Rabinowitsch trick

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 g 0 , g 1 , … , g m ∈ K [ x 0 , x 1 , … , x n ] {\displaystyle 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 , … , x n ) ( 1 − x 0 f ( x 1 , … , x n ) ) + ∑ i = 1 m g i ( x 0 , x 1 , … , x n ) f i ( x 1 , … , x n ) {\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 K [ x 0 , x 1 , … , x n ] {\displaystyle K[x_{0},x_{1},\dots ,x_{n}]} . Since x 0 , x 1 , … , x n {\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 x 0 = 1 / f ( x 1 , … , x n ) {\displaystyle x_{0}=1/f(x_{1},\dots ,x_{n})} that

1 = ∑ i = 1 m g i ( 1 / f ( x 1 , … , x n ) , x 1 , … , x n ) f i ( x 1 , … , x n ) {\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 K ( x 1 , … , x n ) {\displaystyle K(x_{1},\dots ,x_{n})} , the field of fractions of the polynomial ring K [ x 1 , … , x n ] {\displaystyle 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 = ∑ i = 1 m h i ( x 1 , … , x n ) f i ( x 1 , … , x n ) f ( x 1 , … , x n ) r {\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 r and polynomials h 1 , … , h m ∈ K [ x 1 , … , x n ] {\displaystyle h_{1},\dots ,h_{m}\in K[x_{1},\dots ,x_{n}]} . Hence

f ( x 1 , … , x n ) r = ∑ i = 1 m h i ( x 1 , … , x n ) f i ( x 1 , … , x n ) , {\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 f r {\displaystyle f^{r}} lies in the ideal generated by f1,....,fm. This is the full version of the Nullstellensatz for K[x1,...,xn].

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