Rabinowitsch trick

In mathematics, the Rabinowitsch trick, introduced by harvtxt|Rabinowitsch|1929is 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 ["x"₁,..."x"_"n"] vanishes whenever all polynomials "f"₁,....,"f"_"m" vanish. Then the polynomials "f"₁,....,"f"_"m", 1 − "x"₀"f" have no common zeros (where we have introduced a new variable "x"₀), so by the "easy" version of the Nullstellensatz for C ["x"₀, ..., "x"_"n"] they generated the unit ideal of C ["x"₀ ,..., "x"_"n"] .From this an easy calculation (setting "x"₀ = 1/"f" and multiplying by the greatest common denominator therein introduced) shows that some power of "f" lies in the ideal generated by "f"₁,....,"f"_"m", which is the "hard" version of the Nullstellensatz for C ["x"₁,..."x"_"n"] .

References

*springer|id=r/r130010|first=W. Dale|last= Brownawell
*citation|first=J.L.|last= Rabinowitsch|title=Zum Hilbertschen Nullstellensatz|journal= Math. Ann. |volume= 102 |year=1929|pages= 520 |doi=10.1007/BF01782361