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\begin{document}

\title[The leading ideal of a complete intersection]
{The Leading Ideal of a Complete Intersection
of Height Two in a 2-dimensional Regular Local Ring}




%    Information for first author
\author{Shiro Goto}
\address{Department of Mathematics, School of Science and Technology
Meiji University, 214-8571 Japan}

\email{goto@math.meiji.ac.jp}

\thanks{Shiro Goto, Department of Mathematics, School of Science and Technology
Meiji University, 214-8571 Japan, email: goto@math.meiji.ac.jp,
 Shiro Goto is supported by the
Grant-in-Aid for Scientific Researches in Japan (C(2), No.1364044).}

\author{William Heinzer}
\address{Department of Mathematics, Purdue University, West
Lafayette, Indiana 47907}


\email{heinzer@math.purdue.edu}

\thanks{William Heinzer, Department of Mathematics, Purdue University, West
Lafayette, Indiana 47907, email: heinzer@math.purdue.edu}



\author{Mee-Kyoung Kim}
\address{
Department of Mathematics, Sungkyunkwan University Jangangu Suwon
440-746, Korea} \email{mkkim@skku.edu}


\thanks{Mee-Kyoung Kim, Department of Mathematics, Sungkyunkwan University Jangangu Suwon
440-746, Korea, email: mkkim@skku.edu, Mee-Kyoung Kim  was supported
by Faculty Research Fund, Sungkyunkwan University, 2003.}



\date \today

\subjclass{Primary: 13A30, 13C05; Secondary: 13E05, 13H15}

\keywords{ideal of initial forms, associated graded ring,
multiplicity, regular local ring, Hilbert series, minimal
generating set }

\maketitle

\begin{abstract}
\noindent Let $(S,\n)$ be a $2$-dimensional regular local ring and
let $I = (f, g)$ be an ideal in $S$  generated by a regular sequence
$f, g$ of length two. Let $I^*$ be the leading ideal of $I$ in the
associated graded ring $\gr_{\frak{n}}(S)$, and set  $R = S/I$ and
$\frak{m} = \frak{n}/I$.
 In \cite{GHK2}, we prove that
 if $\mu_G(I^*) =n$, then $I^*$ contains a
homogeneous system $\{\xi_i\}_{1 \le i \le n}$ of generators such
that $\deg \xi_i + 2 \le \deg \xi_{i+1}$ for $2 \le i \le n-1$, and
$\mathrm{ht}_G(\xi_1, \xi_2, \cdots, \xi_{n-1}) = 1$, and we
describe precisely the Hilbert series
$\mathrm{H}(\mathrm{gr}_{\fkm}(R), \lambda)$ in terms of the degrees
$c_i$ of the $\xi_i$ and the integers $d_i$,  where $d_i$ is the
degree of $D_i = \GCD(\xi_1, \ldots, \xi_i)$. To the complete
intersection ideal $I = (f, g)S$ we associate a positive integer $n$
with $2 \le n \le c_1+1$, an ascending sequence of positive integers
$(c_1, c_2, \ldots, c_n)$, and a descending sequence of integers
$(d_1 = c_1, d_2, \ldots, d_n =0)$ such that $c_{i+1} - c_i >
d_{i-1} - d_i > 0$ for each $i$ with $2 \le i \le n-1$.  We
establish here that this necessary  condition is also  sufficient
for there to exist a complete intersection ideal $I = (f, g)$ whose
leading ideal has these invariants. We give several examples to
illustrate our theorems.



\end{abstract}

\baselineskip 18 pt


\section{Introduction}

 This paper examines generators of the leading ideal of  a complete
 intersection of height two in a $2$-dimensional regular local ring.  Motivation for our work
 comes from
 a paper of S. C. Kothari \cite{K} that answers several questions raised by  Abhyankar concerning
 the local Hilbert function of a pair of plane curves.
Before going ahead, let us fix some notation, which we shall maintain throughout this paper.


\begin{setting} \label{1.1}
{\em Let $(S,\frak{n})$ be a regular local ring of dimension $2$ and
let $I = (f,g)$ be an ideal in $S$ generated by a regular sequence
$f, g$ of length two. For simplicity we assume that the residue
class field $k = S/\fkn$ is infinite. We put $R=S/I$ and $\fkm =
\fkn/I$. Let
$$\mathrm{R}'(\frak{n}) = \sum_{i \in \Bbb Z}\fkn^i t^i \subseteq S[t,t^{-1}]\  \ \ \text{and}
\ \ \mathrm{R}'(\frak{m}) = \sum_{i \in \Bbb Z}\frak{m}^i t^i \subseteq R[t,t^{-1}]$$
denote the Rees algebras of $\frak{n}$ and $\frak{m}$ respectively, where $t$ is an indeterminate.
We put $$G = \gr_{\frak{n}}(S) = \mathrm{R}'(\fkn)/t^{-1}\mathrm{R}'(\frak{n}) \ \ \ \text{and} \ \
\gr_{\frak{m}}(R) = \mathrm{R}'(\frak{m})/t^{-1}\mathrm{R}'(\frak{m}).$$ For each $0 \ne h \in S$
let $\mathrm{o}(h) = \mathrm{sup}\{i \in \Z \mid h \in \frak{n}^i \}$ and  put $h^* = \overline{ht^n}$,
where $n = \mathrm{o}(h)$ and $\overline{ht^n}$ denotes the image of $ht^n$ in $G$.
 The canonical
map $S \to R$ induces the epimorphism $\varphi : G \to \gr_{\frak{m}}(R)$ of the associated graded rings.
We put  $$I^*= \mathrm{Ker}~(G \overset{\varphi}\to \gr_{\fkm}(R)).$$
Then the homogeneous components
$\{[I^*]_i \}_{i \in \Bbb{Z}}$ of the leading form ideal $I^*$ of $I$ are
given by $$[I^*]_i = \{ \overline{ht^i} \mid  h \in I \cap \fkn^i\}$$
for each $i \in \Bbb{Z}$. We throughout assume that $a=\mathrm{o}(f) \leq b=\mathrm{o}(g)$ and
that $f^* \nmid g^*$ in $G$. The latter part of the condition  is equivalent to
saying that $f^*, g^*$ form a part of a minimal homogeneous system of generators of $I^*$. }
\end{setting}


