# Tagged Questions

Questions about quadratic forms in multiple variables, for example $4x_1^2 + 3x_1x_2 + 5x_1x_3 - 8x_3^2$.

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### A two-variable quadratic form over a field of characteristic 2 with no nontrivial roots

I'm looking for a quadratic form of the form $q(x,y)=ax^2 + bxy + cy^2 \in F[x,y]$, where $F$ has characteristic 2, and $q(x,y)$ has no roots besides the obvious one, $x=y=0$. I've proved the case of ...
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### A nontrivial solution to the quadratic form $x^2 - xy + y^2$ over the finite field $𝔽_p$ with $p ≡ 1 \pmod3$ a prime

I'm trying to prove that when $p ≡ 1 \pmod3$ is a prime, $p$ is reducible over the Eisenstein integers, and I've gotten to the point where, provided $p\,|\,u^2 - u + 1$ for some integer $u$, then $p$ ...
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### Is the squared euclidean norm a measure for the distance of two points?

I like to prove that a measure for the distance $d$ of two points $\vec a$ and $\vec b$ in $R^N$ is given by the squared euclidean norm $$d^2= \sum^N_j (a_j - b_j)^2$$ So far I was able to show ...
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### Vertex Form of Parabola - Why does it work?

Recently, I have been trying to plot parabolas of quadratic equations. First, I have to convert them to vertex form and then we can easily plot them. This makes me wonder why the vertex form of a ...
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### Subtle error with a module endormorphism on $\mathbb{Z}_8 \times \mathbb{Z}_8$

Let $a,b,c$ be arbitrary integers such that $a$ is odd and $(a,b,c)=1$. Let $R = \mathbb{Z}_8$, the set of all integer residues modulo $8$. Define an $R$-module endomorphism $\phi \colon R \times R$ ...
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### Expectation of a special form of quadratic form

Let $\mathbf x$ be a $n\times1$ random variable, $\mathbf s$ be a vector of size $3\times 1$ and $A$, $M_1$, $M_2$ and $M_3$ be $n\times n$ matrices. What is the following expectation with respect to ...
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### A quadratic form is positive-definite iff its set of isotropic vectors is trivial

Considering a quadratic form $Q$ in a finite dimensional vector space $V$ can I say that $\mathscr{I}=\big\{ \vec{o} \big\} \iff Q$ is definite positive ? Where $\mathscr{I}$ is the isotropic ...
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### Is the quadric $3$-fold $v^2 + w^2 + x^2 + y^2 + z^2 = 0$ isomorphic to $P^3$?

The subset of projective $4$-space given by $5$-tuples $[v:w:x:y:z]$ with $v^2 + w^2 + x^2 + y^2 + z^2 = 0$ is birational to projective $3$-space. I think it has the same cohomology as projective $3$-...
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### Classify the surface $x^2 + y^2 - z^2 + 2xy - 2xz - 2yz - y = 0$

I am working on a problem in which I must classify the surface described by the following equation $$x^2 + y^2 - z^2 + 2xy - 2xz - 2yz - y = 0.$$ I have looked at this Stack Exchange discussion (on ...
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### Given $p \equiv q \equiv 1 \pmod 4$, $\left(\frac{p}{q}\right) = 1$, is $N(\eta) = 1$ possible?

Given distinct primes $p$ and $q$, both congruent to $1 \pmod 4$, such that $$\left(\frac{p}{q}\right) = 1$$ and obviously also $$\left(\frac{q}{p}\right) = 1$$ is it possible for the fundamental unit ...
### Help solving $ax^2+by^2+cz^2+dxy+exz+fzy=0$ where $(x_0,y_0,z_0)$ is a known integral solution
Help solving over the integers: $$ax^2+by^2+cz^2+dxy+exz+fzy=0$$ where $(x_0,y_0,z_0)$ is a known integral solution and $a,b,c,d,e,f$ are integral coefficients. I found in Tito Piezas' identities the ...
An integer quadratic form is a function $Q(x,y) = ax^2 + bxy + cy^2$ where the numbers $a,b,c \in \mathbb Z$. Call the set of values a quadratic forms takes on \$V(Q) = \{ Q(x,y) \in \mathbb Z | x,y \...