Rabinowitsch trick
Encyclopedia
In mathematics, the Rabinowitsch trick, introduced by ,
is a short way of proving the general case of the Hilbert Nullstellensatz from an easier special case, by introducing an extra variable.
The Rabinowitsch trick goes as follows. Suppose the polynomial f in C[x1,...xn] vanishes whenever all polynomials f1,....,fm vanish. Then the polynomials f1,....,fm, 1 − x0f have no common zeros (where we have introduced a new variable x0), so by the "easy" version of the Nullstellensatz for C[x0, ..., xn] they generated the unit ideal of C[x0 ,..., xn].
From this an easy calculation (setting x0 = 1/f and multiplying by the greatest common denominator therein introduced) shows that some power of f lies in the ideal generated by f1,....,fm, which is the "hard" version of the Nullstellensatz for C[x1,...,xn].
is a short way of proving the general case of the Hilbert Nullstellensatz from an easier special case, by introducing an extra variable.
The Rabinowitsch trick goes as follows. Suppose the polynomial f in C[x1,...xn] vanishes whenever all polynomials f1,....,fm vanish. Then the polynomials f1,....,fm, 1 − x0f have no common zeros (where we have introduced a new variable x0), so by the "easy" version of the Nullstellensatz for C[x0, ..., xn] they generated the unit ideal of C[x0 ,..., xn].
From this an easy calculation (setting x0 = 1/f and multiplying by the greatest common denominator therein introduced) shows that some power of f lies in the ideal generated by f1,....,fm, which is the "hard" version of the Nullstellensatz for C[x1,...,xn].