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\def\Span{\mathop{\rm Span}\nolimits}
\def\supp{\mathop{\rm supp}\nolimits}
\def\lcm{\mathop{\rm lcm}\nolimits}
\def\Lp{\mathop{\rm Lp}\nolimits}
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\newtheorem{remark}[theoremo]{Remark}
\newtheorem{example}[theoremo]{Example}
\newtheorem{algorithm}[theoremo]{Algorithm}
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\newtheorem{History}[theoremo]{Historical Remark} 
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\begin{document}
\title{Some Comments on Cerlienco--Mureddu Algorithm and Enhanced Lazard
Structural Theorem}
\numberofauthors{2}
\author{
\alignauthor Maria Grazia Marinari\\
\affaddr{DIMA, Universit\`a}\\
\affaddr{Genova, Italy}\\
\email{marinari@dima.unige.it} 
\alignauthor Richard Francis Ree\\
\affaddr{Miskatonic Univ.}\\
\affaddr{Arkham (Mass.), U.S.A.}\\
\email{prometeo@dima.unige.it}
}
\maketitle
%
\begin{abstract}
%
In his seminal 1985 paper, Lazard described the structure of the lexicographical
reduced Gr\"obner basis of any polynomial ideal in $k[X_1,X_2]$; his result was then
subsumed by Gianni-Kalkbrener Theorem describing the lexicographical
reduced Gr\"obner basis of any zero-dimensional polynomial ideal in $k[X_1,\ldots,X_n]$.

Cerlienco and Mureddu in 1990 gave a combinatorial algorithm which, given a finite set of
points in $k^n$ allows to deduce the lexicographical escalier of the radical ideal
 in $k[X_1,\ldots,X_n]$ vanishing in these points.

Recently, Marinari and Mora merged Lazard's result, Cer\-lien\-co--Mu\-red\-du Algorithm and
M\"oller Algorithm in order to give an enhanced Lazard Structural Theorem.

In the meantime, however, Cerlienco and Mureddu extended their results 
to cover any
zero-dimensional ideal whose primary components are translations of monomial primaries at
the origin ({\em CM-ideal}).

The aim of this note is to generalize Marinari--Mora result to this new setting. 
\end{abstract}

\noindent
{\bf Acknowledgements} The authors are indepted to Theo Mora for having
suggested the problem and fruitable discussion.
%
\section{A boring section on notations}
%
Let ${\Cal P} := k[X_1,\ldots,X_n]$,
${\sf m} = (X_1,\ldots,X_n)$ the maximal ideal at the origin,
${\Cal T} := \{X_1^{a_1}\cdots X_n^{a_n} :
(a_1,\ldots,a_n)\in {\Bbb N}^{n}\},$
$<$ the lexicographical ordering on ${\Cal T}$
induced by $X_1 < \cdots < X_n.$

We denote ${\sf k}$ the algebraic closure of $k$; for each zero-di\-men\-sio\-nal ideal
${\sf I} \subset {\Cal P}$,
${\Cal Z}({\sf I}) := \{{\sf a}\in{\sf k}^n : f({\sf a}) = 0, \forall f\in {\sf I}\}\subset
{\sf k}^n;$
for any $\alpha = (b_1,\ldots,b_d)\in {\sf k}^d$, 
$\Phi_\alpha$ the projection 
$\Phi_\alpha : {\Cal P}  \mapsto {\sf k}[X_{d+1},\ldots,X_n]$
defined by 
$$\Phi_\alpha(f) = f(b_1,\ldots,b_d,X_{d+1},\ldots,X_n] \forall
f\in k[X_1,\ldots,X_n].$$

Each element $f\in {\Cal P}$ can be uniquely expressed either as 
$$f = \sum_{i=0}^{\deg(f)} g_i X_n^i \in k[X_1,\ldots,X_{n-1}][X_n],$$
$g_i\in k[X_1,\ldots,X_{n-1}]$, $g_{\deg(f)} \neq 0,$ 
or as a linear combination
$$f = \sum_{t\in{\Cal T}} c(f,t) t = \sum_{i=1}^s c(f,t_i) t_i,$$ $c(f,t_i) \neq 0,$
$t_i \in {\Cal T}, t_1 > \cdots > t_s$
 of terms $t\in{\Cal T}$
with coefficients $c(f,t)$ in $k$; and we will denote 
$\Lp(f) := g_{\deg(f)}$ the {\em leading  polynomial} of $f$, 
${\bf T}(f) := t_1$ its {\em maximal term},
$\lc(f) := c(f,t_1)$ its {\em leading cofficient}. 
%and 
%\item ${\bf M}(f) := c(f,t_1)t_1$ its {\em maximal monomial}.

For each set $G\subset {\Cal P}$, 
${\bf T}\{G\}$ denotes the set $\{{\bf T}(g) : g\in G\},$
and ${\bf T}(G)$ the monomial ideal $\{\tau {\bf T}(g) : \tau\in {\Cal T}, g\in G\}$
it generates.
For each ideal ${\sf I}\subset {\Cal P}$, we denote
${\bf G}({\sf I})$ the minimal basis 
of the monomial ideal ${\bf T}({\sf I}) = {\bf T}\{{\sf I}\}$,
${\bf N}({\sf I}) := {\Cal T} \setminus {\bf T}({\sf I})$  and
\begin{eqnarray*}
{\bf B}({\sf I}) &:=& \{X_h t : 1\leq h \leq n, t\in {\bf N}({\sf I})\}
\setminus {\bf N}({\sf I})\\
&=&  {\bf T}({\sf I}) \cap \left(\{1\} \cup \{X_h t : 1\leq h \leq n, t\in {\bf N}({\sf I})\}\right)
\end{eqnarray*}
and we set $k[{\bf N}({\sf I})] :=\Span_k({\bf N}({\sf I}))$.

For each $f\in{\Cal P},$ there is  \cite{B1,B2, BL} 
a unique {\em canonical form}
$$g := \Can(f,{\sf I})  = \sum_{t\in{\bf N}({\sf I})} \gamma(f,t,<) t 
\in  k[{\bf N}({\sf I})]$$ such that $f - g \in {\sf I}$.
A {\em Gr\"obner basis}  \cite{B1,B2} of ${\sf I}$ is any set 
$G \subset {\sf I}$ such that ${\bf T}(G) = {\bf T}\{{\sf I}\},$ i.e.
${\bf T}\{G\}$ generates the monomial ideal ${\bf T}({\sf I}) = {\bf T}\{{\sf I}\};$
the {\em reduced Gr\"obner basis}  \cite{B1,B2} of ${\sf I}$ is the set 
${\Cal G}({\sf I}) := \{\tau - \Can(\tau,{\sf I}) : \tau \in {\bf G}({\sf I})\};$ 
the {\em border basis} \cite{M$^3$}  of ${\sf I}$ is the set 
${\Cal B}({\sf I}) := \{\tau - \Can(\tau,{\sf I}) : \tau \in {\bf B}({\sf I})\}.$ 

Two sets
${\Bbb L} :=\{\ell_1,\ldots, \ell_s\}\subset 
{\Cal P}^{\ast} :=  \Hom_k({\Cal P}, k),$
and ${\bf q} = \{q_1,\ldots,q_s\}\subset {\Cal P}$ 
are {\em triangular} if  
$\ell_i(q_j) = 0,$ for each $i<j$.

Denoting, for each $k$-vector subspace $L \subset {\Cal P}^{\ast}$,  
$$\Frak{P}(L) := \{g\in {\Cal P} : \ell(g) = 0, \Forall \ell\in L\}$$
and, for each $k$-vector subspace $P \subset {\Cal P}$,  
$$\Frak{L}(P) := \{\ell \in {\Cal P}^{\ast} : \ell(g) = 0, \Forall g\in P\},$$
we recall \cite{Mac1,Mac2, M$^3_2$, AMM, SPES} that
the mutually inverse maps $\Frak{L}(\cdot)$ and $\Frak{P}(\cdot)$ 
give a biunivocal, inclusion reversing, correspondence 
between the set of the zero-dimensional ideals  $P \subset {\Cal P}$ and
the set of the finite  $k$-dimensional ${\Cal P}$-modules $L \subset {\Cal
P}^{\ast}$. 

Denoting, for each $\tau\in {\Cal T}$, 
$M(\tau) : {\Cal P} \rightarrow k$ the morphism defined
by 
$M(\tau) = c(f,\tau)$ for each $f = \sum_{t\in{\Cal T}} c(f,t) t\in {\Cal T}$
and ${\Bbb M} := \{M(\tau) : \tau\in{\Cal T}\}$, then
$\Span_k({\Bbb M})  \subset {\Cal P}^{\ast}$ is the set of the {\em
Noetherian equations} \cite{Mac1,Mac2, SPES} of ${\Cal P}$.
If we set 
$$\ell(\tau) := M(\tau) + \sum_{t\in{\bf T}({\sf I})}
 \gamma(t,\tau,<) M(t)\in\Span_k({\Bbb M}),$$
for each ${\sf m}$-closed ideal ${\sf I}\subset {\Cal P}$ and
each $\tau\in {\bf N}({\sf I})$,
then ${\sf I}$ can be characterized 
\cite{Mac1,Mac2, M$^3_2$, SPES} by its unique
{\em Macaulay basis}
$\{\ell(\tau) : \tau\in {\bf N}({\sf I})\}$, which is a $k$-basis of
the $k$-subspace
$\{\ell \in {\Cal P}^{\ast} : \ell(g) = 0, \forall g\in {\sf I}\}\subset\Span_k({\Bbb M})
  \subset {\Cal P}^{\ast}$ consisting of all the Noetherian equations of ${\sf I}$.

Therefore, each zero-dimensional ideal
${\sf I}\subset{\Cal P}$
can be considered as {\em given} if we know
 the set ${\sf Z}:={\Cal Z}({\sf I})$ and,
for each ${\sf a}\in{\sf Z}$, a Macaulay basis of the corresponding
primary component of ${\sf I}$.
For each ${\sf a}\in{\sf Z}$,  ${\sf a} :=
(a_{1},\ldots,a_{n})$, let us therefore denote $\lambda_{\sf a} : \Cal P \mapsto \Cal P$
the translation $\lambda_{\sf a}(X_i) = X_i+a_i,\Forall i$,
${\frak m}_{\sf a} = (X_1-a_{1},\ldots,X_n-a_{n})$,
${\frak q}_{\sf a}$ the ${\frak m}_{\sf a}$-primary component of ${\sf I}$,
$\{\ell(\upsilon) : \upsilon\in{\bf N}_<(\lambda_{\sf a}({\frak q}_{\sf
a}))\}$ is
the Macaulay basis of $\lambda_{\sf a}({\frak q}_{\sf
a})$,
$\ell_{\upsilon{\sf a}}$ 
for each $\upsilon\in{\bf N}_<(\lambda_{\sf a}({\frak q}_{\sf
a}))$  the Macaulay equation $\ell_{\upsilon{\sf a}} := \ell(\upsilon)$.
Setting
$s := \sum_{{\sf a}\in{\sf Z}} \deg({\frak q}_{\sf a})$
and ${\Bbb L} := \{\lambda_1,\ldots,\lambda_s\}  :=
\{\ell_{\upsilon{\sf a}}\lambda_{\sf a} :
\upsilon\in{\bf N}_<(\lambda_{\sf a}({\frak q}_{\sf
a})), {\sf a}\in{\sf Z}\},$ 
we know that 
$\Span_k({\Bbb L}) = {\Frak L}({\sf I})$ 
and ${\sf I} = {\Frak P}(\Span_k({\Bbb L}));$
moreover \cite{Mac1,Mac2, M, SPES} 
we can wlog assume ${\Bbb L}$ to be ordered so that, for
each $\sigma$,
${\Frak P}(\Span_k(\{\lambda_1,\ldots,\lambda_\sigma\}))$ 
is an ideal.
We also set $${\sf X} := \{{\sf x}_1,\ldots,{\sf x}_s\} :=
\{({\sf a}, \upsilon) :
\upsilon\in{\bf N}_<({\frak q}_{\sf a}), {\sf a}\in{\sf Z}\}$$
enumerated so that
${\sf x}_j = ({\sf a},\upsilon) 
\iff \lambda_j = \ell_{\upsilon{\sf a}}\lambda_{\sf a}$
and we set, for each $j, 1\leq j \leq s$, $M(\lambda_j) := M(\upsilon)\lambda_{\sf a}$
where $\lambda_j = \ell_{\upsilon{\sf a}}\lambda_{\sf a}$.

