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\title
{The Minimal Height of Quadratic Forms of Given Dimension
}

\dedicatory{In Memory of Martin Kneser}

\keywords
{Quadratic forms, Witt indices,
Chow groups, correspondences.
{\em 2000 Mathematical Subject Classifications:}
11E04; 14C25}


\date
{\today}
%{August, 2004}

\author
[J. Hurrelbrink, N. Karpenko, and U. Rehmann]
{Jurgen Hurrelbrink, Nikita A. Karpenko, and Ulf Rehmann}



\address
{Jurgen Hurrelbrink\\
Department of Mathematics\\
Louisiana State University\\
Baton Rouge, LA 70803\\
USA}
\email
{jurgen@math.lsu.edu}


\address
{Nikita A. Karpenko\\
Laboratoire de Math{\'e}matiques de Lens\\
Facult{\'e} des Sciences Jean Perrin\\
Universit{\'e} d'Artois\\
rue Jean Souvraz SP 18\\
62307 Lens Cedex\\
FRANCE}
\address
{currently:
Institute for Advanced Study\\
1 Einstein Dr.\\
Princeton, NJ 08540\\
USA}
\address
{{\it Web page:}
{\tt www.math.uni-bielefeld.de/\~{ }karpenko}}
\email
{karpenko@euler.univ-artois.fr}

\address
{Ulf Rehmann\\
Fakult\"{a}t f\"ur Mathematik\\
Universit\"at Bielefeld\\
33501 Bielefeld\\
Germany}
\email
{rehmann@mathematik.uni-bielefeld.de}


\thanks
{The second and third authors were partially supported by
the European Community's Human Potential
Programme under contract HPRN-CT-2002-00287, KTAGS.
The James D. Wolfensohn Fund and The Ellentuck Fund support
is acknowledged by the second author.}


\begin{document}

\begin{abstract}
Given an arbitrary $n$,
we consider anisotropic quadratic forms of dimension $n$
over all fields of characteristic $\ne2$ and
prove  that
the height of an $n$-dimensional excellent form (depending on $n$ only)
is the (precise) lower bound of the heights of all forms of dimension $n$.

%HR%For odd $n$ (as well as for even $n\leq170$),
%HR%this is already known due to Hurrelbrink-Rehmann.
\end{abstract}


\maketitle

\tableofcontents

\section
{Introduction}

Excellent quadratic forms and their fundamental properties have been
discovered by Kne\-busch \cite{Knebusch-II} in 1977. They do occur over
arbitrary fields. The simplest examples are sums of squares, which are
always excellent. This shows that, over suitable fields, {\em
anisotropic} excellent forms do occur in any dimension.

The definition of excellent forms is closely related to Pfister forms
\cite{Scharlau}.
A quadratic form $\phi$ is a {\em Pfister neighbor} \cite{Knebusch-II},
%if it is similar to a subform of a Pfister form, such that its
%dimension is more than half of the dimension of that Pfister
%form.
if it is isomorphic to a subform of a quadratic form $\pi$ such that
$\pi$ is similar to a Pfister form and $\dim\phi>(\dim\pi)/2$.
%In this case, the isomorphism class of the Pfister form and the similarity class
%of the orthogonal complement of the subform inside the Pfister form are
%uniquely determined.
In this case, the form $\pi$ and the orthogonal complement of $\phi$ inside
$\pi$ are uniquely determined (up to an isomorphism) by $\phi$.
A quadratic form is called {\em excellent}, if
its dimension is $\le 1$ or if it is a Pfister neighbor with an
excellent complement.

If a quadratic form $\phi$ over a field $F$ is anisotropic
and $\dim\phi\geq2$,
its {\em
first higher Witt index} is the Witt index of the form $\phi_{F(\phi)} = \phi
\otimes_F F(\phi)$, where $F(\phi)$ denotes the function field of the quadric
defined by $\phi$.
The anisotropic kernel $(\phi_{F(\phi)})_{an}$ is the first higher
anisotropic kernel of $\phi$
(see \cite{Scharlau} for the definitions of the usual Witt index and of the
usual anisotropic kernel of a quadratic form).
Repeating this construction gives, for each
$\phi$, a sequence of field extensions of $F$
%which is in a certain sense
%uniquely determined by $\phi$ and
which is
%known to be
called the {\em
(generic) splitting tower} of $\phi$. It is generic in the sense that, for each
extension $L$ of $F$, a specialization from one of the layers $K$ of the
splitting tower of $\phi$ to $L$ can be defined such that the
indices of $\phi_K$ and $\phi_L$ are the same.


