Rabinowitsch trick

In mathematics, the Rabinowitsch trick, introduced by George Yuri Rainich and published under his original name 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].

References

Brownawell, W. Dale (2001) [1994], "Rabinowitsch trick", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Rabinowitsch, J.L. (1929), "Zum Hilbertschen Nullstellensatz", Math. Ann. (in German), 102 (1): 520, MR 1512592, doi:10.1007/BF01782361

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