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

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0answers
110 views

Quadratic transformations of vector spaces

Much is known about transformations of the following form $$y_i = L_{ij}x_j \;\;: \;\; x\in\mathcal{R}^n, L\in\mathcal{R}^{n\times n}$$ We can infer a number of geometric properties about the ...
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1answer
102 views

Quadratic spaces and definition of hyperbolic plane

I'm trying to figure out a proof in Lam's book on quadratic forms (he uses this to define a hyperbolic plane). He states that if $(V,q)$ is a 2-dimensional quadratic space over $F$, then the following ...
8
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2answers
488 views

Proving an inequality with Cauchy-Schwarz

In the "User's guide to viscosity solutions" by Crandall, Ishii and Lions (link), they make the following claim (inequality (A.4) p. 58) : Given $x$, $\xi$ $\in \mathbb{R}^n$, $A \in \cal{S}(n)$ ...
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vote
1answer
103 views

Extension to Witt's bases

Let $X$ be a real finite dimensional linear space. Let $B:X\times X \rightarrow \mathbf{R}$ be a bilinear symmetric non-degenerated form. Let $M$, $N$ be totally isotropic subspaces with the same ...
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0answers
162 views

Solutions to equation in matrix form

Suppose $x=[x_1, x_2, \cdots, x_n]^t$, $b=[b_1,b_2,\cdots,b_n]^t$ with $b_i\in K$ and $A\in M_n(K)$, where $K$ is a field. There are well known criteria for the system of equations $Ax=b$, by ...
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4answers
2k views

Determining if a quadratic is always positive

Is there a quick and systematic method to find out if a quadratic equation is always positive or may have positive and negative or always negative for all values to its variables? Say for a quadratic ...
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0answers
236 views

Surface Function Fitting to Spherical Data

I have a set of geographic (longitude,latitude,value) data to which I would like to fit surface functions, specifically, the set of quadratic surfaces: $f(x,y)=Ax^2+Bx^2+Cxy+Dx+Ey+F$ At the moment, ...
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vote
1answer
359 views

Maximal subspace that a quadratic form is non-negative on

Let $q: \mathbb{R^3}\to \mathbb{R},\ q(x_1, x_2, x_3)=-5x_1^2-x_2^2-x_3^2+2x_1x_3+2x_2x_3-4x_1x_2$ be a quadratic form. Find a maximal subspace $W \subseteq \mathbb{R^3}$ such that $\forall w ...
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votes
1answer
88 views

A quadratic space over an algebraically closed field is isotropic

Let $F$ be an algebraically closed field, and let $(V,f)$ be a quadratic space over $F$. How can one show that if $\dim V \geq 2$ then it is an isotropic space? Thanks.
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3answers
260 views

Confused about quadratic forms

My book states: if $q: \mathbb{R}^n \to \mathbb{R}$ is a positive definite quadratic form then there exists a basis $B=(v_1,...,v_n)$ such that $q(x)=x_1^2+...+x_n^2=||x||^2$ for every ...
3
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1answer
73 views

Intersection of Sphere and Line in $\mathbb{R}^n$?

This seems to me as a very simple and basic question, though I'm having trouble with it. The Problem Given a sphere $K\in\mathbb{R}^n$ with radius $r\in\mathbb{R}$ and center ...
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vote
1answer
193 views

Completion of the squares (actually of the quadratic forms)

I have this paragraph in the book I cannot understand (this is not the first time I encounter this thing but I usually move on). $$A^\prime TA+A^\prime QB+B^\prime UA+B^\prime RB = ...
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vote
2answers
261 views

creating the smoothest curve

I have this iPhone app that has an array containing around 50 to 100 points. How do I calculate the smoothest curve that will fit the points? It can be bezier, cubic, quadratic, whatever. It just have ...
3
votes
4answers
229 views

Quadratic vertex form - negative $h$ means it's on the right side of the graph?

