Rabinowitsch trick explained

In mathematics, the Rabinowitsch trick, introduced by,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[''x''₁,...''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[''x''₀, ..., ''x''_''n''] they generate the unit ideal of K[''x''₀ ,..., ''x''_''n'']. Spelt out, this means there are polynomials

g _{0,g} _{1,...,g} _{m} \in K[x _{0,x} _{1,...,x} _{n]} such that1 = g _{0(x} _{0,x} _{1,...,x} _{n)} ( 1 - x _{0} f(x _{1,...,x} _{n))} + \sum ^{m} _{i =1} g _{i(x} _{0,x} _{1,...,x} _{n)} f _{i(x} _{1,...,x} _{n)} 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)} that1 = \sum ^{m} _{i =1} g _{i(1/f(x} _{1,...,x} _{n),x} _{1,...,x} _{n)} f _{i(x} _{1,...,x} _{n)} 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 form1 = (\sum ^{m} _{i =1} h _{i(x} _{1,...,x} _{n)} f _{i(x} _{1,...,x} _{n)})/(f(x _{1 , ... ,x} ^{r} ^{n)}) for some natural number r and polynomials h _{1,...,h} _{m} \in K[x _{1,...,x} _{n]} . Hencef(x _{1,...,x} ^{r} ^{n)} = \sum ^{m} _{i =1} h _{i(x} _{1,...,x} _{n)} f _{i(x} _{1,...,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[''x''₁,...,''x''_''n'']