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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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) \...
Keivan Karai's user avatar
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16 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 ...
Adam Sheffer's user avatar
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8 votes
3 answers
1k views

Integral orthogonal group for indefinite ternary quadratic form

I have the indefinite quadratic form $q(x,y,z) = 19 x^2 + 5 y^2 - z^2.$ It's not my fault. I find, on reflection, that I have no idea how to describe the orthogonal group of this over the integers. ...
Will Jagy's user avatar
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7 votes
1 answer
1k views

When do two non-degenerate quadratic forms give rise to isomorphic Lie algebras?

Let $V$ be a vector space over some number field $k$. (I'm fine with $\mathbb{Q}$.) Let $\phi \colon V \to k$ be a non-degenerate quadratic form. Associated with $\phi$ is the orthogonal group $\...
jmc's user avatar
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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, ...
Will Jagy's user avatar
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3 votes
2 answers
461 views

Indefinite quadratic form universal over negative integers

Here's a question that (I hope) may seem very trivial for you, and I hope one of you may provide me with a reference answering it (unless it's a trivial colloquial knowledge). Let $f$ be an ...
SashaKolpakov's user avatar
3 votes
6 answers
1k views

Isotropic ternary forms

It is well known that some questions about isotropic ternary forms reduces to the study of the special case $f_0(X)=xz-y^2, X=(x,y,z)$, see page 301 of Cassel's "Rational quadratic forms" (Dover, 2008)...
user148455's user avatar
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+...
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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}(...
Jeremy Rouse's user avatar
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20 votes
3 answers
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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 ...
Nik Weaver's user avatar
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14 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:...
darij grinberg's user avatar
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 ...
Will Jagy's user avatar
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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 ...
M.B's user avatar
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8 votes
1 answer
2k views

A spectral inequality for positive-definite matrices

Question. Given a positive-definite $n \times n$ matrix $A = (a_{ij})$ with eigenvalues $$ \lambda_1 \leq \cdots \leq \lambda_n , $$ is there a sharp upper bound for the product $\lambda_2 \cdots \...
alvarezpaiva's user avatar
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7 votes
1 answer
265 views

Is a 8-dimensional quadratic form recognized by its Lie algebra, modulo equivalence and scalar multiplication?

Question. Let $K$ be a field of characteristic zero (large characteristic should be fine too). Let $q,q'$ be two non-degenerate quadratic forms on $K^n$ with $n=8$. Suppose that the Lie algebras $\...
YCor's user avatar
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5 votes
1 answer
616 views

Ternary quadratic form theta series as Hecke eigenforms and class number one

At Simple comparison of positive ternary quadratic form representation counts Jeremy answered: "The reason is that the theta series for the sums of three squares form is an eigenfunction for all the ...
Will Jagy's user avatar
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3 votes
1 answer
570 views

Automorphism groups of indefinite non-unimodular integer lattices

Does anyone know of any papers in which structural aspects of the orthogonal group of some indefinite non-unimodular integral lattice are calculated? The exact lattice isn't so important and they don'...
user36896's user avatar
24 votes
2 answers
847 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 ...
Philip Boyle Smith's user avatar
22 votes
3 answers
1k 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 $R \setminus \{0\}$. ...
Pete L. Clark's user avatar
22 votes
1 answer
2k 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) ...
21 votes
1 answer
12k 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 ...
Qfwfq's user avatar
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21 votes
3 answers
1k 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 ...
Kevin O'Bryant's user avatar
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 ...
Will Jagy's user avatar
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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 ...
Franz Lemmermeyer's user avatar
16 votes
3 answers
4k 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 ...
ReverseFlowControl's user avatar
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 ...
Nick Salter's user avatar
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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 $\tilde q$ ...
Anton Petrunin's user avatar
15 votes
1 answer
3k 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 ...
Will Jagy's user avatar
  • 25.2k
14 votes
3 answers
969 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 ...
David E Speyer's user avatar
14 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 ...
Misha Verbitsky's user avatar
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 (...
Will Jagy's user avatar
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12 votes
0 answers
758 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, ...
Will Jagy's user avatar
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12 votes
1 answer
854 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 $b(X,Y)=\ell(X)\cdot\ell(Y)$ ...
Anton Petrunin's user avatar
11 votes
1 answer
735 views

Positive ternary quadratic forms in the same genus that represent the same numbers

There are three genera of positive, integral, ternary quadratic forms in which both forms (classes...) are regular, so the paired forms represent the same numbers. These pairs (complete genera) are: $...
Will Jagy's user avatar
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10 votes
1 answer
2k views

Sums of three non-zero squares

It is a well-known result of Legendre that a positive integer is sum of three squares unless it is of the form $4^a(8b+7)$. In Grosswald, E.; Calloway, A.; Calloway, J. The representation of ...
Andrés E. Caicedo's user avatar
9 votes
2 answers
2k views

Does a positive binary quadratic form represent a set of primes possessing a natural density

In his answer to my question The Green-Tao theorem and positive binary quadratic forms Kevin Ventullo answers my initial question in the affirmative. What remains is the title ...
Will Jagy's user avatar
  • 25.2k
8 votes
1 answer
566 views

minimizing an integral over integer-coefficient polynomials $\displaystyle \inf_{f \in \mathbb{Z}[x]} \int_a^b f(x)^2 \, dx $ [duplicate]

Let's consider the space $L^2[a,b]$ of functions on the interval and the norm: $$ ||f(x)||^2 = \int_a^b |f(x)|^2 \, dx $$ Now what if we consider only polynomials with integer coefficients: $f(x) \...
john mangual's user avatar
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8 votes
2 answers
504 views

What's in the genus of the cubic lattice?