Under the assumption that 
$\lambda = M(\lambda)$ for each $\lambda\in{\Bbb L}$, --- this is equivalent as
assuming that each
$\lambda_{\sf a}({\frak q}_{\sf
a})$ is a monomial ideal ---
Cerlienco--Mureddu Algorithm (Alg.~\ref{CMA}) associates to each such
sets\footnote{Both \cite{CM1, CM2} and \cite{Iran} express the algorithm and the
structural result in terms of ${\sf X}$. The result reported in \S~\ref{ABUC}
compelled us to express everything in terms of ${\Bbb L}$. Also, the conditions of
Th.~\ref{IDDU} have been suitably relabelled.}
${\Bbb L}$ and ${\sf X}$, an order ideal\footnote{A subset ${\bf
N} \subset {\Cal T}$ is called an order ideal if it satisfies 
$st\in{\bf N} \then t\in{\bf N},$ for each $s,t\in{\Cal T}.$} 
${\bf N} := {\bf N}({\Bbb L})$ and 
 a bijection  $\Phi := \Phi({\Bbb L}):  {\Bbb L} \mapsto {\bf N}$,
which, as we will proof later, satisfies
${\bf N}_<({\Bbb L}) = {\bf N}({\frak P}({\Bbb L}))$
for the lexicographical ordering induced 
by $X_1 < \cdots < X_n$.

\begin{definition} The ordered sets ${\Bbb L}({\sf I}) := {\Bbb L}$ 
and
${\sf X}({\sf I}) := {\sf X}$ are called, respectively, a 
{\em Macaulay representation}\index{Macaulay representation} and a
{\em CM-scheleton} of ${\sf I} := {\frak P}({\Bbb L})$;
each $\lambda = \ell_{\upsilon{\sf a}}\lambda_{\sf a}\in{\Bbb L}$
is called a {\em CM-functional} and each ${\sf x} = ({\sf a},\upsilon)\in{\sf X}$ 
a {\em CM-card}.

If, moreover, for each $\lambda = \ell_{\upsilon{\sf
a}}\lambda_{\sf a}\in{\Bbb L},$
$\lambda = M(\lambda) =  M(\upsilon)\lambda_{\sf a}$, then 
${\sf I}$ is called a {\em CM-ideal},
${\sf X}$ its {\em CM-scheme}, and each
${\sf x} = ({\sf a},\upsilon)\in{\sf X}$ a {\em CM-condition}.
\qed\end{definition}

We  need also to consider, for each $m < n$, the set
\begin{eqnarray*}
{\Cal T}[1,m] &:=& {\Cal T} \cap k[X_1,\ldots,X_m]\\
&=& \{X_1^{a_1}\cdots X_m^{a_m} :
(a_1,\ldots,a_m)\in {\Bbb N}^{m}\},\\
{\Bbb M}[1,m] &:=& \{M(\tau) : \tau\in{\Cal T}[1,m]\} 
\end{eqnarray*}
and the projection  
$$\pi_m : k^n \mapsto k^m,\pi_m(x_{1},\ldots, x_{n}) =  (x_{1},\ldots, x_{m}),$$
which we freely use to denote also the projections
$\pi_m : {\Cal T} \simeq {\Bbb N}^n \mapsto {\Bbb N}^m \simeq {\Cal T}[1,m],$
$\pi_m(X_1^{\alpha_1}\cdots X_n^{\alpha_n}) = X_1^{\alpha_1}\cdots X_m^{\alpha_m}$,
$\pi_m : {\Bbb M} \mapsto {\Bbb M}[1,m],$
$\pi_m(M(\tau)) = M(\pi_m(\tau)),$
and
$\pi_m : k^n \times {\Cal T} \mapsto k^m \times {\Cal T}[1,m],$
$\pi_m({\sf a},\tau) = (\pi_m({\sf a}),\pi_m(\tau)).$
Also, for a CM-condition ${\sf x} = ({\sf a},\upsilon)\in k^m \times {\Cal T}[1,m]$
we also set 
$\pi_{m}(\lambda) := \pi_{m}(M(\upsilon)\lambda_{\sf a}) := 
M(\pi_{m}(\upsilon))\lambda_{\pi_{m}({\sf a})}.$ 

Remark that a  CM-ideal  is radical iff
$\upsilon = 1 \forall ({\sf a},\upsilon)\in{\sf X}$ iff $\#{\sf X} = \#{\sf Z}$.
If this happens, with an abuse of notation, we
simply identify each CM-condition ${\sf x}_i = ({\sf a}_i,1)$ and the corresponding
CM-functional $\lambda_i = \lambda_{{\sf a}_i}$ with ${\sf a}_i.$

Let  ${\sf I}\subset{\Cal P}$ be a
CM-ideal,
and, using the same notation as above,
${\Bbb L} := \{\lambda_1,\ldots,\lambda_s\}$
and 
${\sf X} := \{{\sf x}_1,\ldots,{\sf x}_s\} \subset k^n \times {\Cal T},$  
${\sf x}_i = ({\sf a}_i,\upsilon_i),$
${\sf a}_i := (a_{i1},\ldots,a_{in}),$
%% \upsilon_i =\prod_{l=1}^n X_l^{\alpha_{il}}$$
$\lambda_i =  M(\upsilon_i)\lambda_{{\sf a}_i}$
a Macaulay representation and a CM-scheme of ${\sf I}$;
let us denote ${\bf N} := {\bf N}({\sf X})$ and $\Phi := \Phi({\sf X})$ the result of 
Cer\-lien\-co--Mu\-red\-du Correspondence which satisfies \cite{CM1,CM2}
${\bf N} := {\bf N}({\sf I}).$

Since ${\bf N}$ is an order ideal, 
${\bf T} := {\Cal T}\setminus {\bf N}$ is a monomial ideal
whose minimal basis 
${\bf G} := \{{\sf t}_1,\ldots,{\sf t}_r\}$ will be ordered so that
${\sf t}_1 < {\sf t}_2 < \ldots < {\sf t}_r;$
we also set
$${\bf B} :=\left( \{1\} \cup \{X_i\tau :  \tau\in
{\bf N}\}\right)\setminus {\bf N}.$$

We extend the ordering of ${\sf X}$ to  
${\bf N} =  \{\tau_1,\ldots, \tau_s\}$ enumerating it so that
$\tau_\sigma = \Phi({\sf x}_\sigma),$ for each $\sigma$ 
and let us denote the ordering of ${\sf X}$ and ${\bf N}$ by $\prec$ so that
for each $\alpha, \beta, \tau_\alpha \prec \tau_\beta, {\sf x}_\alpha \prec {\sf x}_\beta 
\iff \alpha < \beta.$


For a term $\tau :=  X_1^{d_1}\cdots X_n^{d_n}\in{\Cal
T}\setminus {\bf N}({\sf X})$ such that ${\bf N}\cup\{\tau\}$ is an
order ideal,
we define, for each $m, 1 \leq m \leq n$: 
\begin{eqnarray*}
{\sf N}_{m}(\tau) &:=&  \{\omega\in{\Cal T}[1,m] :  
\tau > \omega X_{m+1}^{d_{m+1}}\cdots X_n^{d_n}\in {\bf N}\},\\
{\sf A}_{m}(\tau)  &:=& 
\{\Phi^{-1}(\omega X_{m+1}^{d_{m+1}}\cdots X_n^{d_n}) : 
\omega\in {\sf N}_{m}(\tau)\},\\  
{\sf B}_{m}(\tau)  &:=& \pi_{m}({\sf A}_{m}(\tau)),\\
{\sf C}_{m}(\tau)  &:=& 
\{\pi_{m}(\lambda)\in{\sf B}_{m}(\tau) : 
\pi_{m-1}(\lambda)\not\in{\sf B}_{m-1}(\tau)\},
\\{\sf L}_{m}(\tau)  &:=& 
\{\lambda \in{\Bbb L} :
\pi_{m}(\lambda)\in{\sf C}_{m}(\tau)\},\\
{\sf D}_{m}(\tau) &:=& \{{\sf x}_i
\in{\sf X} :
\pi_{m}(\lambda_i)\in{\sf C}_{m}(\tau)\},\\ 
{\sf M}_{m}(\tau)  &:=& 
\{\omega\in {\Cal T}[1,m] : 
\\&&\quad
\omega < X_{m}^{d_{m}},
\omega X_{m+1}^{d_{m+1}}\cdots X_n^{d_n}\in {\bf N}\},\\
{\Frak M}_m(\tau) &:=& \{\omega\in{\sf M}_m(\tau) : \omega \prec \tau\},
\end{eqnarray*}
where, with  slight abuse of notation, we have   
${\sf N}_{n}(\tau) := \{\omega\in{\Cal T} : \omega <\tau\},$
${\sf A}_{n}(\tau) %:= {\sf B}_{n}(\tau) 
:= \{\lambda : \Phi(\lambda) < \tau\},$
${\sf C}_{1}(\tau) := {\sf B}_{1}(\tau).$