The length of the splitting tower of $\phi$ is
called the {\em height} $\Height(\phi)$, and
the respective anisotropic kernels of $\phi$ over the field extensions of
its splitting tower are its {\em higher anisotropic kernels}.

These notions were introduced and extensively studied as well  by Knebusch
in \cite{Knebusch-I}, \cite{Knebusch-II}.
It turned out that, for anisotropic excellent forms $\phi$ of given
dimension, the dimensions of the higher kernels are completely
independent of the underlying excellent form and even of the ground
field $F$.
Knebusch also gave another characterization of excellent forms in
\cite{Knebusch-II}: A quadratic form is excellent if and only if all
of its higher anisotropic kernels are already defined over $F$
itself.
(It should be mentioned here that Knebusch worked over fields of
characteristic $\ne 2$, but recent research by Hoffmann and Laghribi
\cite{Hoffmann-Laghribi}
showed that the same phenomena do occur in characteristic $2$.)

In \cite{HR}, the notion of the {\em splitting pattern} of a quadratic
form $\phi$ was defined as the
%sequence
sequence
of all possible indices of $\phi$
over some field extension.
From the description above it is clear that
all these indices do occur already over the field extensions of
the splitting tower of $\phi$.
%It is clear that
In the present note we define
the splitting pattern of
$\phi$ as the set of the dimensions of its
higher anisotropic kernels, or, equivalently,
as the set of the integers $\dim(\phi_E)_{an}$, where $E$ runs over all
field extension of $F$.

%For an excellent form, it was shown in \cite{HR} that the terms of its
%splitting pattern can be determined by its dimension in an easy way:

Every nonnegative integer $n$ is uniquely representable as
an alternating sum of 2-powers:
$$ n=2^{p_0}-2^{p_1}+2^{p_2}-\dots+(-1)^{r-1}2^{p_{r-1}}+(-1)^r2^{p_r}
\eqno{(*)}$$
with integers $p_0,p_1,\dots,p_r$ satisfying
$p_0>p_1>\dots>p_{r-1}>p_r+1>0$.
For an excellent form, it was shown in \cite{HR} that its
splitting pattern can be determined from $(*)$ in an easy way
(see \cite{HR} for details).
In particular,
the height $\Height(\phi)$ of an anisotropic excellent quadratic form $\phi$ of dimension
$n$ is given by $\Height(\phi) = r$ if $n$ is even and $\Height(\phi) = r-1$ if $n$ is
odd.
% and we therefore define $\Height(n) = r$ for even $n$ (resp. $ = r-1$
%for odd $n$) to be the height of $n$.
%(The dimensions of the higher anisotropic kernels of $\phi$ can be read
%off of the alternating sum $(*)$ as well, see \cite{HR} for details.)

Properties of anisotropic excellent quadratic
forms suggest that the excellent forms should be a kind of
``object of lowest complexity'' among anisotropic forms of a given
dimension.
In some sense they can be considered as generalizations of
Pfister forms to arbitrary dimensions. (Some aspects of their
splitting behavior are described in \cite[cor. 2.14]{HR}, which seem
to support the conjecture that anisotropic excellent forms provide examples
of
%orthogonal groups
quadratic forms
of minimal {\em canonical dimension}
in the sense of \cite{K_3}, for quadratic forms of given dimension.
An object of ``highest complexity'' for forms of given dimension
clearly is the ``generic quadratic form'' of that dimension, whose
height and canonical dimension are maximal for that dimension.)
Hence the authors of \cite{HR} conjectured that, for example, the
height of an anisotropic excellent form should be the minimal possible height
for anisotropic forms of given dimension.

%% This was supported by a result of Hoffmann, which implied that, at
%% every layer $K_i$ in the generic splitting tower of an excellent $\phi$, the
%% ``relative index'' of ${\phi_{K_i}}_{\rm an} \times K_{i+1}$ attains the
%% maximal possible value for the dimension of ${\phi_{K_i}}_{\rm an}$.

