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\begin{document}
\title{Annihilating polynomials for quadratic forms and Stirling numbers of the second kind}
\author{Stefan De Wannemacker}
%\date{September 1st, 2004}
\maketitle

\begin{abstract}
We present a set of generators of the full annihilator ideal for
the Witt ring of an arbitrary field of characteristic unequal to
two satisfying a non-vanishing condition on the powers of the
fundamental ideal in the torsion part of the Witt ring.  This
settles a conjecture of Ongenae and Van Geel.  This result could
only be proved by first obtaining a new lower bound on the 2-adic
valuation of Stirling numbers of the second kind.
\end{abstract}
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%  SECTION : INTRODUCTION                                                                                        %
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\section{Introduction}

In 1937, Witt already observed that the  Witt ring was integral in
the sense that each element was annihilated by a monic integer
polynomial.  Fifty years later, in 1987, Lewis was the first to
give explicit examples of such polynomials \cite{LEW2}. He showed
that the monic polynomial, $p_n(X)$, defined as
\[p_n(X)=(X-n)(X-(n-2))\ldots (X+(n-2))(X+n)\]
annihilates every non-singular quadratic form of dimension $n$
over every field $F$ of characteristic unequal to two.\\ Since
then, many other polynomials in $\mathbb{Z}[X]$ were found
annihilating all or a family of classes of nonsingular quadratic
forms in the Witt ring and we refer the reader to \cite{LEW} for a
nice survey of the main results on this topic.\\

Let $F$ be a field of characteristic not $2$. The object we want
to consider here is the torsion annihilator ideal
\[ A_t(F) = \{f(X) \in \mathbb{Z}[X]\ |\ f(\phi)=0,\ \forall\ \phi \in I_t(F)\}\]
where $I_t(F)=W_t(F)\,\cap\, I(F)$, $\ W_t(F)$ the torsion part of
the Witt ring and $I(F)$ the ideal of all even-dimensional forms
in the Witt ring. Since $A_t(F)$ is an ideal in the noetherian
ring $\mathbb{Z}[X]$, it is finitely generated. The main problem
is to find a set of generators for this ideal.
\ \\
We will prove the following result.\\
\ \\
\emph{For fields $F$ satisfying the conditions that $2^rW_t(F)=0$
and $2^{r-1}(I_t(F))^{2k-1}\neq 0$ with $k$ uniquely determined by
$r$, the torsion annihilator ideal $A_t(F)$ is the ideal generated
by the monomials\\
{\large
\[\lbrace 2^{r-\nu_2((2i)!)}X^{2i}\rbrace_{0 \leq i \leq
k-1}\ \cup\ \lbrace X^{2k}\rbrace,\]} where $\nu_2$ denotes the
$2$-adic valuation function.
}\\
\ \\
In the case of a nonreal field $F$ this theorem was conjectured
(see Corollary \ref{coro_conj} ) by Ongenae and Van Geel
\cite{OVG}. They gave a proof for fields with level $s(F) \leq 16$
and using the same technique, you can check that the theorem holds
for all nonreal fields $F$ with level $s(F)\leq 64$, but a general
method was lacking.
\ \\
The general method, used to prove the theorem, consists in
evaluating a polynomial $f(X) \in \mathbb{Z}[X]$ in the
even-dimensional forms $\bot_{i=1}^{n}\langle\!\langle
a_i\rangle\!\rangle$ with $1 \leq n \leq deg(f).$  This evaluation
can be rewritten as a linear combination of sums of $m$-fold
Pfister forms and the coefficients that appear turn out to be
related to the Stirling numbers of the second kind.\
\ \\
The result about the (torsion) annihilator ideal could only be
proved by first obtaining a new lower bound for the $2$-adic
valuation of all Stirling numbers $S(n,k)$ of the second kind,
namely
\[\nu_2(S(n,k))\geq d(k)-d(n),\quad\text{for}\quad0\leq k\leq n\]
where $d(k)$ the sum of the binary digits in the binary representation of $k$.\\
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%  SUBSECTION : COMBINATORICS                                                                                       %
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\section{Stirling numbers of the second kind}
\subsection{Preliminaries}

Let $n \in \mathbb{N}\,.$ The Stirling numbers $S(n,k)\ (k \in
\mathbb{N})$ of the second kind are given by
\[x^n=\sum_{k=0}^{\infty}S(n,k)(x)_k,\]
where $(x)_k=x(x-1)(x-2)\ldots(x-k+1)$ for $k \in
\mathbb{N}\setminus \{0\}$ and $(x)_0=1.$  Actually $S(n,k)$ is
the number of ways in which it is possible to partition a set with
$n$ elements in $k$\ classes.\\
The Stirling numbers of the second kind can be defined in several
ways.
