Let $ R=K[x_1,…,x_n]$ , for a field $ K$ , and $ I$ is radical zero dimensional ideal of $ R$ . We define $ s=\Sigma_{i=1}^{n}a_ix_i$ , with $ c_i\in K$ , a shape basis generator of $ I$ , if $ (s+I)$ generates the quotient $ R/I$ (as an algebra over $ K$ ).

If we introduce a new variable $ z$ , and let $ I^\prime$ be the ideal of $ R^\prime=K[x_1,…,x_n,z]$ generated by the embedding of $ I$ in $ R^\prime$ and $ (z-s)$ , then, How the minimal reduced Grobner basis of $ I^\prime$ with respect to the lexicographical order with $ x_1>…>x_n>z$ has the form

$ $ <x_1-f_i(z),x_2-f_2(z),…,x_{n}-f_{n}(z),g(z)>$ $ where the $ f_i$ and $ g$ are univariate polynomials in $ K[z]$ .

I expect that there should be a isomorphism $ K[x_1,…,x_n,z]/{I^\prime}\equiv K[z]/{<g(z)>}$ , but how to see that.

Then we can use correspondence between $ x_i\pmod{I}$ and $ f_i(z)\pmod{<g(z)>}$ to get the form, I am not clear about that too. Please help me to understand this. Thank you