Questions tagged [quadratic-forms]
Quadratic forms are homogeneous quadratic (degree two) polynomials in $n$ variables. In the cases of one, two, and three variables they are called unary, binary, and ternary. For example $\quad Q(x)=2x^2\quad $ is called unary quadratic ploynomial, $\quad Q(x,y)= 2x^2+3xy+2y^2\quad$ is called binary quadratic polynomial and $\quad Q(x,y,z)=2x^2+3y^2+z^2+7xy+5yz+9xz\quad$ is called ternary quadratic polynomial.
2,552 questions
1 vote
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Automorphisms of integral quadratic forms
For $i=1,2$, let $B_i\in\mathrm{SL}_n(\mathbb{Z})$ be two symmetric positive definite matrices. We define their automorphism groups as $$\mathrm{Aut}(B_i)=\{g\in\mathrm{GL}_n(\mathbb{Z})\mid\ gB_ig^{\...
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0 votes
2 answers
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Wanted: Simple criterion for isotropic quadratic forms
I am studying binary integral quadratic forms, i.e. polynomials of the form $ax^2+bxy+cy^2$ with fixed coefficients $a,b,c$ and $a,b,c,x,y$ being integers. As I am interested in the automorphisms of ...
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0 votes
1 answer
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Rational points on $ax^2+by^2=c$
Let $a,b,c$ be integers, not all zero. I am looking for an "iff" condition for $ax^2+by^2=c$ to have a rational point. First note that this is equivalently asking for integer solutions $x,y,...
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5 votes
2 answers
263 views
If the matrix $A$ is positive definite and $T$ is its symmetric part, show that $x^T A^{-1} x \leq x^T T^{-1}x$
In Golub & Van Loan's Matrix Computations, I came across the following problem and I am stumped (been at it for a few days now). A matrix $M\in \mathbb{R}^{n\times n}$ (not necessarily symmetric) ...
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2 votes
1 answer
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Surjectivity of $x^2 + xy + y^2$ over finite fields $\mathbb{Z} / p \mathbb{Z}$ with $p > 3$ [closed]
Let $p$ be a prime with $p > 3$, and consider the bivariate polynomial map $g \colon \mathbb{F}_p^2 \to \mathbb{F}_p$ given by $g( x , y ) = x^2 + xy + y^2$. Here $\mathbb{F}_p = \mathbb{Z} / p \...
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6 votes
8 answers
314 views
Rotating/tilting conics
There is a formula for the inclination/tilt $\theta$ of a conic given by a level curve of the $Ax^2+Bxy+Cy^2$, namely: $\cot 2\theta=\frac{A-C}{B}$. This can be derived by a straightforward ...
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1 vote
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Every primitive form is equivalent to a reduced form?
Here are three results quoted from Cox's Primes of the Form $x^2+ny^2$. I do not understand how proposition 2.15 follows from the previous two results. I apologize in advance for my lack of knowledge ...
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0 votes
1 answer
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Sphere average of the second symmetric sum of a matrix
Say given $B$ symmetric and tr $B=0$. Let $P_u$ be the projection onto the tangent plane attached to $u$, i.e., $ P_u=Id-u\otimes u$. Let $\sigma_2(M)$ be the second symmetric sum of eigenvalues of $...
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