An invertible polynomial map, also called a polynomial automorphism, is a polynomial map 
with ![F_i in K[x_1,...,x_n]](/images/equations/InvertiblePolynomialMap/Inline2.svg)
over a field 
that has a polynomial inverse
function 
satisfying
 |
These are polynomial identities in the polynomial ring, rather than merely equalities of
the induced functions on 
. Equivalently, substitution 
defines a ring
automorphism of ![K[x_1,...,x_n]](/images/equations/InvertiblePolynomialMap/Inline7.svg)
fixing every element of 
. Gröbner bases
provide a means to decide whether a given polynomial
map is invertible (Becker and Weispfenning 1993, p. 330).
The chain rule implies that the Jacobian determinant of an invertible polynomial map is a nonzero element of 
.
Its inverse function is again an invertible polynomial
map, and these maps form a group under function
composition. The tame automorphisms form
the subgroup generated by invertible changes of a single
coordinate that are linear in that coordinate. In three variables,
the Nagata automorphism is a wild
automorphism over fields of characteristic
zero.
A sufficient condition for an explicit polynomial inverse function is available for maps 
over 
,
where the components 
are homogeneous polynomials
of the same total degree 
. If the matrix 
of partial derivatives satisfies 
as a polynomial identity,
then 
(Singer 2001, p. 2).