\begin{proposition}
\begin{align*}
S(n,k) &= \frac{1}{k!} \sum_{i=0}^{k}(-1)^i{k\choose k-i}(k-i)^n,\\
S(n,k)&=S(n-1,k-1)+kS(n-1,k)\\
&\text{ with }\ S(n,0)=S(0,k)=0\ \text{ and }\ S(0,0)=1\\
S(n,k)&=\frac{1}{k!} \sum_{n_1,n_2,...,n_k}{n\choose n_1,n_2,...n_k},\\
&\text{where } n_1,n_2,...,n_k \text{ are non-zero and their sum equals } n.\\
\end{align*}
\end{proposition}
\begin{proof}
\ \\
See \cite{KN} and \cite{R}.\\
\end{proof}

\subsection{$2$-adic valuation of Stirling numbers of the second kind}
The $2$-adic valuation of Stirling numbers of the second kind and
other combinatorial numbers has been widely studied, but many
problems in this area are still unsolved.  We will give a new
lower bound for the $2$-adic valuation of all Stirling numbers of
the second kind.\\
\ \\
Denote by $d(n)$ the sum of the digits in the binary
representation of $n$ and define the $2$-adic valuation function
$\nu_2(n)$ for all non-zero integers $n$ by $\nu_2(n)=p$, where
$2^p|n$ and $2^{p+1}\nmid n$.\\
Recall the following properties.
\begin{align*}
\nu_2(n!)&=n-d(n)\qquad\text{ (Legendre)}\\
\nu_2\left(\binom{n}{k}\right)&=d(k)+d(n-k)-d(n)\qquad\text{ (Kummer)}\\
\end{align*}
for all $k,n \in \mathbb{N}$ with $0\leq k\leq n$.\\
A new lower bound on the $2$-adic valuation of Stirling numbers of
the second kind can be obtained as follows.\\
\begin{theorem}\label{new_lower_bound} Let $n,k \in \mathbb{N}$ and $0 \leq k \leq
n$. Then
\[\nu_2\left(S(n,k)\right) \geq d(k) - d(n).\]
\end{theorem}
\begin{proof}
$\text{ }$\\
\ \\
By induction on $n$.\\
\\
For $n=0$, $\nu_2(S(0,0))=\nu_2(1)\geq d(0) - d(0)$.\\
\\
Assume now that the above inequality is true for all $i<n$.  We
will prove the theorem for $n$. Note that for $k=0$ the result
is obviously true.\\
Let $1\leq k \leq n$. The Stirling numbers of the second kind
satisfy the well-known 'vertical' recurrence relation
\[S(n,k)=\sum_{i=k-1}^{n-1}\binom{n-1}{i}S(i,k-1).\]
Combining this with the 'triangular' recurrence relation
\[S(n,k)=S(n-1,k-1)+kS(n-1,k)\]
we obtain
\[kS(n,k) = \sum_{i=k-1}^{n-1}\binom{n}{i}S(i,k-1).\]
Thus
\begin{align*}
\nu_2(kS(n,k))&=\nu_2\left(\sum_{i=k-1}^{n-1}\binom{n}{i}S(i,k-1)\right)\\
&\geq \min_{k-1\leq i\leq n-1}\{ \nu_2\left(\binom{n}{i}\right)+d(k-1) - d(i)\}\\
& \text{\qquad\qquad\qquad\qquad\qquad\qquad(by the induction hypothesis)}\\
&= \min_{k-1\leq i\leq n-1}\{ d(n-i) + d(k-1) - d(n)\}\\
& \text{\qquad\qquad\qquad\qquad\qquad\qquad(by the Kummer identity)}\\
&= d(k-1)-d(n) + 1.
\end{align*}
So,\\
\begin{align*}
\nu_2(S(n,k))&\geq d(k-1) - \nu_2(k) + 1 -d(n)\\
&=d(k)-d(n).
\end{align*}
\end{proof}
\subsection{Relationship between Stirling numbers of the second kind and quadratic forms}
\ \\
We can evaluate \\
\[f(X)=c_dX^d+\ldots+c_1X+c_0 \in \mathbb{Z}[X]\]\\
in classes of quadratic forms $\phi \in W(F)$ by defining\\
\[f(\phi)=c_d\phi^d\ \bot\ \ldots\ \bot\ c_1\phi\ \bot\ c_0\langle1\rangle \,\in W(F)\]\\
where $c_i\phi = sign(c_i)\underbrace{(\phi\ \bot\ \ldots\ \bot\
\phi)}_{|c_i| \ \text{times}}$ and $\phi^i = \underbrace{\phi\
\otimes\ \ldots\
\otimes\ \phi}_{i \ \text{times}}$.\\
\ \\
For arbitrary $k>0$ and $\ a_1,a_2,\ldots,a_k \in
F^{\ast}=F\setminus\{0\}$ denote with $\langle\!\langle
a_1,a_2,\ldots,a_k\rangle\!\rangle$ the $2^k$-dimensional $k$-fold
Pfister form
\[\langle\!\langle
a_1,a_2,\ldots,a_k\rangle\!\rangle :=\langle 1,a_1\rangle \otimes
\langle 1,a_2\rangle \otimes \langle 1,a_k\rangle.\]\\
Stirling numbers of the second kind turn up in a natural way by
making calculations in the Witt ring.
\begin{proposition}\label{eval_proposition} Let $f(X)=c_dX^d+\ldots+c_1X \in \mathbb{Z}[X].$\\
Then
\begin{align*}
f(\bot_{i=1}^{n}\langle\!\langle
a_i\rangle\!\rangle)&=\ \left(\sum_{q=1}^d2^{q-1}1!\ S(q,1)c_q\right)(\bot_{i=1}^n\langle\!\langle a_i\rangle\!\rangle)\\
&\bot\ \left(\sum_{q=2}^d2^{q-2}2!\ S(q,2)c_q\right)(\bot_{i<j}^{n}\langle\!\langle a_i,a_j \rangle\!\rangle)\\
&\bot\ \ldots\\
&\bot\ \left(\sum_{q=n}^d2^{q-n}n!\
S(q,n)c_q\right)\langle\!\langle a_1,\ldots,a_n\rangle\!\rangle.
