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# Rabinowitsch trick

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*[*x*_{1},...*x*_{n}] vanishes whenever all polynomials *f*_{1},....,*f*_{m} vanish. Then the polynomials *f*_{1},....,*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}, ..., *x*_{n}] they generate the unit ideal of *K*[*x*_{0} ,..., *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 \( f^r l \)lies in the ideal generated by *f*_{1},....,*f*_{m}. This is the full version of the Nullstellensatz for *K*[*x*_{1},...,*x*_{n}].

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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