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Please give me a hint for this exercise.

Show that every finite subset of the affine plane, $\mathbb A^2_k$, over an algebraically closed field, $k$, can be determined by two equations.

I am using Shafarevich's book.

Lord_Farin
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Framate
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    The question of how many equations are necessary to define an affine algebraic set in affine space $\mathbb A^n_k$ is one of the most difficult in algebraic geometry and has been studied for more than a century by the best specialists. It is almost impossible for a beginner to "have thoughts on the problem " or "make attempts" to solve it if they don't know the trick I used in my answer, since the sort of "analytic geometry" I use is no longer taught in contemporary algebraic geometry courses. Could critics show their thoughts and attempts on the problem? Needless to say I vote to reopen. – Georges Elencwajg Nov 29 '13 at 11:33

1 Answers1

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First case
If the points $P_i=(a_i,b_i)$ have different abscissae $a_i\neq a_j$, take for one of the equations $y=f(x)$ where $f$ is a polynomial satisfying $b_i=f(a_i)$ [key word: Lagrange interpolation polynomial] and for the other equation take $\prod (x-a_i)=0$.
Second case
If you are unfortunate and some abscissae coincide, make a change of variables so as to reduce to the first case.
One possible such change is $x=x'+y', y=\lambda y'$ with $\lambda \neq \frac {b_i-b_j}{a_i-a_j}$ for all $i\neq j$.
This works as soon as $k$ infinite but not necessarily algebraically closed.

Georges Elencwajg
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