Let $\ell_S(*)$  denote  length over $S$. Kothari in \cite{K}  proves that
$$0 \leq
\dim_k[\mathrm{gr}_{\frak{m}}(R)]_{i} -
\dim_k[\mathrm{gr}_{\frak{m}}(R)]_{i+1} \leq 1$$
for all $i \geq a$
and that $\ell_S(R) \geq ab$; moreover,  one has the equality
$\ell_S(R) = ab$ if and only if $f^*,~g^*$ are coprime in $G$, that
is, $f^*,~g^*$ form a $G$-regular sequence.

  F. Macaulay in a 1904
 paper \cite{M} employs a different method to determine the same
 necessary condition as Kothari on the Hilbert function of a pair of plane
 curves. Using his inverse systems, Macaulay establishes
 the structure of the Hilbert function $H(A)$ of a complete
 intersection quotient $A = k[[x, y]]/(f,g)$ to be of the form
 \begin{equation}\label{11}
 H = (1,2, \ldots , a, t_a, \ldots, t_j, 0),
 \end{equation}
 where $a \ge t_a \ge t_{a+1} \ge \cdots \ge t_j = 1$ and $|t_i -
 t_{i+1}| \le 1$ for all $i$. Thus the Hilbert function $H$ after
an initial rising segment breaks up into platforms and regular
flights of descending stairs, each step of height one.
The structure of $H(A)$ is studied from the point of view of
parametrizations by J. Brian\c{c}on \cite{Br} and by A. Iarrobino
\cite{Ia1} and \cite{Ia2}. These authors prove that every sequence
satisfying  the conditions in Equation \ref{11} is realizable as the
Hilbert function $H(A)$ of some Gorenstein Artin algebra of the form
$A = k[[x, y]]/(f,g)$.

Let
$v(H) = 2 + \#\{ \text{platforms} \}$.
Iarrobino \cite{Ia1}, \cite{Ia2}  proves that $I^*$ needs two initial generators
$f^*, g^*$ and requires a new generator following each platform,
and that $v(H)$ is the minimum possible number of generators of a graded
ideal defining a standard algebra with Hilbert function $H$. In
\cite[Theorem 2.2.A]{Ia1}, Iarrobino characterizes those graded ideals
corresponding to $I^*$ for which $I$ is a complete intersection of
height two. He proves they are exactly the graded ideals with $v(H)$
generators. In the paper \cite{GHK2} we prove the following.

\begin{thm} \label{1.2} \cite[Theorem 1.2]{GHK2} Let notation be as in Setting \ref{1.1} and
assume that   $n = \mu_G(I^*)$. Then $I^*$ contains
a homogeneous system $\{ \xi_i \}_{1 \leq i \leq n}$ of generators
that satisfy the following three conditions.

\begin{enumerate}
\item[$(1)$] $\xi_1 = f^*$ and $\xi_2 = g^*$.
\item[$(2)$] $\mathrm{deg} \xi_i + 2 \leq \mathrm{deg} \xi_{i+1}$ for
each $i$ with $2 \leq i \leq n-1$.
\item[$(3)$] $\mathrm{ht}_G(\xi_1, \xi_2, \cdots, \xi_{n-1}) = 1$.
\end{enumerate}
\end{thm}


Let $\{ \xi_i \}_{1 \leq i \leq n}$ be a homogeneous system of
generators of $I^*$ satisfying conditions (1) and (2) in Theorem
\ref{1.2}. Then the ideals $\{(\xi_j \mid 1 \leq j \leq i )G \}_{1
\leq i \leq n}$ are independent of the particular choice of
the family $\{ \xi_i \}_{1 \leq i \leq n}$ and are uniquely
determined by $I$. We put $D_i = \mathrm{GCD}(\xi_j \mid 1 \leq j
\leq i)$ and $d_i = \mathrm{deg}D_i$. We then have the descending
sequence of integers $$a=d_1 > d_2 > \cdots > d_{n-1} > d_n =0,$$
and we also have  $\displaystyle\frac{\xi_{i+1}}{D_{i+1}} \in
(\frac{\xi_{1}}{D_{i}}, \frac{\xi_{2}}{D_{i}}, \cdots,
\frac{\xi_{i}}{D_{i}})$ for all $1 \leq i \leq n-1$ \cite[Lemma 3.2]{GHK2}.
 Let $c_i = \mathrm{deg}~\xi_i$ and let
$\mathrm{H}(\mathrm{gr}_{\frak{m}}(R), \lambda)=
\sum_{i=0}^{\infty}\dim_k[\mathrm{gr}_{\frak{m}}(R)]_i\lambda^i$
denote the Hilbert series of $\mathrm{gr}_{\frak{m}}(R)$. Theorem \ref{1.3}
explicitly describes  $\mathrm{H}(\mathrm{gr}_{\frak{m}}(R),
\lambda)$ and the difference $\ell_S(R) - ab$  in terms of
the integers  $c_i$ and $d_i$, thus sharpening Kothari's results.

 \begin{thm}  \label{1.3} \cite[Theorem 1.3]{GHK2}
Let notation be as in Setting \ref{1.1} and assume that
$n = \mu_G(I^*)$.  The following assertions hold true.
\begin{enumerate}
\item[$(1)$] $\mathrm{H}(\mathrm{gr}_{\frak{m}}(R), \lambda) =
\frac{\sum_{i=2}^n\lambda^{d_i}(1-\lambda^{d_{i-1}
-d_i})(1-\lambda^{c_i -d_i})}{(1-\lambda)^2}.$
\item[$(2)$] $\ell_S(R) = \sum_{i = 2}^{n} (d_{i-1} - d_i) (c_i - d_i) =
  ab + \sum_{i = 2}^{n-1} d_i{\cdot} [(c_{i+1}- c_i) - (d_{i-1} - d_i)]$.
\item[$(3)$] $c_{i+1} -c_i > d_{i-1} -d_{i} > 0$ for each $i$ with  $2 \leq i \leq n-1$.
\item[$(4)$] {\rm \cite[Corollary1]{K}} $\ell_S(R) = ab$ if
  and only if $n = 2$, i.e., $f^*, g^*$ is a $G$-regular sequence.
\end{enumerate}
\end{thm}


