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 ${\displaystyle K[x_{1},\dots ,x_{n}]}$.

References

1. The 1929 Math. Ann. article credits authorship to J. L. Rabinowitsch in Moscow, but little else is known about the author. According to mathematical folklore, J. L. Rabinowitsch is a pseudonym of G. Y. Rainich. However, this claim has been disputed: https://mathoverflow.net/questions/416577/identity-of-j-l-rabinowitsch-of-rabinowitsch-trick

• Brownawell, W. Dale (2001) [1994], "Rabinowitsch trick", Encyclopedia of Mathematics , EMS Press
• Rabinowitsch, J. L. (1929), "Zum Hilbertschen Nullstellensatz", Math. Ann. (in German), 102 (1): 520, doi:10.1007/BF01782361, MR   1512592