When putting a quadratic equation in vertex form, I am having difficulty understanding why $h$ is negative when the location of the parabola goes to the right... why is this? For example, if I ...
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votes
2answers
188 views

A question about integral quadratic forms

Hi Would you please advise me? Consider the equation below: $$ ax^2+bxy+cy^2=n $$ in which $a, b, c$ and $n$ are integers. We then suppose that $a, b, c$ are constant. Is there any way to find the ...
5
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1answer
233 views

Universal quadratic forms

A quadratic form is a polynomial $p(x_1,\dots,x_n)$ of the form $$ p(x_1,\dots,x_n)=\sum_{i \leq j}a_{ij}x_ix_j. $$ For example, $p_1(x,y,z,w)=x^2+y^2+z^2+w^2$ and $3x^2-5y^2$ are quadratic forms. I'm ...
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1answer
156 views

construction of the Witt group

I've seen a couple of constructions of the so-called Witt group: it seems that most authors start with the commutative monoid of isometry classes of quadratic spaces under direct sum, pass to the ...
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1answer
735 views

Sufficient Conditions for a Bounded Feasible Region in the Linear Programming Problem

I am working on a problem where it would be nice to prove that the feasible region of a LP problem is bounded, but where it is not necessary to solve any particular problem. In particular, given an ...
5
votes
3answers
245 views

Factoring numbers “of the (binary quadratic) form” in two different ways

For some fixed $n$ define the quadratic form $$Q(x,y) = x^2 + n y^2.$$ I think that if $Q$ represents $m$ in two different ways then $m$ is composite. I can prove this for $n$ prime. I was hoping ...
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2answers
179 views

How does the theory of the quadratic number fields relate to the quadratic forms?

As every one knows, the quadratic number fields shares a deep connection with the binary quadratic forms; I have been told this relation when I was a senior high, and now I, learning some difficult ...
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1answer
187 views

About multiplying binary quadratic forms

The quadratic forms with discriminant -23 up to change of variables are: A(x,y): $x^2 + xy + 6 y^2$ B(x,y): $2 x^2 - xy + 3 y^2$ C(x,y): $2 x^2 + xy + 3 y^2$ Viewed as number fields it's ...
4
votes
1answer
104 views

Partitioning polynomials in $\mathbb{Z}[x,y]$ by the primes they represent

Suppose you have a set $S\subset\mathbb{Z}[x,y].$ How can one efficiently partition the polynomials into sets such that the primes represented by the polynomials in any given set are identical? For ...
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0answers
1k views

Proof of Legendre's theorem on the ternary quadratic form

Theorem (Legendre): Let $a,b,c$ coprime positive integers, then $ax^2 + by^2 = cz^2$ has a nontrivial solution in rationals $x,y,z$ iff ...
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3answers
658 views

Existence of solutions to diophantine quadratic form

Is there a general result about the existence of (non-trivial) solutions of the diophantine equation: $$Ax^2 + By^2 = Cz^2$$ for A,B,C known positive integers, pair-wise relatively prime? What if ...
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3answers
513 views

positive symmetric matrices and positive-definiteness

Is a symmetric real matrix with diagonal entries strictly greater than 1 and off-diagonal entries positive but strictly less than 1 necessarily positive-semidefinite?
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1answer
213 views

Finding the minimum value of a quadratic within a range

Given any quadratic equation of the form $y=ax^2+bx+c$, I want to find the minimum value for a specific range of $x$. My programmer brain can do it in a branchy, algorithmic way as follows, but is ...
25
votes
2answers
816 views

Elementary proof that $2x^2+xy+3y^2$ represents infinitely many primes?

We did in class $x^2+y^2$, which was easy, and we had for homework $2x^2+2xy+3y^2$, which I did (its values (if not square) must be divisible by form primes, or of the form $x^2+5y^2$, and clearly ...
4
votes
1answer
351 views

Is a general (non-homogeneous) quadratic equation in $\mathbb{R}^3$ an ellipsoid?

This sounds like a simple problem, but I can't get it done. Given the general equation $ax^2 + by^2 + cz^2 + dxy + exz + fyz + gx + hy + iz + j = 0,$ what are the requirements on the coefficients so ...
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3answers
781 views

How to solve inhomogeneous quadratic forms in integers?

If I have a quadratic form like $y^2 - x^2 - x = k$ none of the techniques I know work because of the nasty $x$. Note that homogenizing doesn't work because a solution of $Y^2 - X^2 - X Z = k Z^{(2)}$ ...