In Section 2, we use the inverse of a transformation considered by
Kothari and described in parts (1) and (2) of \cite[Corollary 2.5]{GHK2}
to establish the existence of examples showing that every
Hilbert series described in Theorem \ref{1.3} is realizable as the Hilbert series
of $\mathrm{gr}_{\frak{m}}(R)$ for some complete intersection ideal $I=(f,g)$.
Thus the conditions  given in Theorem \ref{1.3} are
both necessary and sufficient for there to exist a complete
intersection ideal $I = (f, g)$  with Hilbert series
$$\mathrm{H}(\mathrm{gr}_{\frak{m}}(R), \lambda) =
\displaystyle\frac{\sum_{i=2}^n\lambda^{d_i}(1-\lambda^{d_{i-1}
-d_i})(1-\lambda^{c_i -d_i})}{(1-\lambda)^2}.
$$


\begin{remark} \label{1.10} {\em
In the case where $f$ and $g$ are a regular sequence in
a regular local ring $S$ with $\dim S > 2$, it is still true that
$\mathrm{ht}_G(f^*, g^*) > 1$ implies  $f^*, g^*$ is a $G$-regular
sequence, and therefore  $I^* =(f^*, g^*)G$ also in this case.
 Thus if  $\mathrm{ht}_G(f^*, g^*)
= 1$ and if we set  $D_2 = \mathrm{GCD}(f^*, g^*)$ and $d_2 = \mathrm{deg}
D_2$, then  $f^* = D_2\xi$ and $g^* = D_2\eta$. Notice that $\xi, \eta$ is
a regular sequence in $G$. We have  $b \geq a > d_2 > 0$, and
$\mu_G(I^*) = n \ge 3$. There exists a minimal homogeneous system
$\{ \xi_1, \xi_2, \ldots, \xi_n \}$ of generators of $I^*$ such that
$\xi_1 = f^*$ and $\xi_2 = g^*$, and  $c_i := \deg \xi_i \le \deg
\xi_{i+1} := c_{i+1}$ for each $i \le n-1$. However, the ideal $I^*$
may fail to be perfect, and it is possible to have $D_3 : =
\GCD(\xi_1, \xi_2, \xi_3) = D_2$ as is illustrated in \cite[Example
1.6]{GHK1}. We prove in \cite[Theorem 1.2]{GHK1} that $I^*$ is perfect
if $n = 3$. We also prove  in \cite{GHK1} that $\xi_3 = h^*$, where
$h$  has the form $h = \alpha f + \beta g \in I$ with
$\mathrm{o}(\alpha) = b-d_2$, and $\mathrm{o}(\beta) = a-d_2$, and
that $c_3 := \mathrm{o}(h)
> a + b -d_2$.  Moreover, if $q = \sigma f + \tau g$ is such that
$q^* \not\in (f^*, g^*)G$ and $\mathrm(o)(\sigma) = b-d_2$, then
$\mathrm{o}(q) = \mathrm{o}(h)$ and $(f^*, g^*, h^*)G = (f^*, g^*,
q^*)G$. Thus the ideal $(\xi_1, \xi_2, \xi_3)G$ is independent of
the choice of $\xi_3$. In the case where $n \ge 4$, we also prove
that $c_4 \ge c_3 + 2$  \cite[Proposition 2.4]{GHK1}. However,
examples shown to us by Craig Huneke and Lance Bryant show that it
is possible to have  $c_{i+1} =  c_i$  for $i \ge 4$. This resolves
a question mentioned in  \cite[Discussion 2.5]{GHK1}. If $I^*$ is
perfect, we prove in \cite{GHK2} that $c_{i+1} \ge c_i + 2$ for
each $i$ with $2 \le i \le n-1$. A question raised in \cite{GHK2}
that remains open  is
whether for $I^*$ perfect in this higher dimensional setting, does it
follow that $I^* \subseteq (\xi_1/D_2, \xi_2/D_2)G$.


}
\end{remark}







\section{The Main Results}

We record in  Proposition  \ref{2.1}  behavior of the Hilbert function with
respect to an  inverse of a transformation considered by Kothari and
described in parts (1) and (2) of \cite[Corollary 2.5]{GHK2}.


\begin{prop} \label{2.1} Assume notation as in Setting \ref{1.1} and let $n =
\mu_G(I^*)$. Let $\xi_1, \xi_2, \cdots ,\xi_n$  be a minimal
homogeneous system of generators for $I^*$ satisfying conditions (1)
and (2) in Theorem \ref{1.2}, and let $c_i = \deg \xi_i$ for $1
\leq i \leq n$. Also assume that $\fkn = (x,y)$ and $x^* \nmid f^*$.
Let $m$ be a positive integer and consider the ideal  $J_m := (f,
x^mg)S$. Then the following assertions hold true.
\begin{enumerate}
\item[$(1)$] $\mu_G(J_m^*) = \mu_G(I^*)$.
\item[$(2)$] $J_m^*$ \hskip 2pt  has $\xi_1, X^m\xi_2,  X^m\xi_3, \ldots ,
X^m\xi_n$ as a  minimal homogeneous system of generators, so the
degree sequence for $J_m^*$ is $(c_1', c_2', \ldots, c_n')$, where
$c_1' = c_1$ and $c_i' = c_i + m$ for $i \ge 2$.
\item[$(3)$] With  $D_i' = \GCD(\xi_1, X^m\xi_2, \ldots , X^m\xi_i)$
and $d_i' = \deg D_i'$, we have $D_i' = D_i$ and $d_i' = d_i$ for $1
\le i \le n$.