\end{align*}
\end{proposition}
\begin{proof}
\ \\
\ \\
The evaluation of the polynomial $f(X)=c_dX^d+\ldots+c_1X$ in sums
of $1$-fold Pfister forms $\bot_{i=1}^{n}\langle\!\langle a_i
\rangle\!\rangle$ can be written in the
following way.\\
\begin{align*}
f(\bot_{i=1}^{n}\langle\!\langle a_i \rangle\!\rangle)
=&\ A_1(c_1,\ldots ,c_d)(\bot_{i=1}^n\langle\!\langle a_i\rangle\!\rangle)\\
&\bot\ A_2(c_1,\ldots ,c_d)(\bot_{i<j}^{n}\langle\!\langle a_i,a_j \rangle\!\rangle)\\
&\bot\ \ldots\\
&\bot\ A_{n}(c_1,\ldots ,c_d)\langle\!\langle
a_1,\ldots,a_n\rangle\!\rangle,
\end{align*}\\
where the $A_p(c_1,\ldots,c_d)$ are expressions in
$c_1,\ldots,c_d$ with natural numbers as coefficients, i.e.
\[A_p(c_1,\ldots,c_d)=\sum_{q=1}^d\gamma_{p,q}c_q,\quad\text{with}\quad\gamma_{p,q}\in\mathbb{N}.\]
\\
The natural number $\gamma_{p,q}$ is the coefficient of
$\langle\!\langle a_1,\ldots,a_p\rangle\!\rangle$ in
$(\bot_{i=1}^{p}\langle\!\langle
a_i\rangle\!\rangle)^{q}$.\\
\ \\
For $p\leq q$ we can write\\
\begin{align*}
\gamma_{p,q}&=\sum_{q_1,q_2,...,q_p \geq 1}^{q_1+q_2+...+q_p=q}{q\choose q_1,q_2,...,q_p}2^{q_1-1}2^{q_2-1}...\ 2^{q_p-1}\\
&=2^{q-p}\sum_{q_1,q_2,...,q_p \geq 1}^{q_1+q_2+...+q_p=q}{q\choose q_1,q_2,...,q_p}\\
&=2^{q-p}p!\,S(q,p).
\end{align*}\\
If $p>q$ then clearly no $p$-fold Pfister form can occur in
$(\bot_{i=1}^{p}\langle\!\langle a_i\rangle\!\rangle)^{q}$.\\ So,
\[\gamma_{p,q}=
\begin{cases} 2^{q-p}p!\,S(q,p) & \qquad\text{if} \quad p \leq q,\\
0 &\qquad\text{otherwise.} \end{cases}
\]
Applying this to the coefficients $A_p(c_1,\ldots,c_d)$ we obtain
\begin{align*}
f(\bot_{i=1}^{n}\langle\!\langle
a_i\rangle\!\rangle)&=\ \left(\sum_{q=1}^d2^{q-1}1!\ S(q,1)c_q\right)(\bot_{i=1}^n\langle\!\langle a_i\rangle\!\rangle)\\
&\bot\ \left(\sum_{q=2}^d2^{q-2}2!\ S(q,2)c_q\right)(\bot_{i<j}^{n}\langle\!\langle a_i,a_j \rangle\!\rangle)\\
&\bot\ \ldots\\
&\bot\ \left(\sum_{q=n}^d2^{q-n}n!\
S(q,n)c_q\right)\langle\!\langle a_1,\ldots,a_n\rangle\!\rangle.
\end{align*}
\end{proof}
\begin{corollary}\label{coro}
Let $f(X)=c_dX^d+\ldots+c_1X+c_0 \in \mathbb{Z}[X]$ and $ \phi\
\simeq\ \langle a_1,a_2,\ldots a_n\rangle$ a quadratic form of
dimension $n$. If $f(n):=f(n\langle 1 \rangle)=0$ then
\begin{align*}
f(\phi)&=\ \left(\sum_{q=1}^d\sum_{t=q}^d2^{q-1}1!\ S(q,1){t\choose q}(-n)^{t-q}c_t\right)(\bot_{i=1}^n\langle\!\langle a_i\rangle\!\rangle)\\
&\bot\ \left(\sum_{q=2}^d\sum_{t=q}^d2^{q-2}2!\ S(q,2){t\choose q}(-n)^{t-q}c_t\right)(\bot_{i<j}^{n}\langle\!\langle a_i,a_j \rangle\!\rangle)\\
&\bot\ \ldots\\
&\bot\ \left(\sum_{q=n}^d\sum_{t=q}^d2^{q-n}n!\ S(q,n){t\choose
q}(-n)^{t-q}c_t\right)\langle\!\langle
a_1,\ldots,a_n\rangle\!\rangle
\end{align*}
\end{corollary}
\begin{proof}
\ \\
Note that for all
\[ \phi\ \simeq\ \langle a_1,a_2,\ldots a_n\rangle \]
 we have
\[ \phi\ \simeq\ \bot_{i=1}^{n}\langle\!\langle
a_i\rangle\!\rangle - n\langle 1 \rangle. \] We can also rewrite
$f(X)$ as
\[f(X)=g(X-n)=\sum_{q=1}^d\sum_{t=q}^d{t\choose q}(-n)^{t-q}c_t(X-n)^q.\] The condition $f(n)=0$ implies that the
constant term in $g(Y)$ with $Y=X-n$ vanishes.  The result follows
from the previous proposition applied to $g(Y)$.
