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 , … , g m ∈ K [ x 0 , x 1 , … , x n ] such that

as an equality of elements of the polynomial ring K [ x 0 , x 1 , … , x n ] . Since x 0 , x 1 , … , 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 , … , x n ) that

as elements of the field of rational functions K ( x 1 , … , x n ) , the field of fractions of the polynomial ring K [ x 1 , … , 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

for some natural number r and polynomials h 1 , … , h m ∈ K [ x 1 , … , x n ] . Hence

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].