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Math Amateur

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At present I am focused on Section 15.1 Noetherian Rings and Affine Algebraic Sets ... ...

I am trying to gain a full understanding of the nature of affine algebraic sets ...

If we take an arbitrary subset A of affine space ##\mathbb{A}^n## ... how can we determine whether A is an affine algebraic set ... ?

Are they any methodical approaches ... ?

Do we just have to creatively come up with a polynomial or set of polynomials whose set of zeros equals A?

Any clarifying comments are welcome ...

Peter