Th.~\ref{IDDU} below proves, if ${\sf I}$ is radical, the existence,
for each $m, 1 \leq m\leq n$ and each 
$\tau := X_1^{d_1}\cdots X_n^{d_n}\in{\bf G}$ (resp. ${\bf N}$)
of a suitable polynomial 
$g_{m\tau}\in \Span_k\left(\{X_m^{d_m}\}\cup{\sf M}_{m}(\tau)\right)$
--- resp. $\Span_k\left(\{X_m^{d_m}\}\cup{\Frak M}_{m}(\tau)\right).$
In terms of such polynomials we define
for each
$\tau \in{\bf N}\cup{\bf G}$ and each $m, 1 \leq m\leq n$:
\begin{eqnarray*}
{\sf R}_{m}(\tau) &:=& 
\{{\sf b}\in{\sf C}_{m}(\tau) : \prod_{\nu=1}^{m-1} g_{\nu\tau}({\sf
b})\neq 0\},\cr 
{\sf E}_{m}(\tau) &:=& {\bf N}({\sf
R}_{m}(\tau)),\cr 
{\sf S}_{m}(\tau) &:=&
 \{\pi_m({\sf a})\in{\sf R}_{m}(\tau) : {\sf a} \prec \Phi^{-1}(\tau)\},\cr
{\sf F}_{m}(\tau) &:=&  {\bf N}({\sf S}_{m}(\tau)).
\end{eqnarray*}
Denote for each $\tau\in{\bf N}$
\begin{eqnarray*}
{\Bbb L}(\tau) &:=& \{\lambda\in{\Bbb L} : \lambda \prec \Phi^{-1}(\tau)\} =
\{\lambda\in{\Bbb L} : \Phi(\lambda) \prec \tau\}, 
\\{\Frak X}(\tau) &:=& \{{\sf x}_j : \lambda_j\in {\Bbb L}(\tau)\},
\\{\sf I}({\Bbb L}(\tau)) &:=& {\Frak P}(\Span_k({\Bbb L}(\tau))), 
\end{eqnarray*}
and, for each $\tau\in{\bf N}\cup{\bf B}$:
\begin{eqnarray*}
{\Frak N}(\tau) &:=& \{\omega\in{\bf N} : \omega \prec \tau\},
\\{\Frak M}_m(\tau) &:=& \{\omega\in{\sf M}_m : \omega \prec \tau\}.
\end{eqnarray*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{An instructive section on \\recalls}
%
\begin{algorithm}[Cerlienco--Mureddu]\label{CMA} 
\cite{CM1,CM2}
{\em \  The algorithm is inductive on $s = \#({\Bbb L})$, the only possible
solution, if $s=1$, being ${\bf N} = \{1\}, \Phi({\lambda}_1) = 1$: consider the
subset 
${\Bbb L}' := \{\lambda_1,\ldots,\lambda_{s-1}\}$, the corresponding order ideal 
${\bf N}' := {\bf N}({\Bbb L}')$ and bijection $\Phi' := \Phi({\Bbb L}')$, and 
set 
$$m := \max\left(j : \exists i < s : \pi_j(\lambda_i) = \pi_j(\lambda_s)\right),$$ 
$d := \#\{\lambda_i, i < s : \pi_m(\lambda_i) = \pi_m(\lambda_s), 
\Phi'(\lambda_i)\in {\Cal T}[1,m+1]\}$,
 ${\sf W} := \{\lambda_i : \Phi'(\lambda_i) = 
\omega_i X_{m+1}^{d}, \omega_i\in {\Cal T}[1,m]\} \cup\{\lambda_s\}$,
 ${\sf V} := \pi_{m}({\sf W}),$
$\omega := \Phi({\sf V})(\pi_{m}(\lambda_s)),$
$t_s :=\omega X_{m+1}^{d},$
where ${\bf N}({\sf V})$ and $\Phi({\sf V})$ are the result of the
application of the present algorithm to ${\sf V}$,  which can be inductively applied 
since $\#({\sf V}) \leq s-1$. 
We then define $${\bf N} := {\bf N}' \cup \{t_s\}
\mbox{\rm\ and }
%%  \Phi(\lambda_i) := \left\{\begin{array}{ll}
%%  \Phi'(\lambda_i)& i < s\cr t_s& i = s.\cr\end{array}\right.$$
\Phi(\lambda_i) := \begin{cases}\Phi'(\lambda_i)& i < s\cr t_s& i
=s.\cr\end{cases}\qed$$
%%%%% $\Phi(\lambda_i) := \cases{\Phi'(\lambda_i)& i < s\cr t_s& i =s.\cr}$  
}\end{algorithm}

In the seminal paper \cite{L}, Lazard described the structure of a lexicographical Gr\"obner basis of
a polynomial ideal in the case $n=2$: 
%
\begin{theorem}[Lazard Structural Theorem] \

Let 
${\Cal P} :=k[X_1,X_2]$ and let $<$ be the
lexicographical ordering induced by  $X_1 < X_2$.
Let ${\sf I} \subset {\Cal P}$ be an ideal and let
 $\{f_0,f_1,\ldots,f_k\}$ be a Gr\"obner basis of ${\sf I}$ ordered so that 
${\bf T}(f_0) < {\bf T}(f_1) < \cdots  < {\bf T}(f_k).$
Then
$f_0 = P G_1\cdots G_{k+1},$
$f_j = P H_j G_{j+1}\cdots G_{k+1}, 1 \leq j < k,$
$f_k = P H_k G_{k+1},$ 
where
\begin{eqnarray*}
&&P \mbox{ is the primitive part of }f_0\in k[X_1][X_2];\cr
&&G_i\in k[X_1], 1\leq i \leq k+1;\cr
&&H_i\in k[X_1][X_2] \mbox{is a monic polynomial of degree }d(i), \forall i;\cr
&&d(1) < d(2) < \cdots < d(k);\cr
&&H_{i+1} \in (G_1\cdots G_{i}, 
%H_1 G_{2}\cdots G_{i}, 
\ldots, H_j G_{j+1}\cdots G_i, 
\ldots
%, H_{i-1} G_i
, H_i), \forall i.
\end{eqnarray*}
%
%\begin{description}
%\item $P$ is the primitive part of $f_0\in k[X_1][X_2]$;
%\item $G_i\in k[X_1], 1\leq i \leq k+1$;
%\item $H_i\in k[X_1][X_2]$ is a monic polynomial of degree $d(i), \forall i$;
%\item $d(1) < d(2) < \cdots < d(k)$;
%\item $H_{i+1} \in (G_1\cdots G_{i}, 
%H_1 G_{2}\cdots G_{i}, 
%\ldots, H_j G_{j+1}\cdots G_i, 
%\ldots, H_{i-1} G_i, H_i), \forall i$.
%\end{description}
\qed\end{theorem}
which was subsumed by Gianni \cite{G} and Kalkbrener \cite{K} in 
\begin{theorem}[Gianni---Kalkbrener] 
Let ${\sf I} \subset {\Cal P}$ be an ideal, $<$ be the lexicographical ordering induced by 
$X_1 < \cdots < X_n$;
$G := \{g_1,\ldots,g_s\}$ be a Gr\"obner basis of ${\sf I}$ w.r.t. $<$, enumerated 
in such a way that
$${\bf T}(g_1) < {\bf T}(g_2) < \ldots < {\bf T}(g_{s-1})< {\bf T}(g_{s});$$
denote, for each $d, 1\leq d \leq n$, $\delta\in{\Bbb N}$,
\begin{eqnarray*}
G_d &:=& G \cap k[X_1,\ldots,X_d],\\
G_{d\delta} &:=& \{g\in G, g\in k[X_1,\ldots,X_d],  \deg_i(g) \leq \delta\}
\end{eqnarray*}
and remark that
\begin{eqnarray*}
G_{11} &\subseteq& G_{12} \subseteq \ldots 
 \subseteq G_1 
\subseteq \ldots  \subseteq G_{d-1}
\subseteq \ldots 
\\&\subseteq& G_{d\delta}  \subseteq G_{d\delta+1} \subseteq \ldots  \subseteq G_{d} \subseteq
\ldots
\end{eqnarray*}
 and each $G_{d\delta}$ is a section of both $ G_{d\delta+1}$ and $G_{d}$.

Each $G_d$ and $\Lp_{d\delta}(G) := \{\Lp(g), g\in G_{d\delta}\}$ are
Gr\"obner bases w.r.t. $<$ of, respectively,
${\sf I}_d := {\sf I} \cap k[X_1,\ldots,X_d]$ and
 $\Lp_{d\delta}({\sf I}) := \{\Lp(g), g\in {\sf I}_d, \deg_i(g) \leq \delta\}$.
Moreover, for each $d, 1\leq d \leq n$ and each 
$\alpha := (b_1,\ldots,b_d)\in{\Cal Z}({\sf I}_d)$,
denoting
 $\sigma$ the minimal value such that $\Phi_\alpha(\Lp(g_\sigma)) \neq 0$ and
 $j,\delta$ the value such that 
$$g_\sigma = \Lp(g_\sigma) X_j^{\delta+1} +
\cdots\in k[X_1,\ldots,X_j]\setminus k[X_1,\ldots,X_{j-1}]$$
it holds
\begin{eqnarray*}
&&j = d+1,\cr
&&\mbox{\it for each\ }  g\in G_d, \Phi_\alpha(g) = 0,\cr
&&\mbox{\it for each\ }  g\in G_{d+1\delta}, \Phi_\alpha(g) = 0,\cr
&&\Phi_\alpha(g_\sigma) = \gcd\left(\Phi_\alpha(g) : g\in G_{d+1}\right)
\in{\sf k}[X_{d+1}],\cr 
&&\mbox{\it for each\ }  b\in {\sf k}, 
\\&&\quad (b_1,\ldots,b_d,b)\in{\Cal Z}({\sf I}_{d+1}) \iff 
\Phi_\alpha(g_\sigma)(b) = 0.
\end{eqnarray*}
\qed\end{theorem}