However, a proof of that conjecture so far could not be given with
``classical'' methods of the theory of quadratic forms. But it was observed
by the first and third authors that the results in \cite{i_1} (see
Theorem \ref{i1} here)  could be
used to give a simple proof for forms of odd dimension, and, with  a slight
modification, for all forms of height $\le 4$. In order to give a general
proof, the methods of \cite{i_1} had to be refined, which was done by
the second named author in \cite{neravenstva}, and, in addition, by proving
the content of section \ref{One more property of splitting patterns}.
These ingredients allowed him to assemble a general
proof.  The combined methods yielded the proof given in this note.

\smallskip

We shall write $P(n)$ for the set $\{p_0,p_1,\dots,p_r\}$ occurring in $(*)$
(note that $p_r$ coincides with the $2$-adic order $v_2(n)$ of $n$).
Let us point out that, for $n=0$, our representation is the empty sum, so that
$P(0)=\emptyset$.

The {\em height} $\Height(n)$ of the integer $n$
is the number of positive elements in $P(n)$
(so, $\Height(n)$ is
the number $|P(n)|$ of all elements in the set $P(n)$ for even $n$, while
$\Height(n)=|P(n)|-1$ for odd $n$).

%% Let us recall the definitions of the height $\Height(\phi)$
%% and of some related notions.
%% Let $\phi$ be a non-degenerate quadratic form over a field $F$.
%% The {\em splitting pattern} (this expression appears for the first time
%% in \cite{HR}, although we use it in a slightly different sense here)
%% of $\phi$ is the (finite) set
%% $\{\dim(\phi_E)_{an}\}$ where $E$ runs over all field extensions of $F$
%% (and where $(\phi_E)_{an}$ is the anisotropic part of the quadratic
%% form $\phi_E$, see \cite{Scharlau} for the definition of the quadratic form $\phi_E$
%% as well as for
%% the definition of the anisotropic part of a quadratic form).
%% The height $\Height(\phi)$ of $\phi$ is defined as
%% $\Height(\phi)=|\{\dim(\phi_E)_{an}\}|-1$.
%% Clearly, the splitting pattern and, in particular, the height depend
%% only on the
%% anisotropic part of $\phi$, and we may restrict ourself to anisotropic
%% forms, studying these invariants.


Let us state our Main Theorem:

\begin{thm}
\label{main}
For any anisotropic quadratic form $\phi$ over a field of
characteristic $\ne 2$, one has
$$
\Height(\phi)\geq\Height(\dim\phi)\;.
$$
\end{thm}

For even $\dim\phi$, Theorem \ref{main} is proved in section
\ref{Proof for even dimensions};
for odd $\dim\phi$, see Corollary \ref{HR}.

\smallskip

Sometimes, instead of the splitting pattern
$\{\dim(\phi_E)_{an}\}$
of $\phi$,
it is more convenient to consider the set
$\{\iw(\phi_E)\}$ (with $E$ running over all field extensions of $F$),
where $\iw(\phi_E)$ is the Witt index of $\phi_E$.
%(see \cite{Scharlau} for the definition the Witt index).
The elements of this set are called the {\em absolute higher Witt indices} of
$\phi$
(the notion of the higher Witt indices
has been introduced in \cite{HR}).
Putting them in ascending order we assign to them the numbers
$\jw_0=\iw(\phi)<\jw_1<\dots<\jw_\Height=[(\dim\phi)/2]$
and refer to $\jw_i=\jw_i(\phi)$
as the {\em $i$-th (absolute) Witt index} of $\phi$.

Sometimes it is more convenient to consider the {\em relative
higher Witt indices} defined for $i\in[1,\;\Height]$ as
$\iw_i(\phi)=\jw_i(\phi)-\jw_{i-1}(\phi)$
(for convenience,
we also set $\iw_0(\phi)=\jw_0(\phi)$; this is
the usual Witt index $\iw(\phi)$).

