Rabinowitsch trick

Short description: Mathematical proof of Hilbert's Nullstellensatz

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 $K[x_1,\dots ,x_n]$ vanishes whenever all polynomials $f_1,\dots ,f_m$ vanish. Then the polynomials $f_1,\dots ,f_m,1-x_{0}f$ have no common zeros (where we have introduced a new variable $x_0$), so by the weak Nullstellensatz for $K[x_0,\dots ,x_n]$ they generate the unit ideal of $K[x_0,\dots ,x_n]$. Spelt out, this means there are polynomials $g_0,g_1,\dots ,g_m\in K[x_0,x_1,\dots ,x_n]$ such that

$1=g_0(x_0,x_1,\dots ,x_n)(1-x_{0}f(x_1,\dots ,x_n))+\sum _{i=1}^m{g}_i(x_0,x_1,\dots ,x_n)f_i(x_1,\dots ,x_n)$

as an equality of elements of the polynomial ring $K[x_0,x_1,\dots ,x_n]$. Since $x_0,x_1,\dots ,x_n$ are free variables, this equality continues to hold if expressions are substituted for some of the variables; in particular, it follows from substituting $x_0=1/f(x_1,\dots ,x_n)$ that

$1=\sum _{i=1}^m{g}_i(1/f(x_1,\dots ,x_n),x_1,\dots ,x_n)f_i(x_1,\dots ,x_n)$

as elements of the field of rational functions $K(x_1,\dots ,x_n)$, the field of fractions of the polynomial ring $K[x_1,\dots ,x_n]$. 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

$1=\frac{\sum _{i=1}^m{h}_i(x_1,\dots ,x_n)f_i(x_1,\dots ,x_n)}{f(x_1,\dots ,x_n{)}^r}$

for some natural number $r$ and polynomials $h_1,\dots ,h_m\in K[x_1,\dots ,x_n]$. Hence

$f(x_1,\dots ,x_n{)}^r=\sum _{i=1}^m{h}_i(x_1,\dots ,x_n)f_i(x_1,\dots ,x_n),$

which literally states that $f^r$ lies in the ideal generated by $f_1,\dots ,f_m$. This is the full version of the Nullstellensatz for $K[x_1,\dots ,x_n]$.

References

• Hazewinkel, Michiel, ed. (2001), "Rabinowitsch trick", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=r/r130010 
• Rabinowitsch, J. L. (1929), "Zum Hilbertschen Nullstellensatz" (in de), Math. Ann. 102 (1): 520, doi:10.1007/BF01782361