Rabinowitsch trick

In mathematics, the Rabinowitsch trick, introduced by Template:Harvtxt, 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 $g_0,g_1,\dots ,g_m\in K[x_0,x_1,\dots ,x_n]$ 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,....,f_m. This is the full version of the Nullstellensatz for K[x₁,...,x_n].

References

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