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) \...
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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 ...
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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. ...
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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 $\...
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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, ...
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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 ...
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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)...
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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
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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}(...
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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 ...
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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:...
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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 ...
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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 ...
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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 \...
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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 $\...
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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 ...
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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'...
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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 ...
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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 $R \setminus \{0\}$. ...
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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) ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $\tilde q$ ...
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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 ...
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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 ...
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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 ...
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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 (...
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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, ...
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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 $b(X,Y)=\ell(X)\cdot\ell(Y)$ ...
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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:
$...
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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 ...
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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 ...
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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) \...
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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 ...
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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
$$Q=
\begin{pmatrix}
1 &1/2& & & & \\
1/2& 1 & & & & \\
& & 1 &1/2& & \\
&...
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Fricke Klein method for isotropic ternary quadratic forms
Preface: the most natural way to take one isotropic vector for an indefinite quadratic form and find others is to use stereographic projection. This gives a parametrization in the same $n$ variables ...
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When is the set of numbers represented by certain quaternary quadratic forms completely multiplicative?
Expired by this question
A quadratic form represents all primes except for the primes 2 and 11.
I would like to know some simple sufficient conditions for when the set of numbers integrally ...
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Mass of spinor genus, positive integral quadratic forms
There seems to be general opinion that, for positive integral quadratic forms in at least three variables, spinor genera in the same genus all have the same mass (not representation measures of some ...
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A cubic equation, and integers of the form $a^2+32b^2$
I am trying to determine whether there are any integers $x,y,z$ such that
$$
1+2 x+x^2 y+4 y^2+2 z^2 = 0. \quad\quad\quad (1)
$$
It is clear that $x$ is odd. We can consider this equation as quadratic ...
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Set of quadratic forms that represents all primes
A SPECIFIC CASE:
Any prime number can be classified as either $p \equiv 1 \pmod 3$ or $p \equiv 2 \pmod 3$.
If $p = 3$ or $p = 1 \pmod 3$, then the prime $p$ can be represented by the quadratic form $ ...
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votes
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Computing a Commutator Subgroup
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}$I’m studying the group $\O(5,5,\mathbb{Z})$, the indefinite orthogonal matrices with integer entries. In particular, I ...
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On certain solutions of a quadratic form equation
This is a continuation of this question: A class of quadratic equations
Let $f(x,y) = ax^2 + bxy + cy^2$ be an irreducible and indefinite binary quadratic form. Consider the equation
$$\displaystyle ...
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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 ...
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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-...
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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 ...
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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,\...
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