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[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 lies in the ideal generated by f1,....,fm. This is the full version of the Nullstellensatz for K[x1,...,xn].

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