\end{proof}
\begin{lemma}\label{equations}
Let $f(X)=c_dX^d+\ldots+c_1X \in \mathbb{Z}[X]$.  If $\
f(\bot_{i=1}^{k}\langle\!\langle a_{\sigma
(i)}\rangle\!\rangle)=0,\ $ for all $\ 1\leq k \leq n,\
 \sigma \in S_d$ then
\[\left(\sum_{q=n}^{d}2^{q-n}n!\
S(q,n)c_q\right)\langle\!\langle a_{\tau(1)},\ldots a_{\tau(n)}
\rangle\!\rangle=0,\quad\text{for all}\quad \tau \in S_d.\]
\end{lemma}
\begin{proof}
\ \\
By induction on $n$.\\
For $n=1$,
\begin{align*}
0&= f(\langle\!\langle a_i\rangle\!\rangle)\\
&= c_d\langle\!\langle
a_i\rangle\!\rangle^d+c_{d-1}\langle\!\langle
a_i\rangle\!\rangle^{d-1}+\ldots+c_1\langle\!\langle
a_i\rangle\!\rangle\\
&= (2^{d-1}c_d+2^{d-2}c_{d-1}+\ldots+c_1)\langle\!\langle
a_i\rangle\!\rangle\\
&= \left(\sum_{q=1}^d2^{q-1}1!\ S(q,1)c_q\right)\langle\!\langle
a_i\rangle\!\rangle
\end{align*}
since $S(q,1)=1$ for all $q\geq 1$.\\
Assume now that the lemma is true for all $i<n$.  We will
prove the lemma for $n$.\\
Let $\sigma \in S_d$.
\begin{align*}
0=&\ f(\bot_{i=1}^{n}\langle\!\langle
a_{\sigma(i)}\rangle\!\rangle)\\
&\\
=&\ \left(\sum_{q=1}^d2^{q-1}1!\ S(q,1)c_q\right)(\bot_{i=1}^n\langle\!\langle a_{\sigma(i)}\rangle\!\rangle)\\
&\bot\ \left(\sum_{q=2}^d2^{q-2}2!\ S(q,2)c_q\right)(\bot_{i<j}^{n}\langle\!\langle a_{\sigma(i)},a_{\sigma(j)} \rangle\!\rangle)\\
&\bot\ \ldots\\
&\bot\ \left(\sum_{q=n}^d2^{q-n}n!\
S(q,n)c_q\right)\langle\!\langle
a_{\sigma(1)},\ldots,a_{\sigma(n)}\rangle\!\rangle\\
\end{align*}
by Proposition \ref{eval_proposition}.\\
Note that for every subset $U \subset\{1,\ldots,d\}$ of $k\leq d$
elements, there exists a permutation $\tau \in S_d$ such that
$U=\{\tau(1),\ldots,\tau(k)\}$.\\
So,
\begin{align*}
0=&\ \left(\sum_{q=1}^d2^{q-1}1!\ S(q,1)c_q\right)(\bot_{\tau \in X_1\subset S_d}\langle\!\langle a_{\tau(1)}\rangle\!\rangle)\\
&\bot\ \left(\sum_{q=2}^d2^{q-2}2!\ S(q,2)c_q\right)(\bot_{\tau \in X_2\subset S_d}\langle\!\langle a_{\tau(1)},a_{\tau(2)} \rangle\!\rangle)\\
&\bot\ \ldots\\
&\bot\ \left(\sum_{q=n}^d2^{q-n}n!\
S(q,n)c_q\right)\langle\!\langle
a_{\sigma(1)},\ldots,a_{\sigma(n)}\rangle\!\rangle\\
=& \left(\sum_{q=n}^d2^{q-n}n!\ S(q,n)c_q\right)\langle\!\langle
a_{\sigma(1)},\ldots,a_{\sigma(n)}\rangle\!\rangle\\
\end{align*}
by the induction hypothesis.
\end{proof}
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%  SECTION : Annihilator ideals                                                                                  %
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\section{Polynomials annihilating the Witt Ring}
\subsection{Preliminaries}
%\begin{lemma}\label{unique_k}
%For all $r \in \mathbb{N}$ there exists a unique $k \in
%\mathbb{N}$ such that
%\[\nu_2((2k-2)!)<r\leq \nu_2((2k)!).\]
%\end{lemma}
%\begin{proof}
%See \cite{OVG}.
