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
119
votes
8
answers
35k
views
Zagier's one-sentence proof of a theorem of Fermat
Zagier has a very short proof (MR1041893, JSTOR) for the fact that every prime number $p$ of the form $4k+1$ is the sum of two squares.
The proof defines an involution of the set $S= \lbrace (x,y,z) \...
- 6,224
62
votes
3
answers
5k
views
Can a positive binary quadratic form represent 14 consecutive numbers?
NEW CONJECTURE: There is no general upper bound.
Wadim Zudilin suggested that I make this a separate question. This follows
representability of consecutive integers by a binary quadratic form
where ...
- 25.7k
27
votes
3
answers
1k
views
When does $axy+byz+czx$ represent all integers?
For which $a,b,c$ does $axy+byz+czx$ represent all integers?
In a recent answer, I conjectured that this holds whenever $\gcd(a,b,c)=1$, and I hope someone will know. I also conjectured that $axy+byz+...
user44143
27
votes
4
answers
2k
views
Which quaternary quadratic form represents $n$ the greatest number of times?
Let $Q$ be a four-variable positive-definite quadratic form with integer coefficients and let $r_{Q}(n)$ be the number of representations of $n$ by $Q$. The theory of modular forms explains how $r_{Q}(...
- 20.4k
24
votes
1
answer
694
views
two's and three's survive in gcd of Lagrange
Lagrange's four square_theorem states that every positive integer $N$ can be written as a sum of four squares of integers. At present, let's focus only on positive integer summands; that is, $N=a_1^2+...
- 43.2k
24
votes
2
answers
889
views
Simple conjecture about rational orthogonal matrices and lattices
The following conjecture grew out of thinking about topological phases of matter. Despite being very elementary to state, it has evaded proof both by me and by everyone I've asked so far. The ...
23
votes
1
answer
2k
views
Wanted: Quadratic Space in Characteristic 2 as a Counterexample to a Theorem of Arf
Hi. Peter Roquette sent me an email asking for an example of a quadratic space in characteristic 2 having certain features. I have no idea on this, but maybe someone reading this does.
He would ...
- 50.6k
23
votes
1
answer
3k
views
A list of proofs of the Hasse–Minkowski theorem
I am currently doing a project in which I intend to include the most insightful possible proof of the Hasse–Minkowski theorem (also known as the Hasse principle for quadratic forms, among other names) ...
22
votes
1
answer
13k
views
Non-diagonalizable complex symmetric matrix
This is a question in elementary linear algebra, though I hope it's not so trivial to be closed.
Real symmetric matrices, complex hermitian matrices, unitary matrices, and complex matrices with ...
- 23.3k
22
votes
3
answers
2k
views
Must a ring which admits a Euclidean quadratic form be Euclidean?
The question is in the title, but employs some private terminology, so I had better explain.
Let $R$ be an integral domain with fraction field $K$, and write $R^{\bullet}$ for $R \setminus \{0\}$. ...
- 65.4k
21
votes
3
answers
2k
views
Why are there usually an even number of representations as a sum of 11 squares
Question: Why do so few $n\equiv 3 \bmod 8$ have an odd number of representations in the form $$n=x_0^2 + x_1^2 + \dots + x_{10}^2$$ with $x_i \geq 0$?
Note that $x_i\geq 0$ spoils the symmetry ...
- 9,756
21
votes
1
answer
2k
views
Rationality of intersection of quadrics
Let $X \subset \mathbb{P}^n$ be a complete intersection of two quadrics. It is classical that, if $X$ contains a line, then it is rational. The proof is very simple and basically it is given by taking ...
- 3,779
20
votes
3
answers
2k
views
Primes of the form $x^2+ny^2+mz^2$ and congruences.
This is a sequel of this question where I asked for which positive integer $n$ the
set of primes of the former $x^2+ny^2$ was defined by congruences (a set of primes $P$ is defined by congruences if ...
- 26k
20
votes
3
answers
1k
views
Simultaneous "orthonormalization" in $\mathbb{C}^4$
Let $A$ be a positive, invertible $4 \times 4$ hermitian complex matrix.
So we have a positive sesquilinear form $\langle Av,w\rangle$. Say that a pair $(v,w)$ of vectors in $\mathbb{C}^4$ is good ...
- 42.8k
19
votes
0
answers
1k
views
Does this variant of a theorem of Hasse (really due to Gauss) have an "elementary" proof?
BACKGROUND
Here are 3 theorems of varying difficulty. Let $M$ be the $Z/2$ subspace of $Z/2[[x]]$ spanned by $f^k$, with the $k>0$ and odd, and $f=x+x^9+x^{25}+x^{49}+\cdots$. For $g$ in $M$, let $...
- 5,422
18
votes
2
answers
3k
views
Many representations as a sum of three squares
Let $r_3(n) = \left|\{(a,b,c)\in {\mathbb Z}^3 :\, a^2+b^2+c^2=n \}\right|$. I am looking for the maximum asymptotic size of $r_3(n)$. That is, the maximum number of representations that a number can ...
