Consider the following set of polynomials in two complex variables $z_1$ and $z_2$:
$$a_1+a_2 z_1+a_3 z_1^2~~~~~,~~~~~b_1 +b_2 z_1+b_3 z_2~~~~~,~~~~~c_1 z_1+c_2 z_2$$
the $a_i,b_i,c_i$ are real coefficients which are given explicitly and the above set is assumed to be in Gröbner basis form. Now the claim is that the monomial basis in this Gröbner basis is:
$$B=\{1,z_1\}$$
I wonder how this can possibly be a sufficient monomial basis? For instance, the monomial $z_2$ does not even appear! Could someone explain to me what I am missing here? Thanks for any suggestion.
Okay, I'll write my comment as a short answer.
I think by monomial basis you mean a $\mathbb R$-basis for $\mathbb R[z_1,z_2]/I$ where $I$ is the ideal generated by your three polynomials. The reason that $z_2$ doesn't appear is because of the relation $c_1z_2+c_2z_2=0$: in the quotient it is linearly dependent upon $z_1$, so we dont need it in a basis.
But an aside: are you sure this is the ideal you're working with? I tested it with random coefficients in Macaulay2, and it seems to contain $1$, so that the quotient should be zero.