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The main, broader question is whether a solution set of a system of equations can ever be such that it can't be described by a single equation.

To be more precise, suppose I have a system of $m$ equations in $n$ variables

$f_1(x_1,x_2,...,x_n)=0, f_2(x_1,x_2,...,x_n)=0, ... f_m(x_1,x_2,...,x_n)=0$

(The $f_i$s are assumed to be polynomials for the purpose of this post).

If one restricts the variables and the values of the $f$s to the real number the answer to the above question is no.

The equation $f_1(x_1,x_2,...,x_n)^2+f_2(x_1,x_2,...,x_n)^2+...+f_m(x_1,x_2,...,x_n)^2=0$ has the same solution set as the system of equations.

How about if one restricts the values to complex numbers? Is there any known context where systems of equations are more expressible than single equations in the above sense?

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Yes, such a context does exist. If we allow complex number solutions (or more generally, solutions from any algebraically closed field), then there is a genuine difference between systems of equations and single equations. Here's a precise statement in this direction:

Theorem: Let $\{f_i\}$ be a set of $m$ homogeneous complex polynomials in the variables $x_0,\ldots,x_n$. Then the common zero locus of the $\{f_i\}$ in complex projective space $\mathbf{P}^n$ is an algebraic subset of dimension at least $n-m$.

(We have to allow points in projective space, and hence restrict to homogeneous polynomials, to avoid obvious "cheats", for instance a set of two inconsistent equations.)

In particular if $m=1$, we see that the zero locus of a single homogeneous polynomial $f$ has dimension at least $n-1$ (and in fact equal to $n-1$ as long as $f$ is nonzero). So any algebraic subset of dimension less than $n-1$ cannot be the zero locus of a single homogeneous polynomial in those variables.

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Yes indeed, there is such a context, within Applied Mathematics. It's called the Least Squares Finite Element Method (L.S.FEM) . An article about this numerical method, applied to Ideal Flow in 2-D, is Labrujere's Problem. But if you search for Least Squares Finite Element Methods on the internet, they will tell you a quite different story than this.

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