Rabinowitsch trick
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 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].
[edit] References
- Brownawell, W. Dale (2001), "Rabinowitsch trick", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=r/r130010
- Rabinowitsch, J.L. (1929), "Zum Hilbertschen Nullstellensatz", Math. Ann. 102 (1): 520, doi:10.1007/BF01782361