\item[$(4)$]
 $\mathrm{H}(\mathrm{gr}_{\frak{n}}(S/J_m),
\lambda) =
\displaystyle\frac{\sum_{i=2}^n\lambda^{d_i}(1-\lambda^{d_{i-1}
-d_i})(1-\lambda^m\lambda^{c_i -d_i})}{(1-\lambda)^2}.$

\item[$(5)$] $\ell(S/J_m) - \ell(S/I) = ma$.

\end{enumerate}

\end{prop}


Since the residue field of $S$ is infinite, with notation as in
Setting \ref{1.1} we may choose $x, y$ so that $\fkn = (x, y)$ and
$x^* \nmid f^*$ and $y^* \nmid f^*$. Thus it is possible to obtain
the hypothesis of Theorem \ref{2.2}.


\begin{Thm} \label{2.2}
Assume notation as in Setting \ref{1.1} with  $n = \mu_G(I^*)$, $a <
b$ and $\fkn = (x, y)$ such that $x^* \nmid f^*$ and $y^* \nmid
f^*$. Let $\xi_1, \xi_2, \cdots ,\xi_n$  be a minimal homogeneous
system of generators for $I^*$ satisfying conditions (1) and (2) in
Theorem \ref{1.2} and let $c_i = \deg \xi_i$ for $1 \leq i \leq n$.
Let $m$ be a positive integer and set
$$V_m = (y^mf, \,\,   x^m(f +
g))S.
$$ Then the following assertions hold true.
\begin{enumerate}
\item[$(1)$]$V_m$ is $\fkn$-primary and  $\mu_G(V_m^*) = \mu_G(I^*) + 1$.
\vskip 2pt
\item[$(2)$] $V_m^*$ has $Y^m\xi_1, X^m\xi_1, X^mY^m\xi_2,
X^mY^m\xi_3, \ldots, X^mY^m\xi_n$ as a  minimal homogeneous system
of generators, so the degree sequence for $V_m^*$ is $(c_1', c_2',
\ldots, c_{n+1}')$, where $c_1' = c_1 + m, \,  c_2' = c_1 + m$  and
$c_i' = c_{i-1}  + 2m$ for $i \ge 3$.

\item[$(3)$] With  $D_i' = \GCD(Y^m\xi_1, X^m\xi_1, X^mY^m\xi_2,  \ldots , X^mY^m\xi_{i-1})$
and $d_i' = \deg D_i'$, we have $D_1' = Y^mD_1$, and $D_i' =
D_{i-1}$  for $2 \le i \le n+1$. Thus $d_1' = m + d_1$ and $d_i' =
d_{i-1}$ for $2 \le i \le n+1$.


\item[$(4)$] $\mathrm{H}(\mathrm{gr}_{\frak{n}}(S/V_m),
\lambda) =
\displaystyle\frac{\sum_{i=2}^{n+1}\lambda^{d_i'}(1-\lambda^{d_{i-1}'
-d_i'})(1-\lambda^{c_i' -d_i'})}{(1-\lambda)^2}$

$= \lambda^a(1 + \lambda + \lambda^2 + \cdots + \lambda^{m-1})^2 +
\displaystyle\frac{\sum_{i=2}^n\lambda^{d_i}(1-\lambda^{d_{i-1}
-d_i})(1-\lambda^{2m}\lambda^{c_i -d_i})}{(1-\lambda)^2}.$

\item[$(5)$] $\ell(S/V_m) - \ell(S/I) = m(2a+m)$.

\end{enumerate}
\end{Thm}


\begin{proof} To show that $V_m$ is $\fkn$-primary, we observe that
$y^mf$ is not in any minimal prime $P$ of $x^m(f+g)$. Since $a < b$,
$X^mf^*$ is the leading form of $x^m(f+g)$. Since $y^* \nmid f^*$,
we see that  $y \not\in P$. Since $(f, f+g)S = I$ and $x^* \nmid
f^*$, we see that $f \not\in P$. Thus $V_m$ is $\fkn$-primary. Let
$V_m' = (f+g, y^mf)S$. The part (2) of \cite [Corollary 2.5]{GHK2} implies that
$\mu_G(I) = \mu_G(V_m')$ and the part (3) of \cite [Corollary 2.5] {GHK2} implies
$\mu_G(V_m') + 1 = \mu_G(V_m)$. It also follows from  Corollary 2.5 in \cite
{GHK2} that $Y^m\xi_1, X^m\xi_1, X^mY^m\xi_2, X^mY^m\xi_3,
\ldots, X^mY^m\xi_n$ is a  minimal homogeneous system of generators
for $V_m^*$. The remaining assertions in Theorem \ref{2.2} follow
from this.
\end{proof}



In Theorem \ref{2.3}, we establish the existence of examples to show
that every Hilbert series described in Theorem \ref{1.3} is
realizable as the Hilbert series of $\mathrm{gr}_{\frak{m}}(R)$ for
some complete intersection ideal $I = (f, g)$.

\begin{Thm} \label{2.3}
Let $a$ and $b$ be positive integers with $a \le b$  and consider a
system consisting of
\begin{enumerate}
\item  An integer $n$ with $2 \le n \le a+1$.
\item A sequence  $(a = c_1, b = c_2,  c_3, \ldots, c_n)$ of integers.
 \item A  sequence  $(a = d_1, d_2, \ldots, d_n = 0)$ of integers such
 that \newline
 $c_{i+1} -c_i > d_{i-1} -d_{i} > 0$ for all $i$ with  $2 \leq i
\leq n-1$.
\end{enumerate}
For each system satisfying these conditions,  there exists an ideal
$I = (f, g)$ as in Setting \ref{1.1} such that $(\xi_1 = f^*, \xi_2
= g^*, \xi_3, \ldots, \xi_n)$ is a minimal set of homogeneous
generators of $I^*$, $\deg \xi_i = c_i$, and $d_i = \deg D_i$, where
$D_i = \GCD(\xi_1, \ldots, \xi_i)$, for each $i$ with $1 \le i \le
n$.

