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?