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2
votes
0answers
52 views

Statements generalizing representability of integers by binary quadratic forms to $n$-variable higher homogeneous forms?

Representing integers through the theory of binary quadratic forms is a well studied topic. We know that given $a,b,c\in\Bbb N$, based on discrimant $b^2-4ac$, we can study the representability of ...
10
votes
2answers
314 views
+300

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 ...
5
votes
1answer
115 views

Isotropic subvarieties of $ V(x_1^2+\dots+x_n^2-t_1^2-\dots-t_n^2) $

Let $ K $ be a field, $ \operatorname{char} K = 0, $ let $ Q = x_1^2+\dots+x_n^2-t_1^2-\dots-t_n^2 $ be the totally isotropic form, then the maximal linear isotropic subspaces of $ V(Q) \subset ...
1
vote
1answer
55 views

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 ...
0
votes
0answers
44 views

counting fundamental units of real quadratic fields

For a given real quadratic field $ K$, the units of its ring of integers is $\mathcal{O}_K\cong(\pm1)\times \mathbb{Z}$ by Dirichlet unit theorem. For each $\mathcal{O}_K$, pick the fundamental unit ...
5
votes
4answers
167 views

From a (not positive definite) Gram matrix to a (Kac-Moody) Cartan matrix

Suppose I am given a symmetric matrix $G_{ij}$ with $G_{ii} = 2$. Can I always find an invertible integer matrix $S$ such that $(S^T G S)_{ii}=2$ and $(S^T G S)_{ij} \leq 0$ for $i \neq j$? Is there a ...
3
votes
6answers
376 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, ...
2
votes
1answer
80 views

Real slices of Minkowski space, using a complex quadratic form

Ordinary Minkowski space is $\mathbb{R}^{3,1}:=(\mathbb{R}^4,\phi)$ where $\phi:\mathbb{R}^4\rightarrow\mathbb{R}$ is a quadratic form of signature $(3,1)$. Lying within this is a hyperboloid model ...
4
votes
1answer
180 views

Numbers represented by inhomogeneous forms

I have a family of Diophantine equations that I am trying to solve, and I am trying to figure out what methods could be used to prove existence of solutions. Unfortunately, the equations are ...
4
votes
3answers
370 views

genus 2 Siegel theta series of 3-dimensional lattices

Let $(V,f)$ be a $3$-dimensional positive definite quadratic space over $\mathbf Q$. Let $G(V)$ be a set of representatives of the isometry classes of maximal integral lattices on $V$. To an ...
3
votes
2answers
168 views

Paper of Denis Simon on quadratic equations in dimensions 4, 5?

In several places I have come across references to a 2005-6 preprint of Denis Simon entitled Quadratic equations in dimensions 4, 5, and more This paper gives fast algorithms to find isotropic ...
4
votes
2answers
173 views

Orbits of the maximal compact subgroup on the light cone for $p$-adic groups

It is known that if $Q$ is an indefinite non-degenerate quadratic form on $ \mathbb{R}^n$ with $n \ge 3$, then any maximal compact subgroup $K$ of the orthogonal group $SO(Q)$ acts transitively on the ...
8
votes
3answers
323 views

Deciding a quadratic diophantine equation

Given $a,b\in\Bbb Q_+$, is there an easy way to decide if $$S_{a,b}=\{(x,y)\in\Bbb Z^2:ax^2 + by^2=1\}=\emptyset?$$ I am more interested in seeing if there is a quick way to test for case when ...
4
votes
1answer
215 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 ...
0
votes
0answers
64 views

About bounding values of quadratic forms

It would be helpful if someone can share (either as references) examples of calculations/analysis which achieves bounding of values of quadratic forms in say either of the following situations, ...
0
votes
1answer
99 views

How to minimize this sparse quadratic function?

