criteria for the system of equations $Ax=b$, by ...
4
votes
3answers
118 views
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 $q = (c^2 + nd^2)$,
$$pq = (a^2 + nb^2)(c^2 + nd^2) = (ac ...
4
votes
2answers
169 views
Equivalence of quadratic forms over p-adic fields.
There is a theorem that states that two quadratic forms over $\mathbb{Q}_p$ are equivalent iff they have the same rank, discriminant and the same $\epsilon$ invariant.
(The last is defined as ...
4
votes
2answers
81 views
Proving the multiplicativity of a binary quadratic form
Consider the set $S$ of all integers of the form $x^2+y^2+4xy$, where $x$ and $y$ are integers. How could one prove the set $S$ is closed under multiplication? I have tried the bashy brute force ...
4
votes
1answer
268 views
Is a general (non-homogeneous) quadratic equation in $\mathbb{R}^3$ an ellipsoid?
This sounds like a simple problem, but I can't get it done. Given the general equation
$ax^2 + by^2 + cz^2 + dxy + exz + fyz + gx + hy + iz + j = 0,$
what are the requirements on the coefficients so ...
4
votes
1answer
90 views
Matrix Equation with Quadratic form
I am working in a problem that involves multivariate normal distributions and, at a given point, I need to solve the following matrix equation:
$$x=\sqrt{x^{\prime}\Sigma^{-1}x} \cdot y$$
Where $x$ ...
4
votes
1answer
184 views
Universal quadratic forms
A quadratic form is a polynomial $p(x_1,\dots,x_n)$ of the form $$ p(x_1,\dots,x_n)=\sum_{i \leq j}a_{ij}x_ix_j. $$
For example, $p_1(x,y,z,w)=x^2+y^2+z^2+w^2$ and $3x^2-5y^2$ are quadratic forms. I'm ...
4
votes
1answer
127 views
Help fixing my broken example of Arf invariant
I need help fixing a broken example I've come up with. In particular, I wanted to use the Arf invariant to distinguish two non-homeomorphic surfaces. That's the first part that's broken since there ...
4
votes
1answer
67 views
Two elements $a$ and $b$ of a quadratic space generate a nondegenerate two dimensional subspace if and only if $(a\cdot a)(b\cdot b)\neq(a\cdot b)^2$
I whould like to prove the following statement:
Lemma:
Let $(V,Q)$ be a nondegenerate quadratic vectorspace over a field $\mathbb{F}$ and $a,b\in V\setminus\{0\}$. Then for ...
4
votes
1answer
129 views
Centre of a quadric
I found the following sentence in my linear linear algebra book (affine and projective geometry): $Q:V \to \mathbb{K}$ is a quadric (quadratic function) and $\alpha\in Aff(V)$. $Aff(V)$ is the set of ...
4
votes
1answer
121 views
Map preserving indefinite scalar product must be linear
Let $V$ be a finite dimensional real vector space and $\langle\cdot,\cdot \rangle$ be a positive definite scalar product in $V$.
It is well know that if a map $T:V \to V$ preserves ...
4
votes
2answers
279 views
Etymology of the word “isotropic”
Given a quadratic form $q : V \rightarrow k$, a nonzero vector $v \in V$ is said to be isotropic if $q(v) = 0$. Any subspace of $V$ containing such a vector is also said to be isotropic, and the ...
4
votes
0answers
153 views
Understanding a proof about Hilbert Matrix
EDIT: I asked 3 questions. The first one I was able to solve myself, and the other two I cross-posted to MO.
Lately I've been interested in the Hilbert Matrix (its definition will come later). I went ...
4
votes
0answers
120 views
Proof that the Arf invariant is independent of choice of basis
I'm confused about the proof of the following claim:
Let $V$ be a vector space of dimension $2n$ and let $e_i, f_i$ be a symplectic basis. Let $q: V \to Z_2$ be a non-degenerate quadratic form. ...
4
votes
1answer
96 views
Partitioning polynomials in $\mathbb{Z}[x,y]$ by the primes they represent
Suppose you have a set $S\subset\mathbb{Z}[x,y].$ How can one efficiently partition the polynomials into sets such that the primes represented by the polynomials in any given set are identical? For ...
3
votes
4answers
137 views
Integral solutions of hyperboloid $x^2+y^2-z^2=1$
Are there integral solutions to the equation $x^2+y^2-z^2=1$?
3
votes
3answers
128 views
Question about answer about quadratic forms on MO
I have a question regarding this MO answer:
The answer says that in characteristic $2$, we cannot obtain a quadratic form from a bilinear form. I thought it was the other way around and now I am ...
3
votes
3answers
205 views
Confused about quadratic forms
My book states:
if $q: \mathbb{R}^n \to \mathbb{R}$ is a positive definite quadratic form then there exists a basis $B=(v_1,...,v_n)$ such that $q(x)=x_1^2+...+x_n^2=||x||^2$ for every ...
3
votes
2answers
168 views
Hilbert Symbols when $K =$ the $p$-adic numbers
How can I show that the Hilbert Symbol is bimultiplicative, when the local field is the $p$-adic numbers? Everything I can find just sort of asserts bimultiplicativity without much proof, so I'm ...
3
votes
2answers
86 views
When does a binary quadratic form represent 1 or -1
Let $a,b,c$ be integers. Is there a reasonably concise condition on $(a,b,c)$ which ensures that
$$ax^2+bxy+cy^2=\pm 1$$
has a solution in integers $x,y$?
In addition to direct answers I would also ...
3
votes
2answers
84 views
A standard quadratic minimization problem
Consider the "Complex" Quadratic minimization problem
\begin{align}
\min_{\mathbb{x}\in \mathbb{C}^{N \times 1}}~\mathbf{{x}}^H\mathbf{Q}\mathbf{x}-2~\Re{(\mathbf{x}^H\mathbf{b})}+1
\end{align}
...
3
votes
1answer
120 views
Why does positive semi-definiteness in this inequality imply a convex set?
I was reading a proof that rewrote an inequality in the form:
$$b^Tx +x^T A x \le \alpha$$
for $b,x \in \mathbb{R}^n$ and $\alpha \in \mathbb{R}$, and with $A$ positive semidefinite. It then ...
3
votes
1answer
180 views
Question about $4$-manifolds and intersection forms
This is a question related to an earlier question of mine:
I've been reading about topological invariants. Some of them are defined in terms of quadratic forms.
My current understanding is: we can ...
3
votes
1answer
276 views
When are two diagonal matrices congruent?
This is probably a question that does not admit a simple answer. However, I'd like to know whether there exist criteria that determine when two diagonal matrices are congruent. I have the suspicion ...
3
votes
1answer
71 views
Intersection of Sphere and Line in $\mathbb{R}^n$?
This seems to me as a very simple and basic question, though I'm having trouble with it.
The Problem
Given a sphere $K\in\mathbb{R}^n$ with radius $r\in\mathbb{R}$ and center ...
3
votes
1answer
106 views
Center of SO(V,q)
Let $V$ be finite dimensional vector spaces and $q$ is quadratic form. I'm looking for $Z(SO(V,q))$. where $SO(V,q)$ is special orthogonal group.
If $\operatorname{dim} V$ is odd then ...