%\end{proof}
\ \\
The \textit{fundamental ideal} $I(F)$ of $W(F)$ is the ideal
consisting of all even dimensional forms of $W(F)$. Put
$I_t(F):=I(F)\cap W_t(F)$.\\
A field $F$ is called \textit{nonreal} if $-1$ is a sum of squares
in $F$.  The natural number \[ s(F):= min\{n \in \mathbb{N}
\;\vline\; a_1^2+a_2^2+\ldots+a_n^2=-1,\ a_i \in F\}\]
is called the \textit{level} of $F$.\\
A field $F$ is called \textit{(formally) real} if  $-1$ is not a
sum of squares in $F$. In this case we define the level
$s(F)=\infty$.\\ It is a well-known fact that $s(F)$ is a power of
two, if $F$ is nonreal (\cite{SCH} or \cite{LAM}). The
\textit{height} of $F$ is the smallest two-power $h(F)=2^d$ such
that $2^dW_{t}(F)=0$. If no such power exists then $h(F):=\infty$.
If $F$ is nonreal then $h(F) = 2 s(F)$.\\
%For real fields $F$, $I_t(F)=W_t(F)$ since every torsion is even.
%For nonreal fields $F$, $I_t(F)=I(F)$ since $W_t(F) = W(F)$.
%We will need the following result.
%\begin{lemma}\label{existence_of_elements}
%Let $F$ be a field such that $(I_t(F))^n\neq0$.  Then there exist
%$a_1,\ldots,a_n \in F^{\ast}$ such that \begin{itemize} \item[(i)]
%$\langle\!\langle a_i \rangle\!\rangle\in I_t(F)$,\quad for all\
%$1 \leq i\leq n$,\item[(ii)] $\langle\!\langle
%a_{1},\ldots,a_{n}\rangle\!\rangle\neq0$.\end{itemize}
%\end{lemma}
%\begin{proof}
%For a nonreal field $F$, $I_t(F)=I(F)$. Since $I^n(F)$ is finitely
%generated by the $n$-fold Pfister forms, the non-vanishing
%condition on the power of the fundamental ideal implies the
%existence of elements $a_1,\ldots,a_{n} \in F^{\ast}$ such that
%$\langle\!\langle a_{1},\ldots,a_{n}\rangle\!\rangle\neq 0$.\\
%For a real field $F$, $I_t(F)=W_t(F)$. The torsion ideal $W_t(F)$
%is generated by the $1$-fold Pfister forms $\langle\!\langle a
%\rangle\!\rangle$ where $a$ is totally negative.  The condition on
%the power of this torsion ideal thus implies the existence of
%totally negative elements $a_{1},\ldots,a_{n}$ such that
%$\langle\!\langle a_{1},\ldots,a_{n}\rangle\!\rangle\neq 0$.
%\end{proof}
 \vskip 15pt \noindent We define the \textit{torsion annihilator
ideal} $A_{t}(F)$ in $\mathbb{Z}[X]$ by
\[ A_{t}(F)=\{ f(X)\in \mathbb{Z}[X]\;\vline\; f(\varphi)=0 \;\text{for all}\; \varphi \in I_{t}(F)\}.\]
\ \\
For $F$ nonreal, define the \textit{full annihilator ideal} $A(F)$
in $\mathbb{Z}[X]$ by
\[ A(F)=\{ f(X)\in \mathbb{Z}[X]\;\vline\; f(\varphi)=0 \;\text{for all}\; \varphi \in W(F)\},\]
the \textit{even annihilator ideal} by\\
 \[A_e(F)=\{f(X)\in \mathbb{Z}[X]\; \vline\; f(\varphi)=0 \;\text{for all even dimensional}\; \varphi \in W(F)\}\]
and the \textit{odd annihilator ideal} by\\
\[ A_o(F)=\{ f(X)\in \mathbb{Z}[X]\; \vline\; f(\varphi)=0 \;\text{for all odd dimensional}\;\varphi \in W(F)\}.\]
\ \\
In what follows, let $k=k(r)$ be the natural number uniquely
determined by $\nu_2((2k-2)!) < r \leq \nu_2((2k)!)$ (see
\cite{OVG}).
\ \\
\ \\ We define the ideals\\
\begin{eqnarray*}
J_{e,r} &=& (\{2^{r-\nu_2((2i)!)}X^{2i}\}_{0\leq i\leq k-1})\ +\  (X^{2k})\\
&=& (2^{r},2^{r-1}X^{2},2^{r-3}X^{4},...,2^{r-\nu_2((2i)!)}X^{2i},...,2^{r-\nu_2((2k-2)!)}X^{2k-2},X^{2k}),\\
J_{o,r}&=& (\{2^{r-\nu_2((2i)!)}(X-1)^{2i}\}_{0\leq i\leq k-1})\ +\  ((X-1)^{2k})\\
&=&(2^{r},2^{r-1}(X-1)^{2},2^{r-3}(X-1)^{4},...,2^{r-\nu_2((2i)!)}(X-1)^{2i},...,2^{r-\nu_2((2k-2)!)}(X-1)^{2k-2},(X-1)^{2k})\\
&&\text{and}\\
J_{r} &=& (\{2^{r-\nu_2((2i)!)}X^{2i}(X-1)^{2i}\}_{0\leq i\leq k-1}) \ +\  (X^{2k}(X-1)^{2k})\\
&=&(2^{r},2^{r-1}X^{2}(X-1)^{2},2^{r-3}X^{4}(X-1)^{4},...,\\
&&2^{r-\nu_2((2i)!)}X^{2i}(X-1)^{2i},...,2^{r-\nu_2((2k-2)!)}X^{2k-2}(X-1)^{2k-2},X^{2k}(X-1)^{2k}).\\
\end{eqnarray*}
\begin{example}
\begin{eqnarray*}
J_{e,1}&=&(2,X^2),\\
J_{e,2}&=&(4,2X^2,X^4),\\
J_{e,3}&=&(8,4X^2,X^4),\\
J_{e,4}&=&(16,8X^2,2X^4,X^6),\\
J_{e,5}&=&(32,16X^2,4X^4,2X^6,X^8).\\
J_{e,6}&=&(64,32X^2,8X^4,4X^6,X^8).\\
\end{eqnarray*}
\end{example}
\subsection{Generators for the full annihilator ideal}
Since $\mathbb{Z}[X]$ is noetherian, the ideals $A_e(F), A_o(F),
A(F)$ and $A_t(F)$ are finitely generated.  Under certain
conditions we give a set of generators.\\ The following lemma is
proved in \cite{OVG}.  We will give an alternative proof, using
Stirling numbers of the second kind.

