“Rabinowitsch trick”的意思、由来-开放百科全书

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Rabinowitsch trick

释义

References

In mathematics, the Rabinowitsch trick, introduced by George Yuri Rainich and published under his original name {{harvtxt|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 such that

as an equality of elements of the polynomial ring . Since are free variables, this equality continues to hold if expressions are substituted for some of the variables; in particular, it follows from substituting that

as elements of the field of rational functions , the field of fractions of the polynomial ring . 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

for some natural number r and polynomials . Hence

,

which literally states that lies in the ideal generated by f₁,....,f_m. This is the full version of the Nullstellensatz for K[x₁,...,x_n].

References

{{springer|id=r/r130010|first=W. Dale|last= Brownawell|authorlink=W. Dale Brownawell|title=Rabinowitsch trick}}
{{citation|first=J.L.|last= Rabinowitsch|title=Zum Hilbertschen Nullstellensatz|journal= Math. Ann. |volume= 102 |year=1929|issue=1|pages= 520|mr=1512592|doi=10.1007/BF01782361|language=de}}

1 : Commutative algebra

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