Rabinowitsch trick

This "trick" deduces the general Hilbert Nullstellensatz (cf. Hilbert theorem) from the special case that the polynomials have no common zeros. Indeed, let $f , f _ { 1 } , \dots , f _ { m } \in R : = k [ x _ { 1 } , \dots , x _ { n } ]$, where $k$ is a field. If $f$ vanishes on the common zeros of $f _ { 1 } , \ldots , f _ { m }$, then there are polynomials $a _ { 0 } , a _ { 1 } , \dots , a _ { m } \in R [ x _ { 0 } ]$ such that

\begin{equation*} a _ { 0 } ( 1 - x _ { 0 } f ) + a _ { 1 } f _ { 1 } + \ldots + a _ { m } f _ { m } = 1. \end{equation*}

Substitution of $x _ { 0 } = 1 / f$ into this identity and clearing out the denominator shows that

\begin{equation*} b _ { 1 } f _ { 1 } + \ldots + b _ { m } f _ { m } = f ^ { \mu }, \end{equation*}

where $\mu : = \operatorname { max } \operatorname { deg } _ { x _ { 0 } } a _ { i }$ and $b _ { j } = a _ { j } |_{x _ { 0 } = 1 / f} f ^ { \mu }$. This ingenious device was published in the one(!) page article [a1].

References

[a1]	J.L. Rabinowitsch, "Zum Hilbertschen Nullstellensatz" Math. Ann. , 102 (1929) pp. 520,}