Rabinowitsch trick

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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 K {\displaystyle K} {\displaystyle K} be an algebraically closed field. Suppose the polynomial f {\displaystyle f} {\displaystyle f} in K [ x 1 , … , x n ] {\displaystyle K[x_{1},\dots ,x_{n}]} {\displaystyle K[x_{1},\dots ,x_{n}]} vanishes whenever all polynomials f 1 , … , f m {\displaystyle f_{1},\dots ,f_{m}} {\displaystyle f_{1},\dots ,f_{m}} vanish. Then the polynomials f 1 , … , f m , 1 − x 0 f {\displaystyle f_{1},\dots ,f_{m},1-x_{0}f} {\displaystyle f_{1},\dots ,f_{m},1-x_{0}f} have no common zeros (where we have introduced a new variable x 0 {\displaystyle x_{0}} {\displaystyle x_{0}}), so by the weak Nullstellensatz for K [ x 0 , … , x n ] {\displaystyle K[x_{0},\dots ,x_{n}]} {\displaystyle K[x_{0},\dots ,x_{n}]} they generate the unit ideal of K [ x 0 , … , x n ] {\displaystyle K[x_{0},\dots ,x_{n}]} {\displaystyle K[x_{0},\dots ,x_{n}]}. 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}]} {\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})} {\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}]} {\displaystyle K[x_{0},x_{1},\dots ,x_{n}]}. Since x 0 , x 1 , … , x n {\displaystyle x_{0},x_{1},\dots ,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})} {\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})} {\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})} {\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}]} {\displaystyle K[x_{1},\dots ,x_{n}]}. Moreover, the only expressions that occur in the denominators of the right hand side are f {\displaystyle f} {\displaystyle f} and powers of f {\displaystyle f} {\displaystyle 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}}}} {\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 {\displaystyle r} {\displaystyle 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}]} {\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}),} {\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}} {\displaystyle f^{r}} lies in the ideal generated by f 1 , … , f m {\displaystyle f_{1},\dots ,f_{m}} {\displaystyle f_{1},\dots ,f_{m}}. This is the full version of the Nullstellensatz for K [ x 1 , … , x n ] {\displaystyle K[x_{1},\dots ,x_{n}]} {\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 for certain about the author. According to mathematical folklore, the author and originator of the 'Rabinowitsch trick' was the mathematical physicist George Yuri Rainich (original name: Yuriy Germanovich Rabinovich, Юрий Германович Рабинович). However, this appears to be a misattribution which conflated this article with work in algebraic number theory that Rainich published under the name Georg Rabinowitsch. The author of the 'trick' has been tentatively identified as another mathematician, Yuriy Lazarevich Rabinovich (Юлий Лазаревич Рабинович). For a detailed discussion, see: https://mathoverflow.net/questions/416577/identity-of-j-l-rabinowitsch-of-rabinowitsch-trick