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If we look at the equation $xy=0$, the roots of that equation can be described as two linear equations: $x=0$ and $y=0$.

It seems to me that there must be other equations where the roots can be described as higher order curves, and possibly even surfaces, and I observe that normally, roots are described as 0d values.

Are there formalized techniques or a field for analyzing curves and (hyper)surfaces such that you solve for roots where the roots are described as curves and (hyper)surfaces?

Alan Wolfe
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    I'm certainly not in the know, but I think you're venturing into the territory of algebraic geometry. – pjs36 Apr 25 '17 at 03:38
  • It's quite unclear what you mean. The equation $xy=0$ is equivalent to $x=0$ and $y=0$, but the equation $z=xy$ isn't. More generally, whenever you have an equation with something factorized equal to zero, then it's equivalent to any factor being zero, for example $f(x,y)g(x,y)=0$ if and only if $f(x,y)=0$ or $g(x,y)=0$. – Hans Lundmark Apr 25 '17 at 07:13
  • You are right. I'll fix. Z should never have made an appearance. – Alan Wolfe Apr 25 '17 at 08:06

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