\end{Thm}


\begin{proof} The proof is by induction on $a$. If $a = 1$,
then $n = 2$ and $f = x^a,  g = y^b$ shows the assertion holds in
this case.  Let  $a'  > 1$ be an integer and assume  that the
assertion holds for all positive integers  $a < a'$. Let $(a' =
c_1', b' = c_2', c_3', \ldots, c_{n+1}')$ and $(a' = d_1', d_2', \ldots,
d_{n+1}' = 0)$ be sequences of $n+1$ integers such that
$$c_{i+1}' -c_i' >
d_{i-1}' -d_{i}' > 0 \quad \text{ for all } \quad i \quad
\text{with} \quad 2 \leq i \leq n.
$$ Notice that $d_{i-1}' - d_i'
> 0$ for all $i$ with $2 \le i \le n$ and $a' =
\sum_{i=2}^{n+1}(d_{i-1}'-d_i')$ implies $n+1 \le a'+ 1$. Let $e :=
c_2' - c_1'$.  By Proposition \ref{2.1}, the system consisting of the
integer $n+1$ and the sequences  $(c_1', c_2', \ldots, c_{n+1}')$
and $(a' = d_1', d_2', \ldots, d_{n+1}' = 0)$ is realizable if and
only if the system consisting of $n+1$ and the sequences $(c_1',
c_2' - e, \ldots, c_{n+1}' - e)$ and $(a' = d_1', d_2', \ldots,
d_{n+1}' = 0)$ is realizable. Thus we may assume that $c_1' = c_2' =
a'$.

Let $m = d_1' - d_2'$ and $a = a'-m$. Then $a = d_2'$. Consider the
system consisting of  the positive integer $n$ and the sequences $(a
= c_1, c_2, \ldots, c_n)$ and $(a = d_1, \ldots, d_n)$, where $c_i =
c_{i+1}' - 2m$ and $d_i = d_{i+1}'$ for each $i$ with $2 \le i \le
n$.  We have
$$c_{i+1} -c_i >
d_{i-1} -d_{i} > 0 \quad \text{for all } \quad i \quad \text{with }
\quad 2 \leq i \leq n-1.
$$
Also $a = \sum_{i=2}^{n}(d_{i-1}-d_i)$ implies $n \le a+ 1$. By our
inductive hypothesis the system consisting of the integer $n$ and
the sequences $(a = c_1, c_2, \ldots, c_n)$ and $(a = d_1, \ldots,
d_n = 0)$ is realizable. Moreover, $c_2 = c_3'$ and $c_3' - a' > m$
and $a' = a + m$ implies $c_2 > a$. Therefore by Theorem \ref{2.2}
the system consisting of $n+1$ and the sequences $(c_1', c_2' - e,
\ldots, c_{n+1}' - e)$ and $(a' = d_1', d_2', \ldots, d_{n+1}' = 0)$
is realizable. This completes the proof of Theorem \ref{2.3}.
\end{proof}


\section{Remarks and Examples}

\begin{remark} \label{3.1} {\em
With notation and hypothesis  as in Theorems \ref{1.2} and \ref{1.3}, it follows
from  Theorem \ref{1.3}  that the Hilbert series
$\mathrm{H}(\mathrm{gr}_{\frak{m}}(R), \lambda)=
\sum_{i=0}^{\infty}\dim_k[\mathrm{gr}_{\frak{m}}(R)]_i\lambda^i$ is
uniquely determined by the degree sequence $(c_1, c_2, \ldots, c_n)$
together with  the sequence $(a = d_1 >  d_2 > d_3 > \cdots
> d_n = 0)$. Notice also that
$\dim_k[\mathrm{gr}_{\frak{m}}(R)]_{c_i}=d_{i-1}-1$, for each $i$
with $2 \leq i \leq n$. Usually the  sequence $(c_1, c_2, \cdots,
c_n)$ does not uniquely determine the Hilbert series of
$\mathrm{gr}_{\frak{m}}(R)$, but Theorem \ref{1.3} implies the following.


\begin{enumerate}
\item[$(1)$] $n = \mu_G(I^*) \leq d_{n-1} + (n-1) \leq  \cdots \leq d_3 + 3 \leq d_2 + 2 \leq a + 1$.
\item[$(2)$] $n = \mu_G(I^*) = a + 1 \Longleftrightarrow d_{i-1} - d_i = 1$
for each $i$ with   $2 \le i \le a + 1$.  Therefore, in this case,
$d_i = n-i$  for each $i$ with $1 \le i \le n$, and  the Hilbert
series of $\mathrm{gr}_{\frak{m}}(R)$ is uniquely determined by the
sequence $(c_1, c_2, \cdots, c_n)$.

\item[$(3)$] If  $\mu_G(I^*) = a + 1$, then
$e(\mathrm{gr}_{\frak{m}}(R)) \ge \frac{3a^2-a}{2}$, and equality
holds  $\Longleftrightarrow$ $a = b$ and $c_{i+1}- c_i = 2$ for each
$i$ with $2 \le i \le a$. In this case, the degree sequence of $I^*$
is $(a, a, a+2, a+4, a+6, \cdots,a+(a-1)2)$.
\end{enumerate}


}
\end{remark}

\begin{exam}\label{3.2} {\em
We describe examples of parameter ideals $I =
(f, g)S$ for which $\mu_G(I^*) = 3$. Let
$(a, b, c)$ be a sequence of integers such that  $ 2 \le a \le b < b + 2 \le c$.
If $(a,b,c)$ is the degree sequence of $I^*$, where $I = (f,g)$, then
Theorem \ref{1.3} implies that   $d := d_2 = \deg \GCD(f^*, g^*)$ must satisfy
$c-b > a-d > 0$.

