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A line in three-dimensional space may be described as an intesection of two planes, for example: $$\begin{align}x+y+z=0\tag{1}\\3x+7y=1\tag{2}\end{align}$$ This can be understood as two separate scalar equations or as a single matrix equation. (One may also describe a line parametrically.)

This poses a question: is it possible to express the same line using a single scalar equation? It turns out that it is. The equation $$x^2+y^2=0\tag{3}$$ can be understood as describing the set of all points $(x,y,z)\in\Bbb R^3$ for which $x=0$ and $y=0$. In other words, it describes the $z$-axis.

So, we have described a line in $\Bbb R^3$ using a single scalar equation. But this means any line in $\Bbb R^3$ (or $\Bbb R^n$ for that matter) can be described by a single scalar equation, simply by using an appropriate affine transformation on the equation $(3)$.

As pointed out by Pantelis Damianou in the comment below, this gives us the equation $$(x+y+z)^2+(3x+7y-1)^2=0$$ in the case described above. Note that this tells us exactly the same thing as the equations $(1)$ and $(2)$, since $z^2+w^2=0$ is just another way to say that $z=0$ and $w=0$.

My question is:

Is this point of view ever useful? Does it have any striking applications? Is there an area of mathematics that uses such equations in a fruitful way?

Thanks.

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So, the first example is $(x+y+z)^2+(3x+7y-1)^2=0$. –  PAD Jul 11 '12 at 21:34
    
@PantelisDamianou: indeed! I'll add that. Thanks. –  Dejan Govc Jul 11 '12 at 21:36
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up vote 2 down vote accepted

More generally, we can "pack up" any finite system of equations (over the reals) as a single equation involving a sum of squares. If you want an approximate solution of the system of equations, you can try to minimize that sum of squares: this is the starting point for Least Squares Approximation, which is very useful in data analysis.

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A set of $D$ of natural numbers is Diophantine if there is a polynomial $P(y,x_1,\dots,x_n)$ such that for all all natural numbers $y$, $$y\in D \quad \text{iff} \quad \exists x_1 \cdots\exists x_n(P(y,x_1,\dots,x_n)=0).$$ Here by tradition the $x_i$ range over the natural numbers, which are defined to include $0$.

Your observation shows that the intersection of finitely many Diophantine sets is Diophantine. This is a useful albeit minor step in showing that every recursively enumerable set is Diophantine, which solves Hilbert's Tenth Problem.

The same idea can be used to show that in fact there is no algorithm for determining whether a quartic Diophantine equation (in many variables) has a solution. The idea, roughly, is to reduce an arbitrary Diophantine equation to a system of quadratic equations, and then using the sum of squares trick to obtain a single quartic equation.

The sum of squares trick has a similar minor but useful role in some proofs of Gödel's Incompleteness Theorem, when we are showing that every recursive predicate is representable in a fairly weak version of formal number theory.

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This is indeed a striking application. I've heard of Hilbert's Tenth Problem before, but I think this just made me realize what it's really telling us: that a whole lot of logic can be expressed using tricks with polynomials. Very inspiring, thanks. –  Dejan Govc Jul 11 '12 at 22:11
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