\begin{lemma}\label{incl_lemma}
Let $F$ be a field for which $2^rW_t(F)=0$ then
\ \\
\[J_{e,r} \subseteq A_t(F).\]
\end{lemma}
\begin{proof}
\ \\
Let $f(X)=2^{r-\nu_2((2i)!)}X^{2i}$ be one of the generators of
$J_{e,r}$ and $\phi \simeq \langle a_1,a_2,\ldots , a_n\rangle$ an
arbitrary even dimensional element of $W_t(F)$.\\  Since $n$ is
even, we have $f(n\langle 1 \rangle) =
2^{r-\nu_2((2i)!)+2i}\left(\frac{n}{2}\right)^{2i} \langle 1
\rangle =
2^{r+d(2i)}\left(\frac{n}{2}\right)^{2i} \langle 1 \rangle = 0$.\\
By Corollary \ref{coro}
\begin{align*}
f(\phi)&=\ \left(\sum_{q=1}^d2^{q-1}1!\ S(q,1){2i\choose q}(-n)^{2i-q}2^{r-\nu_2((2i)!)}\right)(\bot_{i=1}^n\langle\!\langle a_i\rangle\!\rangle)\\
&\bot\ \left(\sum_{q=2}^d2^{q-2}2!\ S(q,2){2i\choose q}(-n)^{2i-q}2^{r-\nu_2((2i)!)}\right)(\bot_{i<j}^{n}\langle\!\langle a_i,a_j \rangle\!\rangle)\\
&\bot\ \ldots\\
&\bot\ \left(\sum_{q=n}^d2^{q-n}n!\ S(q,n){2i\choose
q}(-n)^{2i-q}2^{r-\nu_2((2i)!)}\right)\langle\!\langle
a_1,\ldots,a_n\rangle\!\rangle.
\end{align*}
For all $j\leq q$ we have
\begin{align*}
&\nu_2 \left(2^{q-j}j!\ S(q,j){2i\choose q}(-n)^{2i-q}2^{r-\nu_2((2i)!)}\right)\\
&\qquad\qquad=q-j+\nu_2(j!)+\nu_2(S(q,j))+\nu_2\left({2i\choose q}\right)\\
&\qquad\qquad\qquad+(2i-q)\nu_2(n)+r-\nu_2((2i)!)\\
&\\
&\qquad\qquad=q-j+j-d(j)+\nu_2(S(q,j))+d(q)+d(2i-q)-d(2i)\\
&\qquad\qquad\qquad+(2i-q)\nu_2(n)+r-2i+d(2i)\\
&\qquad\qquad\qquad\qquad\text{(by Kummer and Legendre)}\\
&\\
&\qquad\qquad\geq q-d(j)+d(j)-d(q)+d(q)+d(2i-q)\\
&\qquad\qquad\qquad+(2i-q)\nu_2(n)+r-2i\\
&\qquad\qquad\qquad\qquad\text{(by Theorem \ref{new_lower_bound})}\\
&\\
&\qquad\qquad\geq q+d(2i-q)+(2i-q)+r-2i\\
&\qquad\qquad\qquad\qquad\text{(since $n$ is even)}\\
&\\
&\qquad\qquad\geq r
\end{align*}
Since $2^rW_t(F)=0$ it follows that
\[f(\phi)=0\] or equivalently that
\[f(X)\in A_t(F).\]
\end{proof}
\ \\
This brings us to the main result of this paper:\\
\begin{theorem}
Let $F$ be a field such that $2^rW_t(F)=0$ and
$2^{r-1}(I_t(F))^{2k-1}\neq 0$ with $k$ uniquely determined by
\,$\nu_2((2k-2)!)< r \leq \nu_2((2k)!)$\,. Then
\ \\
\ \\
\[J_{e,r} = A_t(F).\]
\end{theorem}
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%  proof                                                                                                 %
%                                                                                                                %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
\ \\
\ \\
Let $F$ be a field such that $2^rW_t(F)=0$ and
$2^{r-1}(I_t(F))^{2k-1}\neq 0$.  Let $k$ be the unique natural
number such that $\nu_2((2k-2)!) < r \leq \nu_2((2k)!)$.\\
\\
Since $X^{2k} \in J_{e,r}$ annihilates every even quadratic form,
we will have to prove that every polynomial of degree $2k-1$,\\
\[f(X)=c_{2k-1}X^{2k-1}+...+c_1X+c_0,\text{ with }c_i \in \mathbb{Z},\]\\
annihilating every even quadratic torsion form, lies in $J_{e,r}.$\\
\\
$I_t(F)$ is generated by the elements $\langle\!\langle a
\rangle\!\rangle \in I_t(F)$ . The condition on the power of the
fundamental ideal implies the existence of elements
$a_{1},\ldots,a_{2k-1} \in F^{\ast}$ such that the form
$2^{r-1}\langle\!\langle a_1,\ldots,a_{2k-1}\rangle\!\rangle$ is
not zero.\\
Fix such elements.\\
\vskip 5pt We will evaluate the polynomial $f(X)$ in the even
quadratic forms of the form
\[ \bot_{i=1}^{n}\langle\!\langle a_{\sigma(i)}\rangle\!\rangle\]
where $\ 1\leq n\leq 2k-1,\ \sigma \in S_{2k-1}$.\\
\\
Since $f(0)=c_0 \langle 1 \rangle $ has to vanish and
$2^rW_t(F)=0$, it follows that $c_0=b_02^r$ for some $b_0 \in
\mathbb{Z}$.
