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In mathematics, the Rabinowitsch trick, introduced by George Yuri Rainich and published under the pseudonym 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 $$g_0,g_1,\dots,g_m \in K[x_0,x_1,\dots,x_n]$$ 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 $$f^r l$$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), "Rabinowitsch trick", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Rabinowitsch, J.L. (1929), "Zum Hilbertschen Nullstellensatz", Math. Ann. 102 (1): 520, doi:10.1007/BF01782361

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