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Just a quick terminology question.

The set of solutions to a linear system of equations with nonunique solutions is known as the "nullspace".

What is the equivalent terminology (if there is one) for the nonunique solutions to a nonlinear system of equations? (or its equivalent Groebner basis)

For example: $$ \begin{align} &x_{1} + x_{2}^{2} + 3x_{3}= 0\\ &x_{3} = 2 \end{align} $$

Is there a name this set of solutions? The solutions of the above system is: $$ \begin{align} &x_{1} = -6 -t^{2}\\ &x_{2} = t\\ &x_{3} = 2 \end{align} $$

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Maybe the locus? If the linearised system always have the same nullity, then you may even talk about the solution manifold. – Willie Wong Apr 26 '11 at 18:10
I'm with Willie; locus sounds about right. Geometrically your nonunique set of solutions would correspond to some curve/surface... – J. M. Apr 26 '11 at 18:19
I'm torn here: I think both Willie's and lhf's answers have merit, and I'm not sure which one I should accept. Willie, if you would care to make your comment and answer, I'd be happy to upvote (and let others do the same). – Gilead Apr 27 '11 at 1:06

Zero set is a usual term. But so is set of solutions. Zero set is applied to sets of functions. Set of solutions is applied to systems of equations.

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