Let
$$
f = x^dy^{a-d} \qquad \text{and}\qquad g = x^b + y^{c+d-a}.
$$
Notice that $c+d-a > b$. We have
$$
I^* = ({x^*}^{d}{y^*}^{a-d}, {x^*}^{b}, {y^*}^{c}) = I_{2}
\left(
\begin{matrix}
{x^*}^{b-d} & {y^*}^{a-d} & 0\\
{y^*}^{c+d-a} & 0 & {x^*}^{d}
\end{matrix}
\right)
$$


}
\end{exam}

\begin{remark} \label{3.3} {\em
Let $I = (f, g)S$ be a parameter ideal such that $\mu_G(I^*) = 3$.
By Theorem $\ref{1.3}$, the Hilbert series of $\mathrm{gr}_{\frak{m}}(R)$ is uniquely determined
by the degree sequence $(a, b, c)$ together with the integer $d := d_2 = \deg \GCD(f^*, g^*)$;
moreover, we must have $c-b > a-d > 0$. Example \ref{3.2} demonstrates that each system
$(a,b,c)$ and $d$ with $c-b > a-d > 0$ is realizable. For integers $(a,b,c)$ with
$2 \le a \le b < b+2 \le c$, the possible values for $d$ are constrained by the
conditions $d \le a-1$ and $d > \min\{0,  a+b-c \}$. Thus the number of possible
values for $d$, and hence the number of different Hilbert series associated
with the degree sequence $(a,b,c)$,   is $\min\{ a-1, c-b-1 \}$.




}
\end{remark}

Example \ref{3.4} illustrates  Theorem \ref{2.3} in the case  where
$\mu_G(I^*) = 4$.

\begin{exam}\label{3.4} {\em
With the notation of Theorem \ref{1.3}, if $I^*$ has degree sequence
$(4, 5, 8, 11)$, then the possibilities for the sequence $(d_1, d_2,
d_3, d_4)$  are
$$ (4, 3, 2, 0), \quad (4, 3, 1, 0), \quad (4, 2, 1, 0). $$



\item[(1)]
This gives,  for appropriate ideals $I^*$,  the following  three
Hilbert series for $G/I^*$.


$$
\aligned
\text{ (i) }   & \mathrm{Q}_{G/I^*}(t) = 1+2t+3t^2+4t^3+4t^4+3t^5+3t^6+3t^7+2t^8+2t^9+2t^{10}+t^{11},\\
\text{ (ii) }   & \mathrm{Q}_{G/I^*}(t) = 1+2t+3t^2+4t^3+4t^4+3t^5+3t^6+3t^7+2t^8+t^9+t^{10},\\
\text{ (iii) } & \mathrm{Q}_{G/I^*}(t) =
1+2t+3t^2+4t^3+4t^4+3t^5+2t^6+2t^7+t^8+t^9+t^{10}.
\endaligned
$$
corresponding to the sequences $\quad (4, 3, 2, 0), \quad (4, 3, 1,
0), \quad (4, 2, 1, 0). $

\item[(2)]
Using Proposition \ref{2.1} and Theorem \ref{2.2}, we may obtain
from the degree sequence $(4,5,8,11)$ of $I^*$, either the degree
sequence $(3,3,6)$ or $(2,2,5)$. Indeed,  $(3,3,6)$ is obtained by
$$
(4,5,8,11) \overset{\text{m=1 in Prop \ref{2.1}}}\longrightarrow (4,4,7,10)
          \overset{\text{m=1 in Thm \ref{2.2}}}\longrightarrow (3,5,8)
         \overset{\text{m=2 in Prop \ref{2.1}}}\longrightarrow
         (3,3,6),
$$
and  $(2,2,5)$ is obtained by
$$
(4,5,8,11) \overset{\text{m=1 in Prop \ref{2.1}}}\longrightarrow (4,4,7,10)
          \overset{\text{m=2 in Thm \ref{2.2}}}\longrightarrow (2,3,6)
         \overset{\text{m=1 in Prop \ref{2.1}}}\longrightarrow (2,2,5).
$$
The sequence  $(3,3,6)$ is the degree sequences associated with
$(x^3+y^5, x^2y)S$
and $(2,2,5)$ is the degree sequence associated with  $(x^2+y^4, xy)S$.\\
$\text{(i)}$ Let $I^{'}_1 = (f_1, g_1)$, where $f_1 = x^3 + y^{5}$,
$g_1 = x^{2}y$. Since $y^* \nmid f^{*}_1$, Proposition  \ref{2.1}
implies that $J^{'}_1 = (f_1, y^2g_1)$  has   degree sequence
$(3,5,8)$.
      Let $\n = (x+y,  y)$. Then $(x^*+y^*) \nmid f^{*}_1$  and  $y^* \nmid
      f^{*}_1$, so
Theorem \ref{2.2} implies that  $V^{'}_1 = (yf_1,
(x+y)(f_1+y^{2}g_1))=(F_1, G_1)$ has  degree sequence $(4,4,7,10)$.
     Since $y^* \nmid G^{*}_1$, Proposition  \ref{2.1} implies that
     $I_1= (yF_1, G_1)=(y^{2}f_1, ( x+y)(f_1+y^{2}g_1)$ has degree sequence $(4,5,8,11)$.
     Also the descending sequence $(d_1, d_2, d_3, d_4)$  of $I^{*}_1$ is $(4,3,2,0)$. Hence
     the Hilbert series of $G/I^{*}_1$ is (i) of part  $(1)$.\\
$\text{(ii)}$ Let $I^{'}_2 = (f_2, g_2)$, where $f_2 = x^3 + y^{4}$,
$g_2 = xy^2$. Since $y^* \nmid f^{*}_2$, Proposition \ref{2.1}
implies that  $J^{'}_2 = (f_2, y^2g_2)$ has   degree
      sequence $(3,5,8)$.
      Let $\n = (x+y,  y)$. Then $(x^*+y^*) \nmid f^{*}_2$  and  $y^* \nmid f^{*}_2$,
      so Theorem \ref{2.2} implies that  $V^{'}_2 = (yf_2, (x+y)(f_2+y^{2}g_2))=(F_2, G_2)$
      has  degree sequence $(4,4,7,10)$.
      Since $y^* \nmid G^{*}_2$, Proposition \ref{2.1} implies that
      $I_2= (yF_2, G_2)=(y^{2}f_2, (x+y)(f_2+y^{2}g_2)$ has
      degree sequence $(4,5,8,11)$.  The descending sequence $(d_1,
      d_2, d_3, d_4)$ of  $I^{*}_2$ is $(4,3,1,0)$. Hence the Hilbert  series of
      $G/I^{*}_2$ is (ii) of part  $(1)$.\\
$\text{(iii)}$ Let $I^{'}_3 = (f_3, g_3)$, where $f_3 = x^2 +
y^{4}$, $g_3 = xy$. Since $y^* \nmid f^{*}_3$, Proposition \ref{2.1}
implies that  $J^{'}_3 = (f_3, yg_3)$ has  degree sequence
$(2,3,6)$.      Let $\n = (x+y,  y)$. Then $(x^*+y^*) \nmid f^{*}_3$
and  $y^* \nmid f^{*}_3$, so Theorem \ref{2.2} implies that $V^{'}_3
= (y^{2}f_3, (x+y)^{2}(f_3+yg_3))=(F_3, G_3)$ has  degree sequence
$(4,4,7,10)$. Since $y^* \nmid G^{*}_3$, Proposition \ref{2.1}
implies that  $I_3= (yF_3, G_3)=(y^{3}f_3, (x+y)^{2}(f_3+yg_3)$ has
degree sequence $(4,5,8,11)$. The descending sequence $(d_1, d_2,
d_3, d_4)$ of $I^{*}_3$ is $(4,2,1,0)$.
     Hence the Hilbert  series of $G/I^{*}_3$ is (iii) of part $(1)$.\\