There is a problem when I'm reading a paper. Equation: $min_p|p-p^*|^2+\alpha |R(p)|^2 + \beta |D(p)-\delta|^2$, where $p, p^*, R(p), D(p), \delta$ are all $M\times N$ matrices, and $p^*, R(), D(), ...
2
votes
1answer
111 views

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 ...
8
votes
2answers
763 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 ...
1
vote
0answers
68 views

Bound of spectral radius of polynomial of a complex matrix

I am trying to prove or disprove the following inequality. $$ ||P(A)||_2\leq 2 \max_{\alpha\in W(A)}| P(\alpha)|,$$ where $P(\cdot)$ is a complex polynomial, $A\in \mathbb{C}^{n\times n}$ and ...
4
votes
0answers
116 views

Two quadratic programming problems always same answer? [closed]

Was exploring quadratic programming optimization and for two types of problems the answers seemed to always equal. Problem 1: Minimize $\tfrac{1}{2} \mathbf{x}^T Q\mathbf{x}$ Subject to $ A ...
0
votes
1answer
63 views

Lower bounds on the rank of a unimodular lattice, given the binlinear pairing of a subset of basis vectors

I have some questions on lower bounds on the rank of unimodular lattices given the bilinear pairing of a subset of its basis is known. Let $\Lambda$ be an odd, unimodular matrix of signature ...
5
votes
0answers
148 views

Is there any study about positive definiteness of some matrix space whose matrices don't have to be positive-definite?

--Updated description-- I'm trying to investigate the stability of tensegrity structures, and this question is related to the second order test. Suppose there is a vector space ...
1
vote
0answers
136 views

Minimal Length of Quadratic Forms

Let $Q(x_1,\dots,x_n)=X'PX$ be a quadratic form with all non-negative and integral coefficients given by $$Q(x_1,\dots,x_n)=\sum_{i=1}^cf_i^+(x_1,\dots,x_n)g_i^+(x_1,\dots,x_n)$$ where $c$ is the ...
8
votes
1answer
317 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) ...
1
vote
1answer
123 views

Cohomological dimension of transcendental p-adic extensions

Let $q$ denote a quadratic form over a field $k$. The u-invariant of a field $u(k)$ is defined by $u(k):=\{ max (\mathrm{rank}(q)) $ | $ q $ is anisotropic over $k\}$. Let $k = \mathbb{Q}_p$ for any ...
2
votes
2answers
127 views

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 ...
1
vote
0answers
77 views

Hlawka inequality for Lorentz quadratic form

Let $K$ be a convex cone in ${\mathbb R}^n$. A continuous function $f:K\rightarrow\mathbb R$ satisfies a Hlawka inequality if $$f(0)+f(x+y)+f(y+z)+f(z+x)\le f(x)+f(y)+f(z)+f(x+y+z),\qquad\forall ...
6
votes
1answer
183 views

Constructing sums of squares identities

Recall that a sum of squares formula for $[r,s,n]$ over a field $F$ is an identity of the form $$ ( x_{1}^{2} + \cdots + x_{r}^{2})( y_{1}^{2} + \cdots + y_{s}^{2}) = ( z_{1}^{2} + \cdots + ...
0
votes
0answers
49 views

Binary motives in the decomposition of a minimal Pfister neighbor

Let $\alpha \in H^n(k,\mu_2)$ and $X_\alpha$ be the respective Pfister quadric. Its well known due to Rost that the Motive $M(X_\alpha)$ decomposes as a sum of twisted Rost motives $R_\alpha$ such ...
2
votes
1answer
116 views

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 ...
0
votes
6answers
647 views

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 ...
6
votes
1answer
272 views

Given a positive-definite integral unimodular Gram matrix, how to find a basis of the associated lattice (over $\mathbf Q$)?

Let $G$ be a $n\times n-$symmetric matrix with integral coefficients and determinant $1$ (i.e. unimodular) such that the associated quadratic form is positive-definite. I am interested in having an ...
5
votes
1answer
381 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 ...
3
votes
1answer
106 views

Simple comparison of positive ternary quadratic form representation counts

Something came up yesterday in a referee request and I was surprised to find that I did not know the facts in full generality. This is about positive quadratic forms in three variables with integer ...
0
votes
1answer
91 views

Action of $(\mathbb{Z}/2g\mathbb{Z})$ on quadratic forms on $\mathbb{Z}/2\mathbb{Z}$-vector space

Let $\mathbb{Z}/2\mathbb{Z}$ the 2 elements field, with additive notation. I need some clarifications on the relation between quadratic forms on a $\mathbb{Z}/2\mathbb{Z}$-vector space (say, of ...
6
votes
0answers
136 views

