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.
26 questions from the last 365 days
1
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0
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70
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Quadratic forms with the same roots over GF(2) for low rank problems
Let $Q_1(x)=x^TA_1x$ and $Q_2(x)=x^TA_2x$ with $x\in GF(2)^n$, $A_i\in GF(2)^{n\times n}, i \in \{1, 2\}$. If $rank(A_1)=rank(A_2)=2$, is it possible that $Q_1(x)$ and $Q_2(x)$ can have the same roots ...
- 63
5
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1
answer
432
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Number of roots of a quadratic form over GF(2)
If $Q(x) = x^T A x$ with $x \in GF(2)^n$ and $A \in GF(2)^{n \times n}$, is there a way to find how many roots $Q(x)$ has based on any properties of $A$ (e.g., rank, etc.)?
- 63
0
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0
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50
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Self-adjoint operators and index of quadratic form associated to it
Let $B$ a bounded self-adjoint operator on a real Hilbert space $H$ with an associated inner product $(\cdot,\cdot).$ Take $V=\operatorname{span}\{f_1, f_2, \ldots, f_n\}$ a finite dimensional ...
- 101
1
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0
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70
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Notion of length in projective space over function field
Given a projective space $\mathbb{P}^n(\mathbb{R})$ and two points $x, y \in \mathbb{P}^n(\mathbb{R})$, the distance between $x$ and $y$ is defined as
$$
d(x, y) = \frac{\|v_x \wedge v_y\|}{\|v_x\| \|...
- 87
3
votes
1
answer
255
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Reflections on affine quadric hypersurfaces
Let $f\colon\mathbb{Z}^n\otimes\mathbb{Z}^n\to\mathbb{Z}$ be a non-degenerate symmetric bilinear form and consider the affine quadric hypersurface $$
X:=\{f(x,x)+2=0\}\subseteq\mathbb{Z}^n.
$$
For ...
- 341
4
votes
1
answer
236
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If a lattice can be embedded into $\mathbb Q^n,\langle-1\rangle^n$, then can it be embedded into $\mathbb Z^n,\langle -1 \rangle^n$?
Given a graph with negative integers on each vertex $\Gamma$ there is a corresponding intersection lattice denoted $Q_\Gamma$, a free $\mathbb Z$ module generated by the vertices, endowed with a ...
- 43
11
votes
1
answer
598
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How to prove this problem about ternary quadratic form?
Is this right? And how to prove it ?
For $n \equiv 1,2 \bmod 4$
$$ \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
1
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0
answers
52
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Genus of binary quadratic forms: $f(x,y), g(x,y)$ in same genus if and only if represent same values in $(\mathbb Z/m\mathbb Z)^\ast$ for all $m$
In David Cox's book: Primes of the form $x^2+ny^2$, second edition, there is a theorem(Theorem 3.21, page 52) characterize whether two binary quadratic forms in the same genus. The contents of the ...
- 287
0
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0
answers
58
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Diophantine approximation of real quadratic forms by forms defined over a real number field
I am interested in a statement of the following form.
Let $K$ be a real number field and consider real numbers $(\xi^-, \xi^+)$.
Assume that there are infinitely many pairs of real algebraic numbers $(...
1
vote
0
answers
40
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Quadratic equations over division rings of dimension 2
Let $\ell$ be a division ring, and let $k$ be a sub division ring.
I know that a quadratic equation $x^2 + ax + b = 0$, with $a, b \in k$ can have more than two solutions in $\ell$, but what if the ...
- 4,547
1
vote
1
answer
133
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Graceful labeling of the complete bipartite graph and its laplacian quadratic form diagonalized
A graceful labeling of a connected simple undirected graph $G=(V,E)$ is a map $f:V\to\lbrace 1,...,|E|+1\rbrace$ such that for all $t\in\lbrace 1,...,|E|\rbrace$ there is a (trivially unique) $\langle ...
- 91
1
vote
1
answer
153
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How to solve for bounds restricting ${\Sigma}$ to symmetric-positive-semi-definiteness?
Scenario
I have a equation for a covariance matrix ${\Sigma}$ where everything but a vector of correlations is known aka $x=(x_{1}, \dots, x_{D})$ for $x_{i}\in [-1, 1]$.
Problem
I know that ${x}$ ...
5
votes
1
answer
259
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Reduced form of Bhargava cubes
Let $A \in \mathbb{Z}^{2} \otimes \mathbb{Z}^{2} \otimes \mathbb{Z}^{2}$ be a Bhargava cube. We have a natural action of $\mathrm{SL}_{2}(\mathbb{Z})^{3}$ on the space of Bhargava cubes, and I wonder ...
- 2,215
0
votes
0
answers
93
views
Orthogonalization of symmetric non-degenerate bilinear forms
It is well-known that given a field $k$ with characteristic different from $2$, every symmetric non-degenerate bilinear form $B$ over a finite-dimensional space can be orthogonalized. This means that ...
- 1,485
1
vote
0
answers
62
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Isotropic vectors of quadratic forms over number fields
By Meyer's theorem, an indefinite quadratic form $Q$ over $Z$ has an integral isotropic vector is the dimension is at least $5$ and this bound is tight. Indeed, in dimension $4$ there are indefinite ...
6
votes
2
answers
424
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About the solutions of $ \dfrac{x^p - y^p}{x - y} = a^2+pb^2 $
I already posted this question on MSE.
Using theorem $IV$ from this article, it is possible to prove that when $p$ is a prime such that $p ≡ 3\bmod4$, $x ≢ y\bmod{p}$ and $\gcd(x,y) = 1$, then the ...
- 387
1
vote
0
answers
85
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Squares in skew fields of dimension 2 over a sub skew field
Let $\ell$ be a skew field (i.e., a division ring), and let $k$ be a sub skew field, such that the dimension of $\ell$ as a left vector space over $k$ is $2$.
Then if $a \in \ell \setminus k$, we can ...
- 4,547
2
votes
0
answers
156
views
Sets represented by integral quadratic forms
Let $f(x) = x^\intercal A x$ be a positive definite integral quadratic form on $d$ variables. A positive integer $n$ is said to be represented by $f$ if $f(x) = n$ for some $x \in \mathbb Z^d$. A set ...
- 974
-4
votes
1
answer
197
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How to express a quadratic polynomial exactly as a power series [closed]
I claim, for $\operatorname{artanh}(\rho) = \frac{1}{2} \ln\left(\frac{1+\rho}{1-\rho}\right)$, i.e., the inverse hyperbolic tangent function, the following holds approximately under assumptions given ...
- 183
0
votes
1
answer
127
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Automorphism groups in class sets of ternary lattices
Let $\Lambda$ be an integral lattice in some definite ternary quadratic space $(V,Q)$ over $\mathbb{Q}$.
Consider the usual class set $\text{Cl}(\Lambda) = O(V)\backslash\text{Gen}(\Lambda)$, i.e. the ...
- 562
1
vote
1
answer
67
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Low rank matrices which preserve maximizers of quadratic form
Suppose $x, y \in \{0,1\}^d$ are binary vectors. For a matrix $M$ consider the quadratic form,
\begin{align}
x^T M y + (\mathbb{1} - x)^T M (\mathbb{1} - y)
\end{align}
Does there exist an $M$ such ...
- 297
6
votes
0
answers
380
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A possible variant of Zagier's one-sentence proof for Fermat's sum of two squares theorem?
Is it possible to modify Zagier's one-sentence proof of Fermat's sum of two squares theorem (see here) to prove certain non-trivial cases of Jacobi's four-square theorem (see here)?
Let $p$ be a prime ...
- 81
0
votes
0
answers
92
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Convergence in probability of quadratic form with positive mean
Let $\boldsymbol{X}_n\in\mathbb{R}^n$ be a sequence of Gaussian random vectors with independent entries, such that $X_{n,i}\sim \mathcal{N}(\mu_i,\sigma^2)$ (that is, all entries of the $n$th vector ...
- 503
5
votes
1
answer
117
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An isomorphic classification of non-associative division octonion algebras
A division octonion algebra over a field $F$ is a $8$-dimensional unital non-associative algebra $A$ over the field $F$, endowed with a quadratic form $N:A\times A\to F$ such that $N(xy)=N(x)N(y)$ and ...
- 41.8k
0
votes
0
answers
111
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Totally isotropic space for bilinear pairing over ring
A duplicate of this:
Consider the following well-known inequality: Let $b$
be a non-degenerate symmetric bilinear pairing over a (finite-dimensional) $\mathbb F$-vector space $V$ and $W$
a totally ...
- 223
2
votes
1
answer
196
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Estimate for the operator $A A_D^{-1}$
Let $O\subset\mathbb{R}^d$
be a bounded domain of the class $C^{1,1}$
(or $C^2$
for simplicity). Let the operator $A_D$
be formally given by the differential expression $A=-\operatorname{div}g(x)\...