Merging Lazard Structural Theorem and Cer\-lien\-co--Mu\-red\-du Algorithm, and using
M\"oller's Algorithm
\cite{BM, FGLM, M$^3$, AMM} as a proving tool, \cite{Iran} proves
%
\begin{theorem}\label{IDDU} 
For a {\em radical} 0-dimensional ideal ${\sf I}$, given by a CM-scheme
${\sf X}$ of CM-conditions, using the same notation as above, it holds
\begin{description}
\item[{\bf (A)}] ${\bf N} := {\bf N}({\sf I}).$
\item[{\bf (B)}] ${\bf G}({\sf I}) = {\bf G} =  
\{{\sf t}_1,\ldots,{\sf t}_r\}, {\sf t}_1 < {\sf t}_2 < \ldots < {\sf t}_r.$
\item[{\bf (C)}] ${\bf B}({\sf I}) = {\bf B}.$
\item[{\bf (D)}] For each $\tau\in{\bf N}$ there is a unique polynomial
$$f_\tau := \tau - \sum_{\omega\in{\Frak N}(\tau)} c(f_\tau,\omega) \omega$$
such that $\lambda(f_\tau) = 0,$ for each $\lambda \in {\Bbb L}(\tau).$
\item[{\bf (E)}] For each $\tau\in{\bf G}$ there is a unique polynomial
$$f_\tau := \tau - \sum_{\omega\in{\bf N}} c(f_\tau,\omega) \omega$$
such that $\lambda(f_\tau) = 0,$ for each $\lambda \in {\Bbb L}.$
\item[{\bf (F)}] For each $\tau\in{\bf B}$ there is a polynomial
$$f_\tau := \tau - \sum_{\omega\in{\Frak N}(\tau)} c(f_\tau,\omega) \omega$$
such that $\lambda(f_\tau) = 0,$ for each $\lambda \in {\Bbb L}.$
\item[{\bf (G)}] The reduced Gr\"obner basis of ${\sf I}$ is
$${\Cal G}({\sf I}) := \{f_\tau : \tau\in {\bf G}\}.$$
moreover, for each $\tau\in{\bf N}$, ${\bf T}(f_\tau) = \tau.$
\item[{\bf (H)}] The border basis of ${\sf I}$ is
$${\Cal B}({\sf I}) := \{f_\tau : \tau\in {\bf B}\}.$$
\item[{\bf (I)}] For each $\nu, 1 \leq \nu < n$, 
\begin{description}
\item[] let $j_\nu$ be the value such that 
${\sf t}_{j_\nu} < X_{\nu+1} \leq {\sf t}_{j_\nu+1};$ then 
$\{f_{{\sf t}_1},\ldots,f_{{\sf t}_{j_\nu}}\}$  is a minimal Gr\"obner basis of both
${\frak P}(\Span_k(\pi_\nu({\Bbb L})))$ and of
${\sf I} \cap k[X_1,\ldots,X_{\nu}];$
\item[] for each
$\delta\in{\Bbb N}$, let $j({\nu\delta})$ be the value
such that ${\sf t}_{j({\nu\delta})} < X_{\nu+1}^{\delta} \leq {\sf t}_{j({\nu\delta})+1};$
then $\{\Lp(f_{{\sf t}_1}),\ldots,\Lp(f_{{\sf t}_{j_{\nu\delta}}})\}$
is a Gr\"obner basis of  ${\sf I}({\sf Y}_{\nu \delta})$.
\end{description}
\item[{\bf (L)}] for each $j, 1 \leq j \leq s$,
$\lambda_j(f_{\tau_j}) \neq 0$ 
so that ${\Bbb L}$ and 
$\{\lambda_j(f_{\tau_j})^{-1} f_{\tau_j}, 1 \leq j \leq s\}$
are triangular.
\item[{\bf (M)}] For each $\tau :=  X_1^{d_1}\cdots X_n^{d_n}\in{\bf N},$ and each
$m, 1
\leq m\leq n$, there are unique polynomials
\begin{eqnarray*}
\gamma_{m\tau} &:=& X_m^{d_m} + \sum_{\omega\in{\sf F}_{m}(\tau)} c(\gamma_{m\tau},
\omega) \omega,\\
\gamma_{m\delta\tau} &:=& X_m + \sum_{\omega\in{\sf F}_{m\delta}(\tau)}
 c(\gamma_{m\tau},\omega) \omega, \quad  1 \leq \delta\leq d_m 
\end{eqnarray*}
such that 
\begin{eqnarray*}
&& \pi_m(\lambda)(\gamma_{m\tau}) = 0,\forall
\lambda\in {\sf Y}_{\nu\delta}(\tau), \lambda
\prec \Phi^{-1}(\tau);
\\&&\pi_m(\lambda)(\gamma_{m\tau}) = 0,\forall \lambda\in {\sf L}_{m}(\tau),
\lambda
\prec \Phi^{-1}(\tau);
\\&& \gamma_{m\tau} = \prod_\delta \gamma_{m\delta\tau}.
\end{eqnarray*}
\item[{\bf (N)}] For each $\tau :=  X_1^{d_1}\cdots X_n^{d_n}\in{\bf G},$ and each $m, 1
\leq m\leq n$, there are unique polynomials
\begin{eqnarray*}
\gamma_{m\tau} &:=& X_m^{d_m} + \sum_{\omega\in{\sf E}_{m}(\tau)}
c(\gamma_{m\tau}, \omega)
\omega,
\\
\gamma_{m\delta\tau} &:=& X_m + \sum_{\omega\in{\sf E}_{m\delta}(\tau)}
 c(\gamma_{m\tau},\omega) \omega, \quad  1 \leq \delta\leq d_m
\end{eqnarray*}
such that 
\begin{eqnarray*}
&& \pi_m(\lambda)(\gamma_{m\tau}) = 0, \forall
\lambda\in {\sf Y}_{m\delta}(\tau), 
\\&& \pi_m(\lambda)(\gamma_{m\tau}) = 0, \forall \lambda\in {\sf L}_{m}(\tau),
\\&& \gamma_{m\tau} = \prod_\delta \gamma_{m\delta\tau}.
\end{eqnarray*}
\item[{\bf (O)}] For each $\tau = X_1^{d_1}\cdots X_\nu^{d_\nu}\in{\bf N},$ 
there are 
$$L_\tau \in k[X_1,\ldots,X_{\nu-1}]$$ and a unique monic polynomial
$$P_\tau = X_\nu^{d_\nu} + \sum_{\omega\in{\sf F}_{\nu}(\tau)} c(P_\tau, \omega) \omega
\in k[X_1,\ldots,X_{\nu-1}][X_\nu]$$
so that $H_\tau  := L_\tau P_\tau$
are such that
\begin{eqnarray*}
&& {\bf T}(H_\tau) = \tau,\Lp(H_\tau) = L_\tau,
\\&&\pi_{\nu-1}(\lambda)(L_\tau) = 0,\forall\lambda\in {\Bbb L}(\tau),
\\&& \pi_{\nu}(\lambda)(P_\tau) = 0,\forall\lambda\in {\sf L}_{\nu}(\tau), 
\\&& \pi_{\nu}(\lambda)(H_\tau) = 0,\forall\lambda\in {\Bbb L} : \lambda
\prec \Phi^{-1}(\tau).
\end{eqnarray*}
\item[{\bf (P)}]
For each $\tau = X_1^{d_1}\cdots X_\nu^{d_\nu}\in{\bf N},$ 
there are 
$$L_\tau \in k[X_1,\ldots,X_{\nu-1}]$$ and a unique monic polynomial
$$P_\tau = X_\nu^{d_\nu} + \sum_{\omega\in{\sf F}_{\nu}(\tau)} c(\rho_\tau, \omega) \omega
\in k[X_1,\ldots,X_{\nu-1}][X_\nu]$$
so that $H_\tau  := L_\tau P_\tau$
are such that
\begin{eqnarray*}
&&{\bf T}(H_\tau) = \tau,\cr
%%\Lp(H_i) = L_i,\cr
&&\pi_{\nu-1}(\lambda)(L_\tau) = 0 \mbox{ for each } \lambda\in {\Bbb L}(\tau),\cr 
&&\pi_{\nu}(\lambda)(P_\tau) = 0 \mbox{ for each } \lambda\in {\sf L}_{\nu}(\tau),\cr
&&\pi_{\nu}(\lambda_i)(H_\tau) = 0 \mbox{ for each } \lambda_i\in {\Bbb L} : {\sf x}_i \prec
\Phi^{-1}(\tau). 
\end{eqnarray*}
\item[{\bf (Q)}] $L_i, P_i, H_i$, $1\leq i \leq r$ satisfy  
\begin{description}
\item[] $\{H_1,\ldots,H_r\}$ is a minimal Gr\"obner basis of ${\sf I},$ 
\item[] for each $\nu, 1 \leq \nu < n,$
$\{H_1,\ldots,H_{j_\nu}\}$ is a  minimal Gr\"obner basis of  
${\sf I} \cap k[X_1,\ldots,X_{\nu}]$
and of ${\sf I}(\pi_\nu({\sf X}));$
\item[] for each $\nu, 1 \leq \nu < n,$ 
$\{L_1,\ldots,L_{j_{\nu\delta})}\}$
is a Gr\"obner basis of 
${\sf I}({\Bbb Y}_{\nu \delta}).$ 
\end{description}
\item[{\bf (R)}] For each $i, 2\leq i \leq r$,
$P_i\in\left(H_j, j< i\right) : L_i$.
\item[{\bf (S)}] For each $j, 1 \leq j \leq s$,
$\lambda_j(H_{\tau_j}) \neq 0$;
${\Bbb L}$ is triangular to
$\{\lambda_j(H_{\tau_j})^{-1} H_{\tau_j}, 1 \leq j \leq s\}$.
\item[{\bf (T)}] for each $\tau :=  X_1^{d_1}\cdots X_n^{d_n}\in{\bf N}$ and each
$m, 1 \leq m\leq n$,  there are polynomials
$$g_{m\tau} := X_m^{d_m} + \sum_{\omega\in{\Frak M}_{m}(\tau)} c(g_{m\tau},\omega) \omega$$
such that $\lambda(g_{m\tau}) = 0,$ 
for each $\lambda\in {\sf L}_{m}(\tau), \lambda \prec \Phi^{-1}(\tau).$
\item[{\bf (V)}] for each $\tau :=  X_1^{d_1}\cdots X_n^{d_n}\in{\bf G},$ and each $m, 1
\leq m\leq n$,  there are polynomials
$$g_{m\tau} := X_m^{d_m} + \sum_{\omega\in{\sf M}_{m}(\tau)} c(g_{m\tau},\omega) \omega$$
such that  $\lambda(g_{m\tau}) = 0,$ 
for each $\lambda\in {\sf L}_{m}(\tau).$
\item[{\bf (W)}] For each $\tau = X_1^{d_1}\cdots X_\nu^{d_\nu}\in{\bf N},$ 
there are 
$$l_\tau \in k[X_1,\ldots,X_{\nu-1}]$$ and a monic polynomial
$$p_\tau = X_\nu^{d_\nu} + \sum_{\omega\in{\Frak M}_{\nu}(\tau)} c(p_\tau,\omega) \omega
\in k[X_1,\ldots,X_{\nu-1}][X_\nu]$$
so that $h_\tau  := l_\tau p_\tau$
are such that
\begin{eqnarray*}
&&{\bf T}(h_\tau) = \tau,\cr
&&\Lp(h_\tau) = l_\tau,\cr
&&l_\tau(\pi_{\nu-1}({\sf a})) = 0 \mbox{ for each } {\sf a}\in {\Frak X}(\tau), \cr
&&p_\tau({\sf a}) = 0 \mbox{ for each } {\sf a}\in {\sf D}_{\nu}(\tau),\cr
&&h_\tau({\sf a}) = 0 \mbox{ for each } {\sf a}\in{\sf X} \mbox{ s. t. } {\sf a} \prec
\Phi^{-1}(\tau).
\end{eqnarray*}
\item[{\bf (X)}] For each $i, 1 \leq i \leq r$,
there are 
$l_i \in k[X_1,\ldots,X_{\nu-1}]$ and a monic polynomial
$$p_i = X_\nu^{d_\nu} + \sum_{\omega\in{\sf M}_{\nu}(\tau)} c(p_i, \omega) \omega
\in k[X_1,\ldots,X_{\nu-1}][X_\nu]$$
so that $h_i  := l_i p_i$
are such that
\begin{eqnarray*}
&&{\bf T}(h_i) = {\sf t}_i = X_1^{d_1}\cdots X_\nu^{d_\nu}\in{\bf G}\cap{\Cal
T}[1,\nu],\cr  
&&l_i(\pi_{\nu-1}({\sf a})) = 0, 
\mbox{ for each } {\sf a}\in \cup_{m=1}^{\nu-1}{\sf D}_{m}({\sf t}_i),\cr
&&p_i({\sf a}) = 0,\mbox{ for each } {\sf a}\in {\sf D}_{\nu}({\sf t}_i),\cr
&&h_i({\sf a}) = 0,\mbox{ for each } {\sf a}\in {\sf X}. 
\end{eqnarray*}
\item[{\bf (Z)}] $l_i, p_i, h_i$, $1\leq i \leq r$ satisfy  
\begin{eqnarray*}
&&\{h_1,\ldots,h_r\} \mbox{ is a minimal Gr\"obner basis of }
{\sf I};\cr 
&&\mbox{for each }\nu, 1 \leq \nu < n,
\{h_1,\ldots,h_{j_\nu}\} \mbox{ is a minimal}\cr
&&\mbox{\quad Gr\"obner basis of }
{\sf I} \cap k[X_1,\ldots,X_{\nu}] \mbox{  and of }
\cr&&\quad {\sf I}(\pi_\nu({\sf X}));\cr
&&\mbox{for each }\nu, 1 \leq \nu < n, 
\{l_1,\ldots,l_{j_{\nu\delta})}\}\mbox{ 
is a Gr\"obner}\cr
&&\mbox{\quad 
basis of }
{\sf I}({\sf Y}_{\nu \delta});\cr
&&\mbox{for each }i, 2\leq i \leq r,
p_i\in\left(h_j, j< i\right) : l_i;\cr
&&\mbox{for each }j, 1 \leq j \leq s,
\lambda_j(h_{\tau_j}) \neq 0;\cr
&&{\Bbb L}({\sf I})\mbox{ is triangular to 
}\{\lambda_j(h_{\tau_j})^{-1} h_{\tau_j}, 1 \leq j \leq s\}.
\end{eqnarray*}
\qed\end{description}
\end{theorem}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{An original section on a result}
%
Since Cerlienco--Mureddu Algorithm applies to a CH-ideal not necessarily radical, it is
natural to query how much of Th.~\ref{IDDU} can be stated without the restriction of
radicality. The proof in \cite{Iran} allows to state that
\begin{proposition}\quad
For a CH-ideal ${\sf I}$, given by a CM-\-scheme ${\sf X}$ of
CM-con\-di\-tions, using the same notation as above, it holds:
\begin{enumerate}
\item ${\sf I}$ satisfies {\bf (A)};
\item if ${\sf I}$ satisfies {\bf (A)}, then it satisfies also {\bf (B-E)};
\item ${\sf I}$ satisfies {\bf (F)};
\item if ${\sf I}$ satisfies {\bf (F)}, then it satisfies also {\bf (G-H)};
\item if ${\sf I}$ satisfies {\bf (F),(P)}, then it satisfies also {\bf (I-L)}
and {\bf (Q-S)};
\item ${\sf I}$ satisfies {\bf (T-V)};
\item if ${\sf I}$ satisfies {\bf (F),(X)}, then it satisfies also {\bf (Z)}.
\end{enumerate}
\end{proposition}
\begin{proof} 1 is the content of \cite{CM2}; 2 is \cite{Iran} Cor.10.2 and Cor.10.3;
3 is a direct consequence of M\"oller Algorithm \cite{M$^3$} (cf. \cite{Iran} \S.11);
4 is \cite{Iran} Cor.10.5; 5 and 7 is \cite{Iran} Cor.10.9, Rem.10.10,  Prop.11.3; 6 is 
 \cite{Iran} Prop.10.6.
\qed\end{proof}
Further
${\bf (M-N)} \then {\bf (O-P)}$
and% \mbox{\rm\ and }
${\bf (T-V)} \then {\bf (W-X)}$, 
if ${\sf I}$ is radical, are a trivial
consequence of the obvious formula 
$\ell_{1{\sf a}}\lambda_{\sf a}(fg) = (fg)({\sf a}) = f({\sf a}) g({\sf a}) = 
\ell_{1{\sf a}}\lambda_{\sf a}(f)\ell_{1{\sf a}}\lambda_{\sf a}(g)$.
This gives the hint on how to redefine the sets ${\sf F}_{\nu}$ and ${\sf E}_{\nu}$ in
order to recover conditions {\bf (M-S)}: apply directly Leibniz Formula \cite{SM}
on  ${\Bbb M}$!
%
\begin{proposition}
\label{LeibLeib} 
For any
$f,g\in {\Cal P}$ and $\omega\in{\Cal T}$ it holds
$$M(\omega)(fg) = \sum_{\upsilon\in{\Cal T}\atop{\upsilon\tau=\omega}}
M(\upsilon)(f) M(\tau)(g)$$
\end{proposition}
%
\begin{proof}
For 
$$\begin{array}{rcccl}
f &=& \sum_{\upsilon\in{\Cal T}} c(f,\upsilon) \upsilon 
  &=& \sum_{\upsilon\in{\Cal T}} M(\upsilon)(f) \upsilon,\\ 
g &=& \sum_{\tau\in{\Cal T}} c(g,\tau) \tau  
  &=& \sum_{\tau\in{\Cal T}} M(\tau)(g)\tau,\\
fg &=& \sum_{\omega\in{\Cal T}} c(fg,\omega) \omega 
 &=& \sum_{\omega\in{\Cal T}} M(\omega)(fg) \omega 
\end{array}$$
and, for each $\omega\in{\Cal T}$, we have
\begin{eqnarray*}
M(\omega)(fg) 
=
c(fg,\omega)
&=&
\sum_{\upsilon\in{\Cal T}\atop\upsilon\tau=\omega}
c(f,\upsilon)  c(g,\tau)
\\&=& 
\sum_{\upsilon\in{\Cal T}\atop\upsilon\tau=\omega}
M(\upsilon)(f)  M(\tau)(g).
\qed\end{eqnarray*}
\end{proof}

