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 ${\displaystyle g_{0},g_{1},\dots ,g_{m}\in K[x_{0},x_{1},\dots ,x_{n}]}$ such that

as an equality of elements of the polynomial ring ${\displaystyle K[x_{0},x_{1},\dots ,x_{n}]}$. Since ${\displaystyle x_{0},x_{1},\dots ,x_{n}}$ are free variables, this equality continues to hold if expressions are substituted for some of the variables; in particular, it follows from substituting ${\displaystyle x_{0}=1/f(x_{1},\dots ,x_{n})}$ that

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