A coordinate polynomial in ![k[x_1,...,x_n]](/images/equations/CoordinatePolynomial/Inline1.svg)
is a polynomial 
that can be completed to a polynomial coordinate
system. There must be 
such that
![k[x_1,...,x_n]=k[f,f_2,...,f_n]](/images/equations/CoordinatePolynomial/NumberedEquation1.svg) |
as a polynomial ring. Equivalently, a 
-algebra automorphism sends
to 
.
For example, 
is a coordinate polynomial, completed by 
.
Every fiber of a coordinate polynomial is affine 
-space and an ambient polynomial automorphism
takes it to a coordinate hyperplane. Having one fiber isomorphic to affine
space does not itself exhibit such an automorphism.
The Abhyankar-Sathaye conjecture
asks whether that quotient condition is sufficient. The stable
coordinate conjecture instead tests coordinates after adjoining independent variables.