We begin by remarking that, since 
$\#({\sf C}_{1}(\tau)) = \#({\sf M}_{1}(\tau))$, the polynomials
$g_{1\tau}$ produced by interpolation in \cite{Iran} are in fact
 unique. We therefore begin our construction by setting
$\gamma_{1\tau} := g_{1\tau}$ and,  inductively, for $m, 1 < m \leq n$,
we set{\tiny 
\begin{eqnarray*}
\zeta_{m\tau} &:=& \prod_{\nu=1}^{m-1} \gamma_{\nu\tau},\cr
{\sf Q}_{m}(\tau) &:=& \{M(\omega)\lambda_{{\sf a}} : \omega\in{\Cal T}[1,m-1],
                       {\sf a}\in{\sf Z}:={\Cal Z}({\sf I}),
%\cr&&\quad\quad
                       M(\omega)\lambda_{{\sf a}}(\zeta_{m\tau})\neq 0\},\cr
{\sf P}_{m}(\tau) &:=& \{M\left(\pi_{m}\left({\upsilon_i\over\omega}\right)\right)\lambda_{{\sf a}_i} : 
%\cr&&\quad\quad
M(\upsilon_i))\lambda_{{\sf a}_i}\in{\sf L}_{m}(\tau),
M(\omega)\lambda_{{\sf a}_i}\in{\sf Q}_{m}(\tau)\},\cr
{\sf R}_{m}(\tau) &:=& 
  \{\left(\pi_m({\sf a}_i),\pi_m\left({\upsilon_i\over\omega}\right)\right) : 
M\left(\pi_{m}\left({\upsilon_i\over\omega}\right)\right)\lambda_{{\sf a}_i}\in{\sf P}_{m}(\tau)\},\cr
{\sf E}_{m}(\tau) &:=& {\bf N}({\sf R}_{m}(\tau)),\cr
{\sf S}_{m}(\tau) &:=&
       \{\left(\pi_m({\sf a}_i),\pi_m\left({\upsilon_i\over\omega}\right)\right)\in{\sf
R}_{m}(\tau)
%\cr&&\quad\quad
({\sf a}_i, \upsilon_i) \prec \Phi^{-1}(\tau)\},\cr
{\sf F}_{m}(\tau) &:=& {\bf N}({\sf S}_{m}(\tau)).
\end{eqnarray*}}
%
\begin{lemma}\label{Etse} With this notation,
the CH-ideal ${\sf I}$ satisfies {\bf (M-P)}
\end{lemma}
%
\begin{proof} (cf. \cite{Iran} Prop.10.6, Cor.10.7)\
\begin{description}
\item[{\bf (N)}] Each $\gamma_{\nu\tau}$ is obtained by interpolation.
\item[{\bf (M)}] One has just to apply {\bf (M)} to ${\Frak X}(\tau)$.
\item[{\bf (O)}] It is sufficient to define
$$H_\tau  := \prod_{m=1}^n \gamma_{m\tau}, 
L_\tau  := \prod_{m=1}^{\nu(\tau)-1} \gamma_{m\tau},%$ and
P_\tau := \gamma_{\nu\tau}$$
and apply Leibniz Formula to prove that
for each
$\lambda := M(\upsilon_i)\lambda_{{\sf a}_i}\in{\sf L}_{m}(\tau)$
$$\lambda(H_\tau) := 
\sum_{\omega\in{\Cal T}\atop \omega\mid\upsilon_i}
M(\omega)\lambda_{{\sf a}_i}({H_\tau\over\gamma_{m\tau}}) 
M({\upsilon_i\over\omega})\lambda_{{\sf a}_i}(\gamma_{m\tau}) = 0.$$
\item[{\bf (P)}] From {\bf (M)} setting 
$H_i := H_{{\sf t}_i}$, $L_i := L_{{\sf t}_i}$, $P_i := P_{{\sf t}_i}$.
\qed\end{description}\end{proof}
%
\begin{corollary} 
Any CM-ideal ${\sf I}$, given by a CM-scheme ${\sf X}$ of
CM-conditions, using the same notation as above, satisfies conditions 
{\bf (A-V)}
\qed\end{corollary} 
%
\section{An illuminating section on an example}\label{sect}
%
Let us set $n = 3$, so that ${\Cal P} := {\Bbb C}[X_1,X_2,X_3]$, 
${\sf a} := (0,0,0)\in{\Bbb C}^3$, ${\sf b} := (1,0,1)\in{\Bbb C}^3$, 
${\sf c} := (0,-1,-1)\in{\Bbb C}^3,$
\begin{eqnarray*}
\lambda_{\sf a}({\frak q}_{\sf a}) &:=& (X_1^4,X_1X_2^2,X_1^2X_2,X_2^3,X_1X_3,X_2X_3,X_3^2)\cr
\lambda_{\sf b}({\frak q}_{\sf b}) &:=& (X_1,X_2^3,X_2X_3,X_3^2)\cr
\lambda_{\sf c}({\frak q}_{\sf c}) &:=& (X_1,X_2^2,X_3^2),\cr
{\sf I} &:=& {\frak q}_{\sf a}\cup{\frak q}_{\sf b}\cup{\frak q}_{\sf c}.
\end{eqnarray*}
so that $s := \deg({\sf I}) = 8+4+4 =16$.

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

\noindent
{\tiny$\begin{array}{l|llllllll}
{\rm i}&1&2&3&4&5&6&7&8\cr
{\sf a}_i&{\sf a}&{\sf a}&{\sf a}&{\sf a}&{\sf a}&{\sf a}&{\sf a}&{\sf a}\cr
\upsilon_i&1&X_1&X_2&X_3&X_1^2&X_1X_2&X_2^2&X_1^3\cr
\Phi(\lambda_i)&1&X_1&X_2&X_3&X_1^2&X_1X_2&X_2^2&X_1^3\cr
\hline
{\rm i}&9&10&11&12&13&14&15&16\cr
{\sf a}_i&{\sf b}&{\sf b}&{\sf b}&{\sf b}&{\sf c}&{\sf c}&{\sf c}&{\sf c}\cr
\upsilon_i&1&X_2&X_3&X_2^2&1&X_2&X_3&X_2X_3\cr
\Phi(\lambda_i)&X_1^4&X_1^2X_2&X_1X_3&X_1X_2^2&X_2^3&X_2^4&X_2X_3&X_2^2X_3\cr
\end{array}$} 

