Rabinowitsch trick

In mathematics, the Rabinowitsch trick, introduced by J. L. Rabinowitsch (1929),^[1] 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 ${\displaystyle K}$ be an algebraically closed field. Suppose the polynomial ${\displaystyle f}$ in ${\displaystyle K[x_{1},\dots ,x_{n}]}$ vanishes whenever all polynomials ${\displaystyle f_{1},\dots ,f_{m}}$ vanish. Then the polynomials ${\displaystyle f_{1},\dots ,f_{m},1-x_{0}f}$ have no common zeros (where we have introduced a new variable ${\displaystyle x_{0}}$), so by the weak Nullstellensatz for ${\displaystyle K[x_{0},\dots ,x_{n}]}$ they generate the unit ideal of ${\displaystyle K[x_{0},\dots ,x_{n}]}$. Spelt out, this means there are polynomials ${\displaystyle g_{0},g_{1},\dots ,g_{m}\in K[x_{0},x_{1},\dots ,x_{n}]}$ such that

${\displaystyle 1=g_{0}(x_{0},x_{1},\dots ,x_{n})(1-x_{0}f(x_{1},\dots ,x_{n}))+\sum _{i=1}^{m}g_{i}(x_{0},x_{1},\dots ,x_{n})f_{i}(x_{1},\dots ,x_{n})}$

as an equality of elements of the polynomial ring ${\displaystyle K[x_{0},x_{1},\dots ,x_{n}]}$. Since ${\displaystyle x_{0},x_{1},\dots ,x_{n}}$ are free variables, this equality continues to hold if expressions are substituted for some of the variables; in particular, it follows from substituting ${\displaystyle x_{0}=1/f(x_{1},\dots ,x_{n})}$ that

${\displaystyle 1=\sum _{i=1}^{m}g_{i}(1/f(x_{1},\dots ,x_{n}),x_{1},\dots ,x_{n})f_{i}(x_{1},\dots ,x_{n})}$

as elements of the field of rational functions ${\displaystyle K(x_{1},\dots ,x_{n})}$, the field of fractions of the polynomial ring ${\displaystyle K[x_{1},\dots ,x_{n}]}$. Moreover, the only expressions that occur in the denominators of the right hand side are ${\displaystyle f}$ and powers of ${\displaystyle f}$, so rewriting that right hand side to have a common denominator results in an equality on the form

${\displaystyle 1={\frac {\sum _{i=1}^{m}h_{i}(x_{1},\dots ,x_{n})f_{i}(x_{1},\dots ,x_{n})}{f(x_{1},\dots ,x_{n})^{r}}}}$

for some natural number ${\displaystyle r}$ and polynomials ${\displaystyle h_{1},\dots ,h_{m}\in K[x_{1},\dots ,x_{n}]}$. Hence

${\displaystyle f(x_{1},\dots ,x_{n})^{r}=\sum _{i=1}^{m}h_{i}(x_{1},\dots ,x_{n})f_{i}(x_{1},\dots ,x_{n}),}$

which literally states that ${\displaystyle f^{r}}$ lies in the ideal generated by ${\displaystyle f_{1},\dots ,f_{m}}$. This is the full version of the Nullstellensatz for ${\displaystyle K[x_{1},\dots ,x_{n}]}$.