By Lemma \ref{equations} we get the following set of equations in the Witt ring:\\
\begin{equation}\label{vanishing_eq}
\left(\sum_{q=n}^{2k-1}2^{q-n}n!\ S(q,n)c_q\right)\langle\!\langle
a_1,\ldots a_n \rangle\!\rangle=0,\ 1 \leq n \leq 2k-1.
\end{equation}
\ \\
For $\ n=2k-1$, this becomes
\begin{align*}
0=& (2k-1)!\ S(2k-1,2k-1)c_{2k-1}\langle\!\langle
a_1,\ldots,a_{2k-1}\rangle\!\rangle\\
=&(2k-1)!\ c_{2k-1}\langle\!\langle
a_1,\ldots,a_{2k-1}\rangle\!\rangle.
\end{align*}
Since
\[2^{r-1}\langle\!\langle
a_1,\ldots,a_{2k-1}\rangle\!\rangle\neq 0,\] and $2^rW_t(F)=0$ it
follows that \[(2k-1)!\ c_{2k-1}=b_{2k-1}2^r\quad\text{for some}\
 b_{2k-1} \in \mathbb{Z},\] and since
$\nu_2((2k-1)!)=\nu_2((2k-2)!)<r$ that
\begin{equation*}
c_{2k-1}=2^{r - \nu_2((2k-1)!)}b_{2k-1}\quad\text{for some}\
b_{2k-1} \in \mathbb{Z}.
\end{equation*}
\\
For $\ n=2k-2\ $, (\ref{vanishing_eq}) becomes
\begin{equation}\label{second_eq}
((2k-2)!\ c_{2k-2}+2(2k-2)!\ S(2k-1,2k-2)c_{2k-1})\langle\!\langle
a_1,\ldots,a_{2k-1}\rangle\!\rangle = 0.
\end{equation}
The second term $2(2k-2)!S(2k-1,2k-2)c_{2k-1}\langle\!\langle
a_1,\ldots,a_{2k-2}\rangle\!\rangle$ vanishes since
\begin{eqnarray*}
\nu_2(2(2k-2)!\ S(2k-1,2k-2)c_{2k-1})&=&1+\nu_2((2k-2)!)\\
&&\ +\nu_2(S(2k-1,2k-2))+\nu_2(c_{2k-1})\\
&=&1+2k-2-d(2k-2)\\
&&\ +\nu_2(S(2k-1,2k-2))+\nu_2(c_{2k-1})\\
&&\qquad\qquad\qquad\qquad\qquad\qquad\text{(by Legendre)}\\
&\geq & 2k-1-d(2k-2)\\
&&\ +d(2k-2)-d(2k-1)+\nu_2(c_{2k-1})\\
&&\qquad\qquad\qquad\qquad\qquad\qquad\text{(by Theorem \ref{new_lower_bound})}\\
&\geq & 2k-1-d(2k-1)+r-(2k-1)+d(2k-1)\\
&=&r\\
\end{eqnarray*}
and $2^rW_t(F)=0.$\\
\ \\
Equation (\ref{second_eq}) is thus equivalent to\\
\begin{equation*}
(2k-2)!\ c_{2k-2}\langle\!\langle a_1,\ldots,a_{2k-2}
\rangle\!\rangle=0
\end{equation*}
and it follows, since $\nu_2((2k-2)!)=\nu_2((2k-1)!)<r$ that
\begin{equation*}
c_{2k-2}=2^{r - \nu_2((2k-2)!)}b_{2k-2}\quad\text{ for some }\
b_{2k-2} \in \mathbb{Z}.
\end{equation*}
\vspace*{2ex} \ \\
Using the same technique and observing that
\begin{align*}
\nu_2(2^{q-n}n!\ S(q,n)c_q)&=q-n+\nu_2(n!)+\nu_2(S(q,p))+\nu_2(c_q)\\
&=q-d(n)+\nu_2(S(q,n))+\nu_2(c_q)\\
&\qquad\qquad\qquad\text{(by Legendre)}\\
&\geq q-d(n)+d(n)-d(q)+r-q+d(q)\\
&\qquad\qquad\qquad\text{(by Theorem \ref{new_lower_bound})}\\
&=r\\
\end{align*}
for all $n<q$ and that $\nu_2(n!)<r$ for all $1\leq n\leq 2k-1$,
 the set of equations (\ref{vanishing_eq})
 is equivalent to the set of equations
\begin{equation*}
n!\ c_n\langle\!\langle a_1,\ldots,a_n\rangle\!\rangle=0\;\text{,
where } n=1,\ldots ,2k-1.