The lex reduced Gr\"obner basis of {\sf I} is 
${\Cal G}({\sf I}) = \{f_i, 1\leq i \leq 9\}$ where
$$\begin{array}{lcl}
f_1 &:=& X_1^5-X_1^4\cr
f_2 &:=& X_1^3X_2-X_1^2X_2\cr
f_3 &:=& X_1^2X_2^2-X_1X_2^2\cr
f_4 &:=& X_1X_2^3\cr
f_5 &:=& X_2^5+2X_2^4+X_2^3\cr
f_6 &:=& X_1^2X_3-X_1X_3\cr
f_7 &:=& X_1X_2X_3-X_1^2X_2\cr
f_8 &:=& X_2^3X_3+2X_2^2X_3+X_2X_3-2X_1X_2^2-X_1^2X_2\cr
f_9 &:=& X_3^2-2X_2^2X_3-4X_2X_3-2X_1X_3-3X_2^4\cr
&+&2X_1X_2^2+4X_1^2X_2+X_1^4 \cr
\end{array}$$
and we have the following factorization of each $f_i$ modulo $(f_1,\ldots,f_{i-1})$:
$$\begin{array}{lcl}
f_1 &=& X_1^4(X_1-1)\cr
f_2 &=& X_1^2(X_1-1)X_2\cr
f_3 &=& X_1(X_1-1)X_2^2\cr
f_4 &=& X_1X_2^3\cr
f_5 &=& X_2^3(X_2+1)^2\cr
f_6 &=& X_1(X_1-1)X_3\cr
f_7 &=& X_1X_2(X_3-X_2)\cr
f_8 &\equiv& X_2(X_2+1)^2(X_3-X_1^2)\cr
f_9 &\equiv& (X_3-X_1^2-2X_2-X_2^2)(X_3+3X_2^2+2X_2^3-X_1^2)\cr 
\end{array}$$
Remark that for
\begin{eqnarray*}
f_2
&&{\sf Q}_{2}({\sf t}_2) = \{M(X_1^2)\lambda_{\sf a},M(X_1)\lambda_{\sf b},
                              M(X_1^2)\lambda_{\sf c}\},
\cr&&{\sf L}_{2}({\sf t}_2) = \{\lambda_5,\lambda_8\},
\cr&&{\sf P}_{2}({\sf t}_2) = \{\lambda_1,\lambda_2\},
\cr&&{\sf E}_{2}({\sf t}_2) = \{1,X_1\};
\cr f_3
&&{\sf Q}_{2}({\sf t}_3) = \{M(X_1)\lambda_{\sf a},M(X_1)\lambda_{\sf b},
                              M(X_1)\lambda_{\sf c}\},
\cr&&{\sf L}_{2}({\sf t}_3) = \{\lambda_2,\lambda_5,\lambda_8\},
\cr&&{\sf P}_{2}({\sf t}_3) = \{\lambda_1, \lambda_2,
                              \lambda_5,\lambda_3\},
\cr&&{\sf E}_{2}({\sf t}_3) = \{1,X_1,X_1^2,X_2\};
\cr f_4
&&{\sf Q}_{2}({\sf t}_4) = \{M(X_1)\lambda_{\sf a},M(1)\lambda_{\sf b},
                              M(X_1)\lambda_{\sf c}\},
\cr&&{\sf L}_{2}({\sf t}_4) = \{\lambda_2,\lambda_5,\lambda_8,\lambda_9\},
\cr&&{\sf P}_{2}({\sf t}_4) = \{\lambda_1, \lambda_2,\lambda_5,\lambda_3,
                              \lambda_9,\lambda_{10},\lambda_{12}\}, 
\cr&&{\sf E}_{2}({\sf t}_4) = \{1,X_1,X_1^2,X_1^3,X_2,X_1X_2,X_2^2\};
\cr f_5
&&{\sf R}_{2}({\sf t}_5) = \{\lambda_1, \lambda_3,\lambda_7,
                              \lambda_{13},\lambda_{15}\}, 
\cr f_6
&&{\sf Q}_{3}({\sf t}_6) = \{M(X_1)\lambda_{\sf a},M(X_1)\lambda_{\sf b},
                              M(X_1)\lambda_{\sf c}\},
\cr&&{\sf L}_{3}({\sf t}_6) = \{\lambda_2,\lambda_5,\lambda_6,\lambda_8\},
\cr&&{\sf P}_{3}({\sf t}_6) = \{\lambda_1, \lambda_2,
                              \lambda_5,\lambda_3\}, 
\cr&&{\sf E}_{3}({\sf t}_6) = \{1,X_1,X_1^2,X_2\};
\cr f_7
&&{\sf Q}_{2}({\sf t}_7) = \{M(X_1)\lambda_{\sf a},M(1)\lambda_{\sf b},
                              M(X_1)\lambda_{\sf c}\},
\cr&&{\sf L}_{2}({\sf t}_7) = \{\lambda_2,\lambda_5,\lambda_8,\lambda_9\},
\cr&&{\sf P}_{2}({\sf t}_7) = \{\lambda_1,\lambda_2,\lambda_5\}, 
\cr&&{\sf E}_{2}({\sf t}_7) = \{1,X_1,X_1^2\};
\cr&&{\sf Q}_{3}({\sf t}_7) = \{M(X_1X_2)\lambda_{\sf a},M(X_2)\lambda_{\sf b},
                              M(X_1)\lambda_{\sf c}\},
\cr&&{\sf L}_{3}({\sf t}_7) = \{
\lambda_6,\lambda_{10},\lambda_{12}\},
\cr&&{\sf P}_{3}({\sf t}_7) = \{\lambda_1,\lambda_9,\lambda_{10}\},
\cr&&{\sf E}_{3}({\sf t}_7) = \{1,X_1,X_2\};
\cr f_8
&&{\sf Q}_{2}({\sf t}_8) = \{M(1)\lambda_{\sf a},M(1)\lambda_{\sf b},
                              M(1)\lambda_{\sf c}\},
\cr&&{\sf L}_{2}({\sf t}_8) = \{\lambda_1,\lambda_{13},\lambda_{14}\},
\cr&&{\sf P}_{2}({\sf t}_8) = \{\lambda_1,\lambda_{13},\lambda_{14},\}, 
\cr&&{\sf E}_{2}({\sf t}_8) = \{1,X_2,X_2^2\};
\cr&&{\sf Q}_{3}({\sf t}_8) = \{M(X_2)\lambda_{\sf a},M(X_2)\lambda_{\sf b},
                              M(X_2^2)\lambda_{\sf c}\},
\cr&&{\sf L}_{3}({\sf t}_8) = \{\lambda_2,\lambda_3,\lambda_5,\lambda_6,
                              \lambda_7,\lambda_8,\lambda_9,\lambda_{10},\lambda_{12}\},
\cr&&{\sf P}_{3}({\sf t}_8) = \{\lambda_1,\lambda_2,\lambda_3,\lambda_9,\lambda_{10}\},
\cr&&{\sf E}_{3}({\sf t}_8) = \{1,X_1,X_2,X_1^2,X_1X_2\}
\end{eqnarray*}
and that each factor is obtained by interpolation as stated in Lemma~\ref{Etse}.
%
\section{An aesthetic section with a \\worthless result}
%
The equivalent factorization of $f_9$ in \S~\ref{sect}, 
which is not implied by {\bf (L)}, was obtained
playing around but has an easy justification: 
${\sf Q} := \{\lambda_1,\lambda_9,\lambda_{13},\lambda_{14}\}$ is obtained
as 
${\sf Q} := 
\bigl\{M({\upsilon_i\over X_3}))\lambda_{{\sf a}_i}, i\in\{4,11,15,16\}\bigr\};$
Cerlienco--Mureddu Correspondence attributes 
${\bf N}({\sf Q}) := \{1,X_1,X_2,X_2^2\}$ to ${\sf Q}$ and interpolatation returns
$X_3-X_1-2X_2-X_2^2$.
Moreover
\begin{eqnarray*}
{\sf P} &:=& \{M({\upsilon_i\over\omega})\lambda_{{\sf a}_i} : 
M(\upsilon_i))\lambda_{{\sf a}_i}\in{\Bbb L},
M(\omega)\lambda_{{\sf a}_i}\in{\sf Q}\}\cr
&:=&
\{\lambda_1,\lambda_2,\lambda_3,\lambda_5,\lambda_9,\lambda_{10},\lambda_{13},\lambda_{14}\},\cr
{\bf N}({\sf P})
&:=&
\{1,X_1,X_1^2,X_1^3,X_2,X_1X_2,X_2^2,X_2^3\}
\end{eqnarray*}
and interpolation returns
$X_3+3X_2^2+2X_2^3-X_1^3$.\footnote{
Remark that
modulo $(f_1,\ldots,f_{6})$ we have also 
$$\begin{array}{lcl}
f_7 &\equiv& X_1X_2(X_3+3X_2^2+2X_2^3-X_1^3)\cr
f_8 &\equiv& X_2(X_2+1)^2(X_3+3X_2^2+2X_2^3-X_1^3).\cr
\end{array}$$}

