Rabinowitsch trick

In mathematics, the Rabinowitsch trick, introduced by (J.L. 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 [math]\displaystyle{ g_0,g_1,\dots,g_m \in K[x_0,x_1,\dots,x_n] }[/math] such that

[math]\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) }[/math]

as an equality of elements of the polynomial ring [math]\displaystyle{ K[x_0,x_1,\dots,x_n] }[/math]. Since [math]\displaystyle{ x_0,x_1,\dots,x_n }[/math] are free variables, this equality continues to hold if expressions are substituted for some of the variables; in particular, it follows from substituting [math]\displaystyle{ x_0 = 1/f(x_1,\dots,x_n) }[/math] that

[math]\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) }[/math]

as elements of the field of rational functions [math]\displaystyle{ K(x_1,\dots,x_n) }[/math], the field of fractions of the polynomial ring [math]\displaystyle{ K[x_1,\dots,x_n] }[/math]. 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

[math]\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} }[/math]

for some natural number r and polynomials [math]\displaystyle{ h_1,\dots,h_m \in K[x_1,\dots,x_n] }[/math]. Hence

[math]\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), }[/math]

which literally states that [math]\displaystyle{ f^r }[/math] lies in the ideal generated by f₁,....,f_m. This is the full version of the Nullstellensatz for K[x₁,...,x_n].

References

• Hazewinkel, Michiel, ed. (2001), "Rabinowitsch trick", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=r/r130010 
• Rabinowitsch, J.L. (1929), "Zum Hilbertschen Nullstellensatz" (in de), Math. Ann. 102 (1): 520, doi:10.1007/BF01782361