Rabinowitsch trick

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In mathematics, the Rabinowitsch trick, introduced by George Yuri Rainich and published under the pseudonym Rabinowitsch (1929), is a short way of proving the general case of the Hilbert Nullstellensatz from an easier special case, by introducing an extra variable.

The Rabinowitsch trick goes as follows. Suppose the polynomial f in C[x1,...xn] vanishes whenever all polynomials f1,....,fm vanish. Then the polynomials f1,....,fm, 1 − x0f have no common zeros (where we have introduced a new variable x0), so by the "easy" version of the Nullstellensatz for C[x0, ..., xn] they generate the unit ideal of C[x0 ,..., xn]. From this an easy calculation (setting x0 = 1/f and multiplying by the greatest common denominator therein introduced) shows that some power of f lies in the ideal generated by f1,....,fm, which is the "hard" version of the Nullstellensatz for C[x1,...,xn].

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