Rabinowitsch trick

Last updated February 05, 2024

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,...x_n] vanishes whenever all polynomials f₁,....,f_m vanish. Then the polynomials f₁,....,f_m, 1 − x₀f have no common zeros (where we have introduced a new variable x₀), so by the weak Nullstellensatz for K[x₀, ..., x_n] they generate the unit ideal of K[x₀ ,..., 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 f₁,....,f_m. This is the full version of the Nullstellensatz for K[x₁,...,x_n].

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References

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

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