Difference between revisions of "Rabinowitsch trick"
From Encyclopedia of Mathematics
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| + | This "trick" deduces the general Hilbert Nullstellensatz (cf. [[Hilbert theorem|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|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*} | ||
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| + | 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 [[#References|[a1]]]. | ||
====References==== | ====References==== | ||
| − | <table>< | + | <table><tr><td valign="top">[a1]</td> <td valign="top"> J.L. Rabinowitsch, "Zum Hilbertschen Nullstellensatz" ''Math. Ann.'' , '''102''' (1929) pp. 520,}</td></tr></table> |
Latest revision as of 17:00, 1 July 2020
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,} |
How to Cite This Entry:
Rabinowitsch trick. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rabinowitsch_trick&oldid=12019
Rabinowitsch trick. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rabinowitsch_trick&oldid=12019
This article was adapted from an original article by W. Dale Brownawell (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article