The Rabinowitsch trick replaces a polynomial nonvanishing condition 
with the equation 
in a new variable 
. At a solution over a field,
this equation holds precisely when 
and 
. It converts polynomial
equations with a nonvanishing condition into equations in one more variable
(Rabinowitsch 1930, p. 520; Milne 2020, p. 62).
Several nonvanishing conditions 
, ..., 
over a field can be imposed
together by taking 
in 
. Over 
, the condition 
excludes only simultaneous vanishing, and permits
some of the 
to be zero. Over 
, cancellation can make 
even when all 
are nonzero.
For an ideal 
in the polynomial
ring ![R=K[x_1,...,x_n]](/images/equations/RabinowitschTrick/Inline17.svg)
over a field 
, a polynomial 
belongs to the ideal
radical 
iff
![<f_1,...,f_s,1-tq>=R[t].](/images/equations/RabinowitschTrick/NumberedEquation1.svg) |
For nonzero 
, an identity expressing 1 in this ideal
gives a positive integer power
of 
in 
by substituting 
and clearing denominators. Conversely, 
implies that 
belongs to the ideal extension
of 
to ![R[t]](/images/equations/RabinowitschTrick/Inline28.svg)
, while 
is divisible by 
(Milne 2020, p. 62).
For example, with 
and 
, the identity
 |
shows that ![<x^2,1-tx>=K[x,t]](/images/equations/RabinowitschTrick/Inline33.svg)
, and hence 
.
Over an algebraically closed field, the trick derives Hilbert's nullstellensatz
from its weak form. Eliminating 
instead computes ideal saturation,
whose zero set can include excluded boundary points.
