Rabinowitsch trick
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In mathematics, the Rabinowitsch trick, introduced by 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 generated the unit ideal of C[x0 ,..., xn]. From this an easy calculation (setting x0 = 1/f) 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].
[edit] References
- Brownawell, W. Dale (2001), “Rabinowitsch trick”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
- Rabinowitsch, J.L. (1929), “Zum Hilbertschen Nullstellensatz”, Math. Ann. 102: 520, DOI 10.1007/BF01782361