}
\end{exam}

Example \ref{3.5} illustrates  Theorem \ref{2.3} in the case  where
$\mu_G(I^*) = 5$.

\begin{exam}\label{3.5} {\em
Assume notation as in Theorems  \ref{1.2} and  \ref{1.3} and also assume that
$\Char(k) \ne 2$. If $I^*$ has degree sequence $(6, 7, 12, 15, 18)$,
then the possibilities for the sequence $(d_1, d_2, d_3, d_4, d_5)$
are
$$
\aligned
&(6,5,4,3,0), \quad (6,5,4,2,0), \quad (6,5,3,2,0), \quad (6,5,3,1,0),\\
&(6,4,3,2,0), \quad (6,4,3,1,0), \quad (6,4,2,1,0), \quad
(6,3,2,1,0).
\endaligned
$$
The following eight parameter ideals $I_i = (f_i, g_i)S$ have
associated degree sequence $(6, 7, 12, 15, 18)$ and each of the
eight possible sequences $(d_1, d_2, d_3, d_4, d_5)$.
\begin{enumerate}

\item[(i)] $I_1=(y^{}y^{}(x+y)(f_1+y^{2}g_1),\quad (x+2y)^{}(x+y)^{}(f_1+y^{2}g_1+y^{4}y^{}f_1)) \\
             \text{where} \quad  f_1 = x^4 + y^{6} \quad \text{and}\quad  g_1 = x^{3}y$,  \\
\item[(ii)] $I_2=(y^{}y^{}(x+y)(f_2+y^{2}g_2), \quad (x+2y)^{}(x+y)^{}(f_2+y^{2}g_2+y^{4}y^{}f_2)), \\
             \text{where} \quad f_2 = x^4 + y^{5} \quad \text{and}\quad  g_2 = x^{2}y^{2},\\$
\item[(iii)] $I_3=(y^{}y^{}(x+y)^{2}(f_3+y^{}g_3),\quad  (x+2y)^{}(x+y)^{2}(f_3+y^{}g_3+y^{4}y^{2}f_3)),\\
             \text{where} \quad  f_3=x^{3}+y^5 \quad \text{and}\quad  g_3=x^2y,\\$
\item[(iv)] $I_4=(y^{}y^{2}(x+y)(f_4+y^{2}g_4),\quad (x+2y)^{2}(x+y)^{}(f_4+y^{2}g_4+y^{3}y^{}f_4)),\\
             \text{where} \quad  f_4 = x^3 + y^{5} \quad \text{and}\quad  g_4 = x^{2}y,\\$
\item[(v)] $I_5=(y^{}y^{}(x+y)^{2}(f_5+y^{}g_5),\quad (x+2y)^{}(x+y)^{2}(f_5+y^{2}g_5+y^{4}y^{2}f_5)),\\
             \text{where} \quad  f_5=x^3+y^4 \quad \text{and}\quad  g_5=xy^{2},\\$
\item[(vi)] $I_6=(y^{}y^{2}(x+y)(f_6+y^{2}g_6), \quad (x+2y)^{2}(x+y)^{}(f_6+y^{2}g_6+y^{3}y^{}f_6)),  \\
              \text{where} \quad  f_6 = x^3 + y^{4} \quad \text{and}\quad  g_6 = xy^{2},\\$
\item[(vii)] $I_7=(y^{}y^{2}(x+y)^{2}(f_7+y^{}g_7),\quad (x+2y)^{2}(x+y)^{2}(f_7+y^{}g_7+y^{3}y^{}f_7)), \\
             \text{where} \quad  f_7 = x^2 + y^{4} \quad \text{and}\quad  g_7 = xy,\\$
\item[(viii)] $I_8=(y^{}y^{3}(x+y)(f_8+y^{2}g_8),\quad (x+2y)^{3}(x+y)^{}(f_8+y^{2}g_8+y^{2}y^{}f_8)), \\
              \text{where} \quad  f_8 = x^2 + y^{4} \quad \text{and}\quad  g_8 = xy.$
\end{enumerate}
To obtain these parameter ideals we reason as follows. Using
Proposition \ref{2.1} and Theorem \ref{2.2}, we may obtain from the
degree sequence $(6, 7, 12, 15, 18)$ of $I^*$ one of the following
three degree sequences $(4,4,7)$, $(3,3,6)$, or $(2,2,5)$. Indeed,
$(4,4,7)$ is obtained by
$$
\aligned &(6,7,12,15,18) \overset{\text{m=1 in Prop
\ref{2.1}}}\longrightarrow (6,6,11,14,17)
               \overset{\text{m=1 in Thm \ref{2.2}}}\longrightarrow (5,9,12,15)\\
&\overset{\text{m=4 in Prop \ref{2.1}}}\longrightarrow (5,5,8,11)
                \overset{\text{m=1 in Thm \ref{2.2}}}\longrightarrow (4,6,9)
                \overset{\text{m=2 in Prop \ref{2.1}}}\longrightarrow (4,4,7).
\endaligned
$$
The sequence  $(3,3,6)$ is obtained in two ways:
$$
\aligned &(6,7,12,15,18) \overset{\text{m=1 in Prop
