# 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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### 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 ...
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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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### Find the transitional matrix that would transform this form to a diagonal form.

Let the quadratic form $F(x,y,z)$ be given as below $F(x,y,z)=2x^2+3y^2+5z^2-xy-xz-yz$ Find the transitional matrix that would transform this form to a diagonal form. I got the symmetric ...
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### Finding the matrix of a quadratic form

I want to find the matrix of quadratic form $Q= \sum^p_{i=1} (y_i - \bar y)^2$. Please help me finding it. For example I have found the quadratic form matrix for $Q= p\bar y^2$ as follows: ...
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### Sinha's Theorem for Equal Sums of Like Powers $x_1^7+x_2^7+x_3^7+\dots$

Sinha’s theorem can be stated as, excluding the trivial case $c = 0$, if, $$(a+3c)^k + (b+3c)^k + (a+b-2c)^k = (c+d)^k + (c+e)^k + (-2c+d+e)^k\tag{1}$$ for $\color{blue}{\text{both}}$ $k = 2,4$ ...
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### Compute the dimension of the space of quadratic forms

We were asked the following: "Compute the dimension of the space of quadratic forms on $V=\mathbb{R^2}.$ Compute also the dimension of the space of symetric forms on $\mathbb{R^2}$, ...
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### How can I find values for which a given expression gives a perfect square?

There have been several posts on this topic on math.se, such as this one with the same title. However all the posts I found contained coefficients to $x^2$, that were perfect squares. I am looking for ...
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### Quartic polynomial in ten variables

I have a quartic form, i.e. a homogeneous 4-th degree polynomial, in ten real variables and the inequality: $f(x_1,\ldots, x_{10}) \geq c$, for some $c>0$, which I believe that geometrically ...
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### $q_I$ primitive as a quadratic form?

Let $I \subset \mathcal{O}_k$ be an ideal, $N(I) = [\mathcal{O}_K : I] = |\mathcal{O}_K/I|$. Define $q_I$ be $q_I(x) = N_{K/\mathbb{Q}}(x)/N(I)$. Is $q_I$ primitive as a quadratic form?
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### Optimizing the sum of powers of positive quadratic functions

In my research I have come across the following optimization problem. \begin{array}{c} maximize \hspace{1cm} \sum \limits_{n=1}^{N}\left(\mathbf{x}^{T}A_n \mathbf{x}\right)^{k}\\ ...
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### How to find max and min of quadratic form matrix

By spectral decomposition, I figured out that the eigenvalues of the sigma matrix (1+p, 1-p, 1) and corresponding eigenvectors (1/sqrt(2), 1/sqrt(2),0) , (1/sqrt(2), -1/sqrt(2),0) , ...
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### Matrix of quadratic form (in Serre's general notion)?

I am currently reading Serre's book on arithmetic. In chapter four (page 27) he defines a general notion of the quadratic form as: Let $V$ be a module of a commutative ring $A$. A function $Q$ is ...
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### Solution of quadratic diophantine equations

Is there any algorithm so that solution to the following equation can be found? $(x+a)^2-y^2=c$ where $c$ and $a$ is a constant. It is similar to Pells eqution with a variation where $D=1$. I am new ...
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### Vieta Jumping and Hurwitz 1907

Today I proved finiteness for the problem here: Is it true that $f(x,y)=\dfrac{x^2+y^2}{xy-t}$ has only finitely many distinct integer values with $x,y$ positive integers? namely: IF we have ...
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### Finding integers of the form $3x^2 + xy - 5y^2$ where $x$ and $y$ are integers, using diagram via arithmetic progression

So the diagram drawn looks like this: We begin at the edges labeled $3$ and $-5$ because we are using those as the bases for $x$ and $y$, respectively. The way we obtain the values of the 2 ...
For each $T$, let $A_T$ be a $T\times T$ matrix of real numbers. let $e_T$ be the $T\times 1$ vector of ones. Assume that the sum of all entries of the matrix $A_T$ divided by $T^2$ is limited as $T$ ...