\end{equation*}
\ \\
The solutions are,
\[c_{n} = 2^{r-\nu_2(n!)}b_{n} \quad\text{ for some }\ b_{n}\in\mathbb{Z}.\]\\
\vspace*{3ex} We can thus rewrite,
\[f(X) = c_{2k-1}X^{2k-1}+...+c_1X+c_0\]
as
\begin{eqnarray*}
f(X)&=&2^{r-\nu_2((2k-1)!)}b_{2k-1}X^{2k-1}+2^{r-\nu_2((2k-2)!)}b_{2k-2}X^{2k-2}+\\
&&\ \ldots+2^{r-\nu_2((2j+1)!)}b_{2j+1}X^{2j+1}+b_{2j}2^{r-\nu_2((2j)!)}X^{2j}+\ldots+2^{r-\nu_2(1!)}b_1X+2^{r-\nu_2(0!)}b_0\\
&=&2^{r-\nu_2((2k-2)!)}X^{2k-2}(b_{2k-1}X+b_{2k-2})+\ldots+2^{r-\nu_2((2j)!)}X^{2j}(b_{2j+1}X+b_{2j})+\ldots+2^r(b_1X+b_0)\\
\end{eqnarray*}
\vspace*{2ex}
or equivalently,\\
\[ f(X) \in J_{e,r}\] i.e.
\ \\
\[ A_t(F) \subset J_{e,r}\] and using the other inclusion of the previous lemma
\ \\
\[ A_t(F) = J_{e,r}.\]
\end{proof}

\begin{corollary}\label{coro_conj}
Let $F$ be a nonreal field of finite level $s(F)=2^{r-1}$ such
that $s(F)(I(F))^{2k-1}\neq0$, where $k=k(r)$ is uniquely
determined by $\nu_2((2k-2)!)<r\leq \nu_2((2k)!)$. Then
\[ A(F) = J_{r}.\]
\end{corollary}
\begin{proof}
Since $s(F)=2^{r-1}$, we have that $2^rW(F)=0$.  Moreover, if $F$
is a nonreal field, $I_t(F) = I(F)$.  The non-vanishing condition
on the power of the fundamental ideal implies that
\[ A_e(F) = J_{e,r}.\]
If $\phi$ is an odd-dimensional form in $W(F)$, then $\phi \perp
-\,\langle 1 \rangle$ is an even-dimensional form in $W(F)$.  This
implies that
\[ A_o(F) = J_{o,r}.\]
And finally, since $J_{e,r}$ and $J_{o,r}$ are comaximal ideals,
we get
\[A(F) = A_e(F) \cap A_o(F) = J_{e,r} \cap J_{o,r} = J_{e,r} \cdot J_{o,r} = J_{r}.\]
\end{proof}

\begin{corollary}\label{coro_real}
Let $F$ be a real field of finite height $h(F)=2^{r}$ such that
$\frac{1}{2}h(F)(W_t(F))^{2k-1}\neq0$, where $k=k(r)$ is uniquely
determined by $\nu_2((2k-2)!)<r\leq \nu_2((2k)!)$. Then
\[ A_t(F) = J_{e,r}.\]
\end{corollary}
\begin{proof}
By the definition of the height $h(F)=2^{r}$, we have that
$2^rW(F)=0$. Moreover, if $F$ is a real field, $I_t(F)=W_t(F)$.
The non-vanishing condition on the power of the torsion ideal
implies that
\[ A_t(F) = J_{e,r}.\]
\end{proof}
\begin{remark}
In \cite{OVG}, examples are given of non-real fields $F$
satisfying the conditions of Corollary \ref{coro_conj}.\\
Examples of real fields of arbitrary height $h(F)=2^r$ satisfying
the conditions of Corollary \ref{coro_real} can be found. These
results will be published later.
\end{remark}
\begin{remark}
One can show that, for a field $F$, satisfying $2^rW(F)=0$, but
not satisfying the non-vanishing condition
$2^{r-1}(I_t(F))^{2k-1}\neq 0$, the torsion annihilator ideal
$A_t(F)$ always differs from $J_{e,r}.$\\ A set of generators for
the ideal $A_t(F)$ in these cases is, in general, not known.
\end{remark}
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%                                                                                                                %
%  THE BIBLIOGRAPHY                                                                                              %
%                                                                                                                %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{thebibliography}{999}
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%{\bf239},2001, \mbox{pp. 150--160}.
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D.~W.~Lewis, {Witt Rings as Integral Rings}, Invent. Math.
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\bibitem{OVG}
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\end{thebibliography}
\vspace*{4ex} \noindent
Stefan De Wannemacker \\
Department of Mathematics \\
University College Dublin \\
Belfield, Dublin \\
Ireland \\
e-mail: {\tt sdwannem@maths.ucd.ie}
\end{document}

