If have computed this Gröbner Basis with Buchberger's algorithm for Degree-Lexicographic-Ordering:
$\{ x^²y+x+1,xy^2+y+1,x-y \} $
I want to to transform it into a unique representation form called Reduced Gröbner Basis. Therefore I remove the leading powers which are divided by other leading powers. In this case $x \mid x^2y$ and $x \mid xy^2$ so the resulting base is
$\{x-y\}.$ This is not the reduced base yet but a form called Minimal Gröber Basis.
I checked the result with Singular. The reduced base computed by Singular ist:
$\{x-y, y^3+y+1\}$
Nevertheless
$ \langle x-y, y^3+y+1 \rangle \neq \langle x-y\rangle.$
Any ideas?