In mathematics, the Rabinowitsch trick, introduced by J.L. Rabinowitsch (1929),
is a short way of proving the general case of the
Hilbert Nullstellensatz
from an easier special case (the so-called
weak Nullstellensatz), by introducing an extra variable.
The Rabinowitsch trick goes as follows. Let K be an algebraically closed field. Suppose the polynomial f in K[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 weak Nullstellensatz for K[x0, ..., xn] they generate the unit ideal of K[x0 ,..., xn]. Spelt out, this means there are polynomials
such that
![{\displaystyle 1=g_{0}(x_{0},x_{1},\dots ,x_{n})(1-x_{0}f(x_{1},\dots ,x_{n}))+\sum _{i=1}^{m}g_{i}(x_{0},x_{1},\dots ,x_{n})f_{i}(x_{1},\dots ,x_{n})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f6af8d2a4d402b3c62a8733e27f19eca53cc013)
as an equality of elements of the polynomial ring
. Since
are
free variables
, this equality continues to hold if expressions are substituted for some of the variables; in particular, it follows from substituting
![{\displaystyle x_{0}=1/f(x_{1},\dots ,x_{n})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28fa0cd3575af4648cde3409f0c7e5054593000a)
that
![{\displaystyle 1=\sum _{i=1}^{m}g_{i}(1/f(x_{1},\dots ,x_{n}),x_{1},\dots ,x_{n})f_{i}(x_{1},\dots ,x_{n})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6097f24dbae4c5fe691f2f08bd9ae8c87868a518)
as elements of the field of rational functions
, the field of fractions of the polynomial ring
. Moreover, the only expressions that occur in the denominators of the right hand side are f and powers of f, so rewriting that right hand side to have a common denominator results in an equality on the form
![{\displaystyle 1={\frac {\sum _{i=1}^{m}h_{i}(x_{1},\dots ,x_{n})f_{i}(x_{1},\dots ,x_{n})}{f(x_{1},\dots ,x_{n})^{r}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/134c5248d43f22c0d431edfe00a4c723d9553936)
for some natural number r and polynomials
. Hence
![{\displaystyle f(x_{1},\dots ,x_{n})^{r}=\sum _{i=1}^{m}h_{i}(x_{1},\dots ,x_{n})f_{i}(x_{1},\dots ,x_{n}),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/694bff63da7c2bd2af2fa5f9709a10d906281246)
which literally states that
lies in the ideal generated by f1,....,fm. This is the full version of the
Nullstellensatz
for
K[
x1,...,
xn].
References