\smallskip
{\bf Illustration:} Let $\phi$ be an anisotropic excellent form of
dimension $n=42$. Then $\Height(\phi)=\Height(n)=5$ and the splitting pattern of
$\phi$ is given by the set $\{\dim (\phi_E)_{an}\} = \{
42,22,10,6,2,0\}$.
Accordingly the set $\{j(\phi_E)\}$ of absolute Witt indices
is given by $\{0,10,16,18,20,21\}$, and the
sequence of relative Witt indices is given by $10,6,2,2,1$.
\smallskip

The splitting pattern $\{n_0>n_1>\dots>n_\Height\}$ of $\phi$ is
reconstructed from the relative higher Witt indices and the parity of
$\dim\phi$
by the formulae
$n_{i-1}=n_i+2\iw_i$ ($i\in[1,\;\Height]$),
taking into account that $n_\Height$ is $0$ or $1$
depending on the parity of $\dim\phi$
(note that all $n_i$ have the same parity as $\dim\phi$).
Vice versa, $\iw_i=(n_{i-1}-n_i)/2$ for $i\in[1,\;\Height]$.

The determination of possible values of splitting patterns
is one of the main problems in the modern theory of quadratic forms.
It is known that the splitting pattern of any quadratic form satisfies the
following restriction:

\begin{thm}[{\cite{i_1}}]
\label{i1}
For any integer $i\in[1,\;\Height]$,
there exists some $m$ such that
$2^m<n_{i-1}$, $\iw_i\in[1,\;2^m]$, and $\iw_i\equiv n_{i-1}\pmod{2^m}$.
\end{thm}

\begin{rem}
\label{interpretation}
There is the following interpretation
of Theorem \ref{i1} in terms of the dyadic expansion of $n_{i-1}$:
for an odd $n_{i-1}$, the integer $\iw_i$ is a proper binary
suffix of $n_{i-1}$;
for an even $n_{i-1}$, the integer $\iw_i$ is a proper binary
suffix or a proper $2$-power divisor of $n_{i-1}$.
\end{rem}

\begin{rem}[{\cite{Knebusch-I}}]
\label{knebuschevshchina}
It is useful to keep in mind that if
$\{n_0>n_1>\dots>n_\Height\}$
with $\Height\geq1$
is the splitting pattern of an anisotropic
quadratic form $\phi$ over a field $F$, then
$\{n_1>n_2>\dots>n_\Height\}$
(and, consequently, also $\{n_i>n_{i+1}>\dots>n_\Height\}$ for
any
$i\leq\Height$)
is also the splitting pattern of some quadratic form, namely, of the
quadratic form
$(\phi_{F(\phi)})_{an}$.
%, where $F(X)$ is the function field of the projective quadric
%given by $\phi$.
In particular, it suffices to announce Theorem \ref{i1}
for $i=1$ only.
\end{rem}


It turns out
(see Corollary \ref{HR}), that
Theorem \ref{main} in odd dimensions is an easy consequence of
Theorem \ref{i1}.
In even dimensions the situation is more complicated.
So, before beginning with the proof of Theorem \ref{main} in even dimension,
we establish
one more property of splitting patterns (see section
\ref{One more property of splitting patterns}).

\medskip
\noindent
{\sl Acknowledgments:}
This article was partially completed when the second author was visiting the
University of Western Ontario in London, Canada.  He is extremely
grateful to the University and personally to J{\'a}n
Min{\'a}$\check{\rm{c}}$ and his wife Leslie for the support,
hospitality, and excellent working conditions. The authors would like
to acknowledge also valuable suggestions by S.~Gorenflo,
W.~Jarnicki, and A.~Laghribi.


\section
{Relations between heights}

In this section, the set
$\{n_0>n_1>\dots>n_\Height\}$ is the splitting pattern of some anisotropic
quadratic form $\phi$. The main lemma, due to the third author, is:


\begin{lemma}
\label{what happens}
For an arbitrary integer $i\in[1,\;\Height]$,
the
difference $\Dh(i)=\Height(n_{i-1})-\Height(n_i)$ is as follows:
\begin{enumerate}
\item[(I)]
If the dimension of $\phi$ is odd, then
$|\Dh(i)|=1$.
\item[(II)]
If the dimension of $\phi$ is even, then
$|\Dh(i)|\leq2$ and
\begin{enumerate}
\item[(+2)]
if