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).
For an ideal 
in the polynomial
ring ![R=K[x_1,...,x_n]](/images/equations/RabinowitschTrick/Inline7.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/Inline18.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/Inline23.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.