I'll write $\mathbf{Z}^n$ for the integral quadratic form $x_1^2 + \cdots + x_n^2$. For which values of $n$ is $\mathbf{Z}^n$ unique in its genus, i.e. isolated in Kneser's graph? In particular can ...
David Treumann's user avatar
7 votes
3 answers
708 views

How many solutions are there to $\sum_{i=1}^3 x_i^2+x_iy_i+y_i^2=k$?

Let $k$ be a positive integer. Let $$Q= \begin{pmatrix} 1 &1/2& & & & \\ 1/2& 1 & & & & \\ & & 1 &1/2& & \\ &...
emiliocba's user avatar
  • 2,279
7 votes
3 answers
1k views

Fricke Klein method for isotropic ternary quadratic forms

Preface: the most natural way to take one isotropic vector for an indefinite quadratic form and find others is to use stereographic projection. This gives a parametrization in the same $n$ variables ...
Will Jagy's user avatar
  • 25.2k
7 votes
0 answers
448 views

When is the set of numbers represented by certain quaternary quadratic forms completely multiplicative?

Expired by this question A quadratic form represents all primes except for the primes 2 and 11. I would like to know some simple sufficient conditions for when the set of numbers integrally ...
Will Jagy's user avatar
  • 25.2k
7 votes
2 answers
739 views

Mass of spinor genus, positive integral quadratic forms

There seems to be general opinion that, for positive integral quadratic forms in at least three variables, spinor genera in the same genus all have the same mass (not representation measures of some ...
Will Jagy's user avatar
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6 votes
3 answers
661 views

A cubic equation, and integers of the form $a^2+32b^2$

I am trying to determine whether there are any integers $x,y,z$ such that $$ 1+2 x+x^2 y+4 y^2+2 z^2 = 0. \quad\quad\quad (1) $$ It is clear that $x$ is odd. We can consider this equation as quadratic ...
Bogdan Grechuk's user avatar
6 votes
1 answer
1k views

Set of quadratic forms that represents all primes

A SPECIFIC CASE: Any prime number can be classified as either $p \equiv 1 \pmod 3$ or $p \equiv 2 \pmod 3$. If $p = 3$ or $p = 1 \pmod 3$, then the prime $p$ can be represented by the quadratic form $ ...
Consider Non-Trivial Cases's user avatar
6 votes
1 answer
464 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
  • 397
6 votes
2 answers
435 views

On certain solutions of a quadratic form equation

This is a continuation of this question: A class of quadratic equations Let $f(x,y) = ax^2 + bxy + cy^2$ be an irreducible and indefinite binary quadratic form. Consider the equation $$\displaystyle ...
Stanley Yao Xiao's user avatar
5 votes
2 answers
354 views

What is $\mathbb{E} [\max_{\sigma \in \{ \pm 1\}^n} \sigma^T Z \sigma]$ for a random Gaussian matrix $Z$?

Given an $n \times n$ random matrix $\mathbf{Z}$ with each entry i.i.d. $\mathcal{N} (0,1)$, what is $\mathbb{E} [\max_{\sigma \in \{ \pm 1\}^n} \sigma^T Z \sigma]$ as $n \to \infty$? If this is too ...
Television's user avatar
5 votes
1 answer
427 views

Constructing groups of Type E7 with certain Tits Index

In a new survey on $E_8$, namely Skip Garibaldi - E8 the most exceptional group , the author gives an example (Example 8.4., page 15) on how to construct a group of type E8 with a prescribed Tits-...
nxir's user avatar
  • 1,397
5 votes
1 answer
244 views

Positive Primes represented by an indefinite binary form, reducing poly degree from 8 to 4

In his lovely answer at Positive primes represented by indefinite binary quadratic form Noam found that a (positive) odd prime $p$ is represented by the indefinite form $x^2 + 13 x y - 9 y^2$ if and ...
Will Jagy's user avatar
  • 25.2k
5 votes
2 answers
224 views

Bounded version of linear and quadratic Hasse--Minkowski theorem

The Hasse-Minkowski theorem states that if $$Q(x_1,\ldots,x_n) = \sum_{i,j=1}^n a_{ij} x_ix_j$$ is a quadratic form with $a_{ij} \in \mathbb Z$ and $\det (a_{ij}) \neq 0$, then the equation $$Q(x_1,\...
Turbo's user avatar
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