Upper bound on the number of ismorphism classes of bilinear forms on $\mathbb{Z}^n$

$\DeclareMathOperator{\Hom}{Hom}$A symmetric, positive definite bilinear form on $\mathbb{Z}^n$ is any mapping $$b : \mathbb{Z}^n \times \mathbb{Z}^n \to \mathbb{Z}$$ satisfying $b$ is bilinear, ...
1
vote
1answer
105 views

Dimension of binary motives of a quadric

Let $Q$ be a anisotropic quadric of dimension $d$ over $k$. We work in the category of effective Chow-Motives over $k$. Let $T$ be the Tate-Motive. For a motive $M$ we write $M(l)$ for its $l$-th ...
2
votes
2answers
138 views

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 my previous question that $Z$ has ...
4
votes
1answer
209 views

orthogonal group in characteristic 2

Let $O(2,\mathbb{Z}_2)$ be the orthogonal group of order two matrices. On $\mathbb{Z}_2$ there should exist just one odd quadratic form, hence the stabilizer subgroup $O^-$ of an odd quadratic for ...
4
votes
1answer
406 views

Are constant connection coefficients uniquely determined by the (1,3) curvature coefficients?

Suppose that on a certain coordinate system the coefficients $\Gamma^i_{jk}$, $i,j,k=1,\cdots, n$, of a linear connection are constant. We do not require compatibility with a metric, however I am ...
1
vote
1answer
130 views

$p$-adic analogues of $SO(3)$

I read in the paper " From Laplace to Langlands via representations of orthogonal groups" by Benedict Gross and Mark Reeder that there are, up to isomorphism, two orthogonal groups of the ...
4
votes
1answer
194 views

Automorphisms of SO_n(k,f)

Let $k$ be a field, $n\in\mathbb{N}$ and $f:k^n\times k^n\to k$ a non-degenerate symmetric bilinear form. Let $$O_n(k,f):=\{ g\in GL_n(k) \mid \forall x,y\in k^n : f(x,y)=f(g.x,g.y) \}$$ and ...
7
votes
2answers
129 views

space of reduced positive definite quadratic forms

What is the highest dimension for which the space of reduced positive definite quadratic forms (or the fundamental domain of $SL_n(\mathbb{R})/SL_n(\mathbb{Z})$) has been explicitly calculated? I know ...
1
vote
0answers
90 views

What is classified by $H^1(\mathbb{R},SO(p,q))$ and by $H^1(\mathbb{R},SU(p,q))$?

We denote by $F^{\mathbb{R}}_{p,q}$ the quadratic form over the field ${\mathbb{R}}$ $$ F^{\mathbb{R}}_{p,q}(x)=x_1^2+\dots+x_p^2-(x_{p+1}^2+\dots+x_{p+q}^2) $$ on the vector space ...
6
votes
1answer
327 views

Easiest way to distinguish $E_8 \oplus E_8$ from $E_{16}$

For $n$ an integer divisible by $8$, let me denote by $E_n$ the "usual" even non-degenerate positive definite integral symmetric bilinear form over $\mathbf Z^n$. It is well known that in dimension ...
4
votes
3answers
553 views

Class number for binary quadratic forms discriminant $\Delta$ to class number $\mathbb Q(\sqrt \Delta)$

Jyrki Lahtonen has suggested I write a blog post relating binary quadratic forms to quadratic field class numbers, ...
1
vote
1answer
199 views

The Chebotarev Density Theorem and the representation of infinitely many numbers by forms

Let $ax^{2}+bxy+cy^{2}$ be a primitive positive definite quadratic form of discriminant $\Delta<0$. It is well known that $ax^{2}+bxy+cy^{2}$ represents infinitely many prime numbers. One of the ...
5
votes
1answer
191 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 ...
8
votes
2answers
560 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, ...
5
votes
1answer
184 views

Rank four quadratic Form with non trivial discriminant in I(k)

Im sure this is a beginners question. Let $k$ be a field and $I(k)$ the fundamental ideal in the Witt-ring W(k). The Arason-Pfister-Hauptsatz states: "If $\varphi$ is any anisotropic class in ...