This remark allows to devise the following computation:
for each 
$\tau :=  X_1^{d_1}\cdots X_n^{d_n},$
for each $\nu \leq n$, iteratively for increasing $\delta \leq d_\nu$, with initial value
$${\sf P}_{\nu d_n+1}(\tau) := {\sf P}_{\nu-1} := {\sf P}_{\nu-1 2}$$ we compute
{\tiny
\begin{eqnarray*}
{\sf Y}_{\nu\delta}(\tau) &:=& \{\pi_\nu({\sf x}) : 
\exists \omega\in {\Cal T}[1,\nu] : \Phi({\sf x}) = \omega X_{\nu+1}^{\delta}, 
{\sf x}\in {\sf P}_{\nu\delta+1}(\tau)\}\cr
{\sf E}_{\nu\delta}(\tau) &:=& {\bf N}({\sf Y}_{\nu\delta}(\tau))\cr
{\sf P}_{\nu\delta}(\tau) &:=& \{M\left(\pi_{\nu}\left({\upsilon_i\over\omega}\right)\right)\lambda_{{\sf a}_i} : 
\cr&&\quad\quad
M(\upsilon_i))\lambda_{{\sf a}_i}\in{\sf L}_{\nu}(\tau),
M(\omega)\lambda_{{\sf a}_i}\in{\sf Y}_{\nu\delta}(\tau)\},\cr%
\end{eqnarray*}
}%
so that
\begin{corollary}\label{RFRee}
For each $\tau :=  X_1^{d_1}\cdots X_n^{d_n}\in{\bf G},$ each $m, 1
\leq m\leq n$, and each $\delta\leq d_m$ there is a unique polynomial
$$\gamma_{m\delta\tau} := X_m + \sum_{\omega\in{\sf E}_{\nu\delta}(\tau)} c(\gamma_{m\tau},\omega) \omega$$
such that 
$\pi_m(\lambda_i)(\gamma_{m\tau}) = 0,$ for each $\lambda_i\in {\sf Y}_{\nu\delta}(\tau);$
also $\gamma_{m\tau} = \prod_\delta \gamma_{m\delta\tau}$.
\qed\end{corollary}
%
\begin{example}{\em If, in the example of \S~\ref{sect}, we now add, where ${\sf d}=(1,0,0)$,
$$\begin{array}{rclrcl}
\lambda_{17} &:=& M(X_3^2)\lambda_{\sf a}
&\Phi(\lambda_{17}) &=& X_3\cr
\lambda_{18} &:=& M(1)\lambda_{\sf d}
&\Phi(\lambda_{18}) &=& X_1X_3
\end{array}$$ 
the corresponding lex reduced Gr\"obner basis is
$\{f_i, 1\leq i \leq 8\}\cup\{f_{10},f_{11}\}$
where
\begin{eqnarray*}
f_{10} &:=& X_2X_3^2 + 2X_2X_3 +2X_2^4 + 3X_2^3 - 3X_1^2X_2\cr
&\equiv&
X_2(X_3-1-4X_2-2X_2^2)
\\&\cdot&
(X_3-X_1^2+3X_2^2+2X_2^3);\cr
f_{11} &:=& X_3^3 - 2X_1X_3^2 + 3X_2^2X_3 + 6X_2X_3 + X_1X_3\cr
&\equiv&  X_3(X_3-X_1-2X_2-X_2^2)
\\&\cdot&
(X_3-X_1^2+3X_2^2+2X_2^3).\cr 
\end{eqnarray*}
The factorization is justified by
\begin{eqnarray*}
f_{10}
&&{\sf Q}_{2}({\sf t}_{10}) = \{M(1)\lambda_{\sf a},M(1)\lambda_{\sf b},
                              M(1)\lambda_{\sf c},M(1)\lambda_{\sf d}\},
\cr&&{\sf L}_{2}({\sf t}_{10}) = \{\lambda_1\},
\cr&&{\sf P}_{2}({\sf t}_{10}) = \{\lambda_1\},
\cr&&{\sf E}_{2}({\sf t}_{10}) = \{1\},
\cr&&\gamma_{2{\sf t}_{10}} = X_2;
\cr&&{\sf Q}_{3}({\sf t}_{10}) = \{M(X_2)\lambda_{\sf a},M(1)\lambda_{\sf b},
                              M(1)\lambda_{\sf c},M(1)\lambda_{\sf d}\},
\cr&&{\sf L}_{3}({\sf t}_{10}) = \{\lambda_i, i\leq 18, 1\neq i\neq 4\},
\cr&&{\sf P}_{3}({\sf t}_{10}) = \{\lambda_i, i\notin\{4,5,6,7,8,18\}\},
%%  \cr&&{\sf E}_{3}({\sf t}_{10}) = \{1\},
\cr&&{\sf Y}_{32}({\sf t}_{10}) = \{\lambda_{9},\lambda_{13},\lambda_{14},\lambda_{18}\},
\cr&&{\sf E}_{32}({\sf t}_{10}) = \{1,X_1,X_2,X_2^2\},
\cr&&\gamma_{32{\sf t}_{10}} = X_3-1-4X_2-2X_2^2,
\cr&&{\sf P}_{32}({\sf t}_{10}) = \{\lambda_i, i\in\{1,2,3,9,10,13,14\}\}, 
\cr&&{\sf Y}_{31}({\sf t}_{10}) = \{\lambda_i, i\in\{1,2,3,9,10,13,14\}\},
\cr&&{\sf E}_{31}({\sf t}_{10}) = \{1,X_1,X_2,X_1^2,X_1X_2,X_2^2,X_2^3\},
\cr&&\gamma_{31{\sf t}_{10}} = X_3-X_1^2+3X_2^2+2X_2^3,
\cr f_{11}
&&{\sf P}_{3}({\sf t}_{11}) = \{\lambda_i, i\leq 18\}\},
\cr&&{\sf Y}_{33}({\sf t}_{11}) = \{\lambda_{1},\lambda_{18}\},
\cr&&{\sf E}_{33}({\sf t}_{11}) = \{1,X_1,\},
\cr&&\gamma_{33{\sf t}_{11}} = X_3,
\cr&&{\sf P}_{33}({\sf t}_{11}) = \{\lambda_i, i\notin\{6,7,8\}\},
\cr&&{\sf Y}_{32}({\sf t}_{11}) = \{\lambda_{1},\lambda_{9},\lambda_{13},\lambda_{14}\},
\cr&&{\sf E}_{32}({\sf t}_{11}) = \{1,X_1,X_2,X_2^2\},
\cr&&\gamma_{32{\sf t}_{11}} = X_3-X_1-2X_2-X_2^2,
\cr&&{\sf P}_{32}({\sf t}_{11}) = \{\lambda_i, i\in\{1,2,3,9,10,13,14\}\}, 
\cr&&{\sf Y}_{31}({\sf t}_{11}) = \{\lambda_i, i\in\{1,2,3,9,10,13,14\}\},
\cr&&{\sf E}_{31}({\sf t}_{11}) = \{1,X_1,X_2,X_1^2,X_1X_2,X_2^2,X_2^3\},
\cr&&\gamma_{31{\sf t}_{11}} = X_3-X_1^2+3X_2^2+2X_2^3,
\end{eqnarray*}
}\end{example}
%
We agree with the reader that this section is really quite poor, being nothing more than a trivial interpolation game --- the more so since
there cannot be any reasonable appplication. We added it just for its aesthetical value.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{A conclusing section with an assertion}\label{ABUC}
%
While this paper was under refereeing, we were finally able, in cooperation with
T. Mora, to state Cerlienco--Mureddu Correspondence \cite{CM1,CM2} and the
(generalizable part of) Enhanced Lazard Structural Theorem \cite{Iran} without the
restriction of ${\sf I}$ being a CM-ideal. Our result will be submitted in
\cite{Cuba}

For each $j = 1,\ldots,n$,
$\sigma_{j} := \sigma_{X_j} : \Span_k({\Bbb M}) \mapsto
\Span_k({\Bbb M})$ is  
the linear map such that
$$\sigma_{X_j}(M(\tau)) =
\begin{cases}\Phi'({\sf x}_i)& \tau = X_j\omega\cr 
0& X_j\nmid \tau.\cr\end{cases}
\Forall \tau\in{\Cal T};$$
since, for each
$i,j$, $\sigma_{X_j} \sigma_{X_i} = \sigma_{X_i} \sigma_{X_j}$, a
linear map  $\sigma_t : \Span_k(\Bbb M) \mapsto
\Span_k(\Bbb M)$ is inductively defined for each $t \in {\Cal M}$ by
$\sigma_{X_jt} := \sigma_{X_j}\sigma_t$
so that for each $\tau,\omega\in{\Cal T}$ we have
$$\sigma_\tau(M(\omega)) =
\begin{cases} M(\upsilon)& \omega = \tau\upsilon\cr
0& \tau \nmid \omega.\end{cases}$$
For each Noetherian equation 
$$\ell(\tau) := 
M(\tau) + \sum_{t\in{\bf T}({\sf I})}
\gamma(t,\tau,{\bf N}({\sf I})) M(t),$$
we set
$$\pi_m(\ell(\tau)) 
:=
(\sigma_{X_{m}^{d_{m}}\cdots X_n^{d_n}}(\ell(\tau)))(X_1,\ldots,X_m)
\in {\Bbb M}[1,m].
$$
Finally, for a CM-functional $\lambda = \ell_{\upsilon{\sf a}}\lambda_{\sf a}$
we set 
$$\pi_{m}(\lambda) := \pi_{m}(\ell_{\upsilon\sf a}\lambda_{\sf a}) := 
\pi_m(\ell_{\upsilon\sf a})\lambda_{\pi_{m}({\sf a})}.$$ 

Given a Macaulay representation 
${\Bbb L} := \{\lambda_1,\ldots,\lambda_s\}$,
let us therefore consider the subset
${\Bbb L}' := \{\lambda_1,\ldots,\lambda_{s-1}\}$, 
the corresponding order ideal 
${\bf N}' := {\bf N}({\Bbb L}')$ and bijection $\Phi' := \Phi({\Bbb L}')$.
Let us also denote, 
for each $\nu, 1 \leq \nu < n,$ and each $\delta\in{\Bbb N},$
$${\sf Y}_{\nu\delta} := \Span_k\{\pi_\nu(\lambda) : 
\lambda\in{\Bbb L}', \Phi'(\lambda) = \omega X_{\nu+1}^{\delta},
\omega\in {\Cal T}[1,\nu]\}.$$
With this notation, let us set
\begin{eqnarray*}
m &:=& \max\left(j : \pi_j(\lambda_s)\in\Span_k(\pi_j({\Bbb L}')\right),\\
d &:=& \min\{\delta : \pi_{m}(\lambda_s)\not\in{\Bbb Y}_{m\delta}\},\\
{\Bbb V} &:=& \{\pi_m(\lambda) : \Phi'(\lambda) = \omega X_{m+1}^{d},\omega\in
{\Cal T}[1,m]\},\\
{\Bbb W} &:=& {\Bbb V} \cup\{\pi_{m}(\lambda_s)\},\\
\omega &:=& \Phi({\Bbb W})(\pi_{m}(\lambda_s)),\\
t_s &:=& \omega X_{m+1}^{d}
\end{eqnarray*}
where ${\bf N}({\Bbb W})$ and $\Phi({\Bbb W})$ are the result of the
application of the present algorithm to ${\Bbb W}$,  which can be inductively applied 
since $\#({\Bbb W}) \leq s-1$.
We then define ${\bf N} := {\bf N}' \cup \{t_s\}$ and
$\Phi(\lambda_i) := \begin{cases}\Phi'(\lambda_i)& i < s\cr t_s&
i=s.\cr\end{cases}$

With this notation
\begin{theorem} \cite{Cuba} For a zero-dimensional ideal
 ${\sf I}$, given by a Macaulay representation ${\Bbb L}$
\begin{description}
\item conditions {\bf (A-L)} hold;
\item if moreover ${\sf I}$ is a CM-ideal it satisfies also {\bf (M-V)}
\item if ${\sf I}$ is radical it satisfies also {\bf (W-Z)}
\end{description}
\end{theorem}

\begin{example}\label{E2}{\em Let us now consider 
${\sf a} := (0,0,0)\in{\Bbb C}^3,$ ${\sf b} := (0,0,1)\in{\Bbb C}^3,$
\begin{eqnarray*}
\lambda_{\sf a}({\frak q}_{\sf a}) &:=& (X_1^2,X_2+X_1,X_3)\cr
\lambda_{\sf b}({\frak q}_{\sf b}) &:=& (X_1^2,X_2-X_1,X_3)\cr
{\sf I} &:=& {\frak q}_{\sf a}\cup{\frak q}_{\sf b}\cr
&=& (X_1^2,X_2+X_1,X_3)\cap(X_1^2,X_2-X_1,X_3-1)\\
&=& (X_1^2,X_1X_2,X_2^2
,%)\\&+&(
X_1X_3-{1\over 2}X_1-{1\over 2}X_2)
\\&+&(X_2X_3-{1\over 2}X_1-{1\over 2}X_2, X_3^2-X_3)\cr
\pi_2({\sf I}) &=& (X_1^2,X_1X_2,X_2^2)
\end{eqnarray*}}
%
{\em The point is that
$\lambda_{\sf a}({\frak q}_{\sf a})$ is described by $$\{M(1),M(X_2)-M(X_1)\}$$
and
$\lambda_{\sf b}({\frak q}_{\sf b})$ by $\{M(1),M(X_2)+M(X_1)\}$;
since $$\pi_2({\sf a}) = \pi_2({\sf b}) = (0,0),$$ 
we have 
$\pi_2({\sf I}) = (X_1^2,X_1X_2,X_2^2)$ be
${\sf m}$-primary given by
\begin{eqnarray*}
&&\{M(1),M(X_2)+M(X_1),M(X_2)-M(X_1)\}
\\&=&\{M(1),M(X_1),M(X_2)\}.
\end{eqnarray*}
}\end{example}

\begin{example}\label{E1}{\em
The construction delined suggests that, apart of CM-ideals, one
cannot hope in factorization results. Here, in any case we have an example.
The ${\sf m}$-primary ideal
$${\sf I} := \{X_2X_3-X_1^2,X_1^3,X_2X_1^2,X_3^2,X_1^2X_3,X_2^2\}$$
in $k[X_1,X_2,X_3]$
has, as Macaulay basis, the set
$$\{M(1),M(X_1),M(X_2),M(X_3),M(X_1X_2),M(X_1X_3),\ell_{X_1^2}\}$$
where $\ell_{X_1^2} := M(X_1^2)+M(X_2X_3);$
%% we know \cite{M$^3_2$} that the Leibniz Formula related to $\ell_{X_1^2}$  is
%% \begin{eqnarray*}
%% \ell_{X_1^2}(fg) &=& M(1)(f)\ell_{X_1^2}(g)+M(X_1)(f)M(X_1)(g)
%% \\&+&M(X_2)(f)M(X_3)(g)+M(X_3)(f)M(X_2)(g)
%% \\&+&\ell_{X_1^2}(f)M(1)(g)
%% \end{eqnarray*}
%% but 
we failed to deduce any factorization 
$$X_2X_3-X_1^2 \cong (X_2+\cdots)(X_3+\cdots)$$
modulo $\{X_1^3,X_2X_1^2,X_3^2,X_1^2X_3\}.$}
 