- 1,072
18
votes
3
answers
5k
views
What is known about primes of the form $x^2-2y^2$?
David Cox's book Primes of The Form: $x^2+ny^2$ does a great job proving and motivating a lot of results for $n>0$. I was unable to find anything for negative numbers, let alone the case I am ...
18
votes
3
answers
2k
views
A Priori proof that Covering Radius strictly less than $\sqrt 2$ implies class number one
It turns out that each of Pete L. Clark's "euclidean" quadratic forms, as long as it has coefficients in the rational integers $\mathbb Z$ and is positive, is in a genus containing only one ...
- 25.7k
18
votes
2
answers
2k
views
History of "no positive definite ternary integral quadratic form is universal"?
I am currently leading a graduate student research group on geometry of numbers (henceforth GoN) and its Diophantine applications, especially to quadratic forms. In an early lecture I gave a bit of a ...
- 65.4k
17
votes
2
answers
2k
views
Quaternary quadratic forms and Elliptic curves via Langlands?
The content of this note was the topic of a lecture by Günter Harder at the School on Automorphic Forms, Trieste 2000. The actual problem comes from the article
A little bit of number theory by ...
- 32.6k
17
votes
2
answers
4k
views
On Siegel mass formula
I have asked this question exactly here. The question is as follows:
I am interested deeply in the following problem:
Let $f$ be a (fixed) positive definite quadratic form; and let $n$ be an ...
16
votes
0
answers
588
views
The number 1680 and Lagrange's four-square theorem
The number $1680$ has the factorization $2^4\times3\times5\times7$. Rather to my surprise, I found that this number has certain mysterious connection with Lagrange's four-square theorem.
QUESTION: ...
- 15.6k
15
votes
3
answers
10k
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, ...
- 253
15
votes
2
answers
1k
views
Positive quadratic polynomial
Let $S$ be solutions of a system of quadratic polynomials on $\mathbb{R}^n$.
Suppose $q$ is another quadratic polynomial such that $q|_S\geqslant 0$.
Is it possible to find a polynomial $\tilde q$ ...
- 45k
15
votes
2
answers
2k
views
Primes and $x^2+2y^2+4z^2$
A few months ago, I have asked a question about primes represented by ternary quadratic forms. I got two wonderful answers, which showed me how the theory was way richer and more complex that I ...
- 26k
15
votes
1
answer
4k
views
The Green-Tao theorem and positive binary quadratic forms
Some time ago I asked a question on consecutive numbers represented integrally by an integral positive binary quadratic form. It has occurred to me that, instead, the Green-Tao theorem may include a ...
- 25.7k
15
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 ...
- 9,237
15
votes
2
answers
2k
views
Clifford PBW theorem for quadratic form
$\DeclareMathOperator\Cl{Cl}$Update Feb 3 '12: now with a question 2 which is much more elementary (and should be well-known!).
Let $k$ be a commutative ring with $1$. Let $L$ be a $k$-module, and $g:...
- 33.8k
15
votes
2
answers
1k
views
Is there an approach to understanding solution counts to quadratic forms that doesn't involve modular forms?
Given a quadratic form Q in k variables, there is an associated theta series $$\theta_Q(z) = \sum_{x\in \mathbb{Z}^k}q^{Q(x)}$$ where $q = e^{2\pi i z}$ which is a modular form of weight $k/2$. Thus ...
- 2,830
15
votes
1
answer
1k
views
Quadratic forms and $p$-adic integers
I want to prove a result on equivalences of quadratic forms over $\mathbb{Q}_p$, with a control on the height of the change-of-basis matrix.
(I am more generally interested in hermitian forms over ...
- 1,500
14
votes
1
answer
2k
views
Do almost all systems of quadratic equations have solutions?
If I have a system of linear equations, $A x = c$, with $A$ an $n\times n$ complex matrix, it is relatively easy to see that the set of matrices $A$ for which there is no (complex) solution has ...
- 342
14
votes
3
answers
985
views
Achieving consecutive integers as norms from a quadratic field
This question is inspired by my inability to make any progress on Will Jagy's question.
Giving a positive answer to this question should be strictly easier than proving Jagy's conjectures.
Suppose ...
- 156k
14
votes
0
answers
476
views
Cohomological interpretations of quadratic form invariants over rings?
The standard approach to classifying of quadratic forms over $\Bbb Q$ is to use the Hasse (local-global) principle together with a system of standard invariants of quadratic forms over the local ...
- 141
13
votes
2
answers
1k
views
Upper bound on answer for Pell equation
A user on MSE, @martin , asked https://math.stackexchange.com/questions/1611411/pell-equations-upper-bound about an upper bound for $x$ in $x^2 - p y^2 = 1,$ when $p$ is prime. I checked, it appears ...
- 25.7k
13
votes
1
answer
990
views
Is -1 a sum of 2 squares in a certain field K?