\ref{2.1}}}\longrightarrow (6,6,11,14,17)
               \overset{\text{m=1 in Thm \ref{2.2}}}\longrightarrow (5,9,12,15)\\
 &\overset{\text{m=4 in Prop \ref{2.1}}}\longrightarrow (5,5,8,11)
               \overset{\text{m=2 in Thm \ref{2.2}}}\longrightarrow (3,4,7)
               \overset{\text{m=1 in Prop \ref{2.1}}}\longrightarrow (3,3,6).
\endaligned
$$
and
$$
\aligned &(6,7,12,15,18) \overset{\text{m=1 in Prop
\ref{2.1}}}\longrightarrow (6,6,11,14,17)
              \overset{\text{m=2 in Thm \ref{2.2}}}\longrightarrow (4,7,10,13)\\
&\overset{\text{m=3 in Prop \ref{2.1}}}\longrightarrow (4,4,7,10)
               \overset{\text{m=1 in Thm \ref{2.2}}}\longrightarrow (3,5,8)
               \overset{\text{m=2 in Prop \ref{2.1}}}\longrightarrow (3,3,6).
\endaligned
$$
The sequence  $(2,2,5)$ is  also obtained in two ways:
$$
\aligned &(6,7,12,15,18) \overset{\text{m=1 in Prop
\ref{2.1}}}\longrightarrow (6,6,11,14,17)
               \overset{\text{m=2 in Thm \ref{2.2}}}\longrightarrow (4,7,10,13)\\
&\overset{\text{m=3 in Prop \ref{2.1}}}\longrightarrow (4,4,7,10)
               \overset{\text{m=2 in Thm \ref{2.2}}}\longrightarrow (2,3,6)
               \overset{\text{m=1 in Prop\ref{2.1}}}\longrightarrow (2,2,5).
\endaligned
$$
and
$$
\aligned &(6,7,12,15,18) \overset{\text{m=1 in Prop
\ref{2.1}}}\longrightarrow (6,6,11,14,17)
              \overset{\text{m=3 in Thm \ref{2.2}}}\longrightarrow (3,5,8,11)\\
 &\overset{\text{m=2 in Prop \ref{2.1}}}\longrightarrow (3,3,6,9)
               \overset{\text{m=1 in Thm \ref{2.2}}}\longrightarrow (2,4,7)
               \overset{\text{m=2 in Prop \ref{2.1}}}\longrightarrow (2,2,5).
\endaligned
$$
We describe a  procedure for obtaining the parameter ideal $I_8$.
Let $I^{''}_8 = (f_8, g_8)$, where $f_8 = x^2 + y^{4}$, $g_8 = xy$.
Then  $I^{''}_8$ has degree sequence $(2, 2, 5)$. Since $y^* \nmid
f^{*}_8$, Proposition \ref{2.1} implies that $J^{''}_8 = (f_8,
y^{2}g_8)$ has degree sequence $(2,4, 7)$.  Let $\n = (x+y, y)$.
Then $(x^*+y^*) \nmid f^{*}_8$ and $y^* \nmid f^{*}_8$, so Theorem
\ref{2.2} implies that  $V^{''}_8 = (y^{}f_8,
(x+y)^{}(f_8+y^{2}g_8))=(F^{'}_8, G^{'}_8)$ has degree sequence $(3,
3,6, 9)$.   Since $y^* \nmid G^{'^{*}}_8$, Proposition \ref{2.1}
implies that  $J^{'}_8= (y^{2}F^{'}_8, G^{'}_8)$ has degree sequence
$(3, 5, 8, 11)$.  Since $\n =(x+2y, y)=(x+y, y)=(x, y)$ and
$(x+2y)^{*} \nmid G^{'^{*}}_8$
      and $y^* \nmid G^{'^{*}}_8$,  Theorem \ref{2.2} implies that
      $V^{'}_8=(y^{3}G^{'}_8, (x+2y)^{3}(G^{'}_8+y^{2}F^{'}_8))=(F_8, G_8)$
      has degree sequence $(6,6,11,14, 17)$.
       Since $y^{*} \nmid G^{*}_8$, Proposition \ref{2.1} implies
       that    $I_8=(yF_8, G_8)=(y^{4}G^{'}_8, (x+2y)^{3}(G^{'}_8 +y^{2}F^{'}_8)=(y^{4}(x+y)(f_8+y^{2}g_8),
       (x+2y)^{3}(x+y)(f_8+y^{2}g_8+y^{3}f_8))$, has degree sequence
       $(6,7,12,15,18)$.

       Similar reasoning is used to obtain the parameter ideals
       $I_i$, for $1\leq i \leq 7$.



}
\end{exam}





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\end{document}
