Rabinowitsch Trick

In mathematics, the Rabinowitsch trick, introduced by George Yuri Rainich and published under his original name ,
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 ''f''₁,....,''f''_''m'' vanish. Then the polynomials ''f''₁,....,''f''_''m'', 1 − ''x''₀''f'' have no common zeros (where we have introduced a new variable ''x''₀), 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,\dots,g_m \in K _0,x_1,\dots,x_n/math> such that
: $1 = g_0(x_0,x_1,\dots,x_n) (1 - x_0 f(x_1,\dots,x_n)) + \sum_^m g_i(x_0,x_1,\dots,x_n) f_i(x_1,\dots,x_n)$
as an equality of elements of the polynomial ring K _0,x_1,\dots,x_n/math>. 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
: $1 = \sum_^m g_i(1/f(x_1,\dots,x_n),x_1,\dots,x_n) f_i(x_1,\dots,x_n)$
as elements of the field of rational functions $K(x_1,\dots,x_n)$, the field of fractions of the polynomial ring K _1,\dots,x_n/math>. 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
: $1 = \frac$
for some natural number ''r'' and polynomials h_1,\dots,h_m \in K _1,\dots,x_n/math>. Hence
: $f(x_1,\dots,x_n)^r = \sum_^m h_i(x_1,\dots,x_n) f_i(x_1,\dots,x_n),$
which literally states that $f^r$ lies in the ideal generated by ''f''₁,....,''f''_''m''. This is the full version of the Nullstellensatz for ''K'' 'x''₁,...,''x''_''n''

References

*
*{{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

Commutative algebra