Consider the field of fractions $K$
of the quotient algebra $\mathbb{R}[x,y,z,t]/(x^2+y^2+z^2+t^2+1)$,
where $\mathbb{R}$ is the field of real numbers and $x,y,z,t$ are variables.
Clearly $-1$ is a ...
- 14.1k
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 (...
- 25.7k
12
votes
2
answers
741
views
A criterion for real-rooted polynomials with nonnegative coefficients
Let $P \in \mathbb{R}[X]$, with $\deg P = n$. Is it true that
$P$ has only real roots $\quad \Longleftrightarrow \quad P\cdot P'' + (\frac{1}{n}-1)P'^2 \leq 0$ ?
The direct implication can be shown ...
- 155
12
votes
1
answer
902
views
Positive 4-form
Denote by $W$ the space of all symmetric bilinear forms on $\mathbb{R}^n$.
Let $Q$ be a quadratic form on $W$.
Suppose that $Q(b)\geqslant 0$ for any $b\in W$ such that $b(X,Y)=\ell(X)\cdot\ell(Y)$ ...
- 45k
12
votes
1
answer
499
views
A diophantine equation in $\mathbb{N}$
While I was working on a paper on graph theory, I encountered a problem which I think is a number-theory-problem. I don't know if there are any tools to answer the question.
Find all natural numbers $...
- 351
12
votes
2
answers
3k
views
On the positive definiteness of a linear combination of matrices
In my work in PDE, the following problem in linear algebra came up. Any help in this direction is appreciated.
QUESTION:
Let $m,n\in\mathbb{N}$ and let $A_1,\ldots, A_m\in M_n(\mathbb{R})$ be real, ...
- 895
12
votes
1
answer
2k
views
Orthogonal group of quadratic form
Orthogonal group of the quadratic form over fields, somehow, is well-studied. Indeed
E. Cartan has proved for quadratic forms over the reals or complexes that any
orthogonal transformation is a ...
- 2,508
12
votes
2
answers
499
views
"Pythagoras number" for integral matrices
It is classically known that every positive integer is a sum of at most four squares of integers, i.e. every sum of squares of integers is a sum of four squares of integers. Now consider a symmetric $...
- 3,031
12
votes
1
answer
529
views
Quadrics in the Grothendieck ring
Let $\mathcal{Q}$ be an irreducible quadric in $\mathbb{P}^n(k)$, with $n \geq 2$ and $k$ a finite field. Let $K_0(V_k)$ be the Grothendieck ring of $k$-varieties. It is well known (it appears) that ...
- 4,547
12
votes
3
answers
790
views
2-dimensional sublattices with all vectors having very big square (in absolute value)
QUESTION: Let $\Lambda\times\Lambda\rightarrow {\Bbb Z}$ be a lattice,
that is, ${\Bbb Z}^n$ with a non-degenerate integer quadratic form, not
definite, not necessarily unimodular, $n>2$. I want ...
- 9,237
12
votes
2
answers
2k
views
Integral positive definite quadratic forms and graphs
Let me start with a question for which I know the answer. Consider a symmetric integral $n\times n$ matrix $A=(a_{ij})$ such that $a_{ii}=2$, and for $i\ne j$ one has $a_{ij}=0$ or $-1$. One can ...
- 13.1k
12
votes
1
answer
775
views
Would a closed universe with special relativity violate causality? Does the universe have to be simply connected?
This question may be more appropriate for physics.stackexchange.com, but it would be helpful to get feedback from experts in Minkowski geometry.
The classic twin paradox is a false thought experiment ...
- 3,359
12
votes
0
answers
767
views
What numbers are integrally represented by $4 x^2 + 2 x y + 7 y^2 - z^3$
This is related to my first MO question and Kevin Buzzard's conjecture at
Integers not represented by $ 2 x^2 + x y + 3 y^2 + z^3 - z $
In December 2010 my question appeared in the M.A.A. Monthly, ...
- 25.7k
11
votes
2
answers
1k
views
Intersection of two quadrics that have a common inscribed sphere
This is related to a question I asked here on math.stackexchange. It didn't receive an answer there (except for my answer), and my question here is a generalization of that one, anyway.
Suppose I ...
- 649
11
votes
2
answers
1k
views
Positive primes represented by indefinite binary quadratic form
Neil Sloane asked me about commands in computer languages to find the (positive) primes represented by indefinite binary quadratic forms. So I wrote something in C++ that works. This is for the OEIS, ...
- 25.7k
11
votes
1
answer
598
views
How to prove this problem about ternary quadratic form?
Is this right? And how to prove it ?
For $n \equiv 1,2 \bmod 4$
$$ \Bigg|\ \mathbb Z^3\cap\Big\{(a_1,a_2,a_3)\ \Big|\
a_1^2+a_2^2+a_3^2=n \Big\}\Bigg| \\
= \frac12\Bigg|\mathbb Z^3\cap\Big\{(a_1,...
- 113