vanishing set

Pick some polynomials and ask where they all equal zero at once: that locus is the vanishing set. It is the most elementary operation in algebraic geometry — the bridge that turns a list of equations into a geometric figure — and every affine variety arises this way.

Given a set S of polynomials in k[x_1, ..., x_n], its vanishing set is V(S) = { a in A^n : f(a) = 0 for all f in S }. Only the ideal generated by S matters, so V(S) = V((S)), and adding or removing polynomials that are already algebraic consequences changes nothing. The map V reverses inclusions: a bigger set of equations cuts out a smaller locus.

The vanishing sets are precisely the closed sets of the Zariski topology, which is why V is the geometric half of the algebra-geometry correspondence. On the prime spectrum Spec R the same notation V(I) means the set of primes containing I, recovering the classical zero locus on maximal ideals while extending it to all primes. One must remember that distinct ideals can share a vanishing set; only their common radical is detected by V.