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.
585 questions
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Spectral gap for a second-variation operator on a jet Hilbert space
Let $J_0$ be a (separable) complex Hilbert space with scalar product $\langle \cdot,\cdot\rangle_{J_0}$ and norm $\|\cdot\|_{J_0}$.
Suppose we are given a real, symmetric, non-negative bilinear form
$$...
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Weakening of the Idoneal Number condition
This is about the following Math.SE Q&A: "What is the largest integer $d$ such there is a congruence relation on primes p so that $x^2 + dy^2 = p^2$ has a non-zero integral solution?".
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Can $w^2+bx^2+cy^2+dz^2$ be universal over a sparse subset of $\mathbb N$?
Let $b,c,d\in\mathbb N=\{0,1,2,\ldots\}$ with $1\le b\le c\le d$. For a subset $S$ of $\mathbb N=\{0,1,2,\ldots\}$, if
$$\{w^2+bx^2+cy^2+dz^2:\ w,x,y,z\in S\}=\mathbb N$$
then we say that $w^2+bx^2+cy^...
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Approximations to pairs of algebraic numbers by quadratic elements over a number field
Question / conjecture
Let $K$ be a real number field and consider a pair of real numbers $(x, x')$.
Assume that there are real numbers $\epsilon > 0$, $C > 0$
and infinitely many pairs of real ...
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Conics and Hermitian curves over $\mathbb{F}_{q^2}$
Let $\mathcal{H} \subset \mathbb{P}^2(\mathbb{F}_{q^2})$ be the Hermitian curve, defined (up to projective equivalence) by
$$
X^{q+1} + Y^{q+1} + Z^{q+1} = 0.
$$
Its automorphism group is $\mathrm{PGU}...
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Find integer coefficients of polynomials from approximate irrational roots [duplicate]
Let take quadratic equations
$$x^2+ax+b=0$$
assume here $a,b$ both are integer and the roots of the equation are irrational if I give you one root in irrational form then is there any method to find $...
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The quadratic forms corresponding to canonical (Neron-Tate) height of an elliptic curve
Surely this is well-known in the arithmetic geometry community, which I (unfortunately) do not regard as a member of, so I will include some basic exposition for the laymen (including myself).
We deal ...
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Is there an elementary proof for this identity involving a simplex, quadratic forms and determinants?
In a recent project we found a curious identity for simplices (Theorem 5.6).
Let $\Delta\subset\Bbb R^d$ be a $d$-simplex with facets $F_0,...,F_d$, $v_i\in\Bbb R^d$ the vertex opposite to $F_i$, $u_i\...
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$\left\{\frac{x(ax+b)}2+\frac{y(ay-b)}2:\ x,y=0,1,2,\ldots\right\}$ and asymptotic bases of order 2
A subset $A$ of $\mathbb N=\{0,1,2,\ldots\}$ is called an asymptotic base of order $h$ if any sufficiently large $n\in\mathbb N$ belongs to the set
$$hA=\{a_1+\ldots+a_h:\ a_1,\ldots,a_h\in A\}.$$
...
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3
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Minimum L2 norm of polynomial and the Hilbert matrix
Hilbert introduced his famous matrix when he studied the following problem. How small can the integral
$$\int_{a}^b|p(x)|^2dx $$
become for a non-zero polynomial $p$ with integer coefficients? He ...
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Orthogonalization of quadratic forms over a $p$-adic Banach space
Let $X$ be an arbitrary set. Let $H = c_0(X, \mathbb{Q}_p)$ be the $p$-adic Banach space with sup norm. Let $\langle \cdot, \cdot \rangle$ be a symmetric, nondegenerate $\mathbb{Q}_p$-bilinear form on ...
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Existence of rank 3 lattice of signature (1,2) containing two copies of $U$ intersecting in a positive vector
Let $U = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ denote the standard hyperbolic plane. I am trying to construct, for a given natural number $N$, an explicit example of a rank 3 lattice $...
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Reference request: analogue of Cramér's conjecture for integers represented by binary quadratic form
The Prime Number Theorem asserts that $\pi(N) \sim N/\log N$ as $N \to \infty$ where $\pi(N)$ is the prime counting function. Colloquially, the (average) density of primes $\le N$ is like $1/\log N$. ...
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Prime Inheritance and Prime-Generating Subsequence Trees in Class Number 1 Quadratic Polynomials [closed]
This question is inspired by the classical behavior of Euler’s polynomial
$$
\mathbf{f(x) = x^2 - x + 41},
$$
which is well-known for producing prime values for integer inputs $x = 0$ to $39$, and is ...
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Prime inheritance in class number 1 quadratic polynomials
This conjecture is based on computational exploration of quadratic polynomials associated with imaginary quadratic fields of class number one.
Let us define:
• For a polynomial $f(x) \in \mathbb{Z}[...
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