{\em Theo Mora pointed us to Noether's\cite{N} notion of {\em
reduced} primary representation of an ideal, a notion stronger of the one
of {\em irredundant} primary representation, which however lose
uniqueness; as Gr\"obner \cite{Gr} proved, reduced
primary representations depend on the frame of coordiantes; with respect
to each such frame are unique and computable (cf. \cite{SPES}).}

{\em The reduced decomposition
\begin{eqnarray*}
{\sf I} &:=&  (X_2X_3-X_1^2,X_1^3,X_2X_1,X_3^2,X_1X_3,X_2^2)
\\&\cap&(X_1^2,X_2^2,X_3)\cap(X_1^2,X_2,X_3^2)
\end{eqnarray*}
gives the trivial factorization
$$X_2X_3-X_1^2\equiv(X_3\pm X_1)(X_3\mp X_2)$$
modulo $(X_1^3,X_2X_1,X_1X_3,X_2^2).$}
%
{\em Remark that  no linear form gives a non-derogatory multiplication matrix \cite{SM} for
$$(X_2X_3-X_1^2,X_1^3,X_2X_1,X_3^2,X_1X_3,X_2^2).$$
}\end{example}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibliographystyle{abbrv}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{1}
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%
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%
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%
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%
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%
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%
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%
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%
\bibitem{M$^3$} 
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%
\bibitem{M$^3_2$} 
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%
\bibitem{Iran} 
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\newblock A remark on a remark by Macaulay or Enhancing Lazard Structural
Theorem. 
\newblock \bibemphic{Bull. Iranian Math. Soc.}~\bibemph{29} (2003),
1-45.
%
\bibitem{M}  
\bibsc{M\"oller H.M.}   
\newblock Systems of Algebraic Equations Solved by Means of
Endomorphisms.
\newblock \bibemphic{L. N. Comp. Sci.}~\bibemph{673} (1993), 43--56
%
\bibitem{BM}
\bibsc{M\"oller H.M., Buchberger B.} 
\newblock The
construction of Multivariate Polynomials with Preassigned Zeros. 
\newblock \bibemphic{L. N. Comp. Sci.}~\bibemph{144} (1982), 24--31.
%
\bibitem{SPES}  
\bibsc{Mora T.} 
\newblock \bibemph{Solving Polynomial Equation Systems II}.
\newblock Cambridge Univ. Press,
to appear.
%
\bibitem{N} 
\bibsc{Noether E.}
\newblock Idealtheorie in Ringbereichen.
\newblock \bibemphic{Math. Annales}~\bibemph{83} (1921), 25--66.
%
\bibitem{SM} 
\bibsc{Stetter H.J., M\"oller H.M.}
\newblock Multivariate Polynomial Equations With Multiple Zeros Solved by Matrix 
Eigenproblems 
\newblock \bibemphic{Numer. Math.}~\bibemph{70} (1995), 311--329.

\end{thebibliography}


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\section{A rambling section with no \\conclusion}
%
Can one state Cerlienco--Mureddu Correspondence \cite{CM1,CM2} and the 
Enhanced Lazard Structural Theorem \cite{Iran} without the restriction of ${\sf I}$ being
a CM-ideal?
We fear not as it is shown by the following examples.
%
\begin{example}\label{E1}{\em
The ${\sf m}$-primary ideal
$${\sf I} := \{X_2X_3-X_1^2,X_1^3,X_2X_1^2,X_3^2,X_1^2X_3,X_2^2\}$$
in $k[X_1,X_2,X_3]$
has, as Macaulay basis, the set
$$\{M(1),M(X_1),M(X_2),M(X_3),M(X_1X_2),M(X_1X_3),\ell_{X_1^2}\}$$
where $\ell_{X_1^2} := M(X_1^2)+M(X_2X_3);$
we know \cite{M$^3_2$} that the Leibniz Formula related to $\ell_{X_1^2}$  is
\begin{eqnarray*}
\ell_{X_1^2}(fg) &=& M(1)(f)\ell_{X_1^2}(g)+M(X_1)(f)M(X_1)(g)
\\&+&M(X_2)(f)M(X_3)(g)+M(X_3)(f)M(X_2)(g)
\\&+&\ell_{X_1^2}(f)M(1)(g)
\end{eqnarray*}
but we failed to deduce any factorization 
$$X_2X_3-X_1^2 \cong (X_2+\cdots)(X_3+\cdots)$$
modulo ${\{X_1^3,X_2X_1^2,X_3^2,X_1^2X_3\}}.$
}\end{example}

%
\begin{example}\label{E2}{\em Let us now consider 
${\sf a} := (0,0,0)\in{\Bbb C}^3,$ ${\sf b} := (0,0,1)\in{\Bbb C}^3,$
\begin{eqnarray*}
\lambda_{\sf a}({\frak q}_{\sf a}) &:=& (X_1^2,X_2+X_1,X_3)\cr
\lambda_{\sf b}({\frak q}_{\sf b}) &:=& (X_1^2,X_2-X_1,X_3)\cr
{\sf I} &:=& {\frak q}_{\sf a}\cup{\frak q}_{\sf b}\cr
&=& (X_1^2,X_2+X_1,X_3)\cap(X_1^2,X_2-X_1,X_3-1)\\
&=& (X_1^2,X_1X_2,X_2^2
,%)\\&+&(
X_1X_3-{1\over 2}X_1-{1\over 2}X_2)
\\&+&(X_2X_3-{1\over 2}X_1-{1\over 2}X_2, X_3^2-X_3)\cr
\pi_2({\sf I}) &=& (X_1^2,X_1X_2,X_2^2)
\end{eqnarray*}}
%
{\em The point is that
$\lambda_{\sf a}({\frak q}_{\sf a})$ is described by $$\{M(1),M(X_2)-M(X_1)\}$$
and
$\lambda_{\sf b}({\frak q}_{\sf b})$ by $\{M(1),M(X_2)+M(X_1)\}$;
since $$\pi_2({\sf a}) = \pi_2({\sf b}) = (0,0),$$ 
we have 
$\pi_2({\sf I}) = (X_1^2,X_1X_2,X_2^2)$ be
${\sf m}$-primary given by
\begin{eqnarray*}
&&\{M(1),M(X_2)+M(X_1),M(X_2)-M(X_1)\}
\\&=&\{M(1),M(X_1),M(X_2)\}.
\end{eqnarray*}
A combinatorial appoach as the one proposed by Cerienco and Mureddu and pursued in
\cite{Iran} seems unable to deal with such phenomena.
}\end{example}
%
\begin{remark}{\em In connection with the last example, Theo Mora remarked
that probably the {\em crux} here is that the two points have different
frame of coordinates and pointed us to Noether's\cite{N} notion of {\em
reduced} primary representation of an ideal, a notion stronger of the one
of {\em irredundant} primary representation, which however lose
uniqueness; as Gr\"obner \cite{Gr} proved, reduced
primary representations depend on the frame of coordiantes; with respect
to each such frame are unique and computable (cf. \cite{SPES}).
 }\end{remark}

\begin{example}\label{E3}{\em 
Mora's remark in fact allows to fix Example~\ref{E2}; it is sufficient in fact to perform the easy change of coordinates
$$Y_1 := X_1+X_2+X_3, Y_2 := X_2+X_3, Y_3 := X_3$$
in orden to obtain
${\sf a} := (0,0,0),$ ${\sf b} := (1,1,1),$
\begin{eqnarray*}
\lambda_{\sf a}({\frak q}_{\sf a}) &:=& (Y_1,Y_2^2,Y_3),\cr
\lambda_{\sf b}({\frak q}_{\sf b}) &:=& ((Y_1-1)^2,2Y_2-Y_1-1,Y_3-1),\cr
{\sf I} &:=& {\frak q}_{\sf a}\cup{\frak q}_{\sf b}\cr
&=& \pi_2({\sf I})+(Y_2^2-Y_1,Y_3+Y_1^2-2Y_1)\cr
\pi_2({\sf I}) &=& (Y_1(Y_1-1)^2,2Y_1Y_2-Y_1^2-Y_1,Y_2^2-Y_1)
\end{eqnarray*}
Here, in fact we have 
$\pi_2({\sf a}) \neq \pi_2({\sf b})$.

We point here the non-trivial factorization
$$Y_2^2-Y_1 \equiv Y_2(Y_1Y_2-{1\over 2}Y_1^2-{1\over 2}Y_1$$
modulo ${Y_1(Y_1-1)^2,2Y_1Y_2-Y_1^2-Y_1}.$
}\qed\end{example}

\begin{example}\label{E3}{\em 
We failed to fix Example~\ref{E1} by means of a change of coordinates, but Mora's remark to Noether's reduced primaries, suggested to
compute the reduced decomposition of
$${\sf I} := \{X_2X_3-X_1^2,X_1^3,X_2X_1^2,X_3^2,X_1^2X_3,X_2^2\}$$

which is:
\begin{eqnarray*}
{\sf I} &:=&  (X_2X_3-X_1^2,X_1^3,X_2X_1,X_3^2,X_1X_3,X_2^2)
\\&\cap&(X_1^2,X_2^2,X_3)\cap(X_1^2,X_2,X_3^2)
\end{eqnarray*}
in order to obtain the trivial factorization
$$X_2X_3-X_1^2\equiv(X_3\pm X_1)(X_3\mp X_2)$$
modulo $(X_1^3,X_2X_1,X_1X_3,X_2^2).$

Remark that  no linear form gives a non-derogatory multiplication matrix \cite{SM} for
$$(X_2X_3-X_1^2,X_1^3,X_2X_1,X_3^2,X_1X_3,X_2^2).$$
}\qed\end{example}

\begin{Conjecture} Let ${\sf I}\subset{\Cal P}$ be a zero-dimensional ideal, given by
means of its Macaulay representation and its
{\em CM-scheleton}.
${\sf I}$ satisfies  conditions {\bf (A-H)}, {\bf (T-Z)} of Theorem~\ref{IDDU} 
provided that the frame of coordinates $\{X_1,\ldots,X_n\}$ is generic.

If moreover, ${\sf I}$ is the intersection of reduced primary components, each corresponding to different
maximal ideals, then also conditions {\bf (I-S)} of Theorem~\ref{IDDU} and Corollary~\ref{RFRee} hold.
\end{Conjecture}