An illuminating section on an example

Let us set n = 3, so that $\cal {P}$ : = $\mathbb {C}$[X1, X2, X3], $\sf a$ : = (0, 0, 0)∈$\mathbb {C}$3, $\sf b$ : = (1, 0, 1)∈$\mathbb {C}$3, $\sf c$ : = (0, -1, -1)∈$\mathbb {C}$3,
λ$\scriptstyle \sf a$($\displaystyle \mathfrak$q$\scriptstyle \sf a$) : = (X14, X1X22, X12X2, X23, X1X3, X2X3, X32)$\displaystyle \cr$λ$\scriptstyle \sf b$($\displaystyle \mathfrak$q$\scriptstyle \sf b$)  

so that s : = deg($\sf I$) = 8 + 4 + 4 = 16.

In the table below we properly list the sets $\sf X$($\sf I$), $\mathbb {L}$($\sf I$) and the result $\bf N$($\sf X$) of Cerlienco–Mureddu Correspondence.

i 1 2 3 4 5 6 7 8$\cr$$\sf a_{i}^{}$

The lex reduced Gröbner basis of I is $\cal {G}$($\sf I$) = {fi, 1≤i≤9} where

f1 : = X15 - X14$\displaystyle \cr$f2

and we have the following factorization of each fi modulo (f1,…, fi-1):

f1 = X14(X1 -1)$\displaystyle \cr$f2

Remark that for
f2   $\displaystyle \sf Q_{{2}}^{}$($\displaystyle \sf t_{2}^{}$) = {M(X12)λ$\scriptstyle \sf a$, M(X1)λ$\scriptstyle \sf b$, M(X12)λ$\scriptstyle \sf c$},$\displaystyle \cr$  

and that each factor is obtained by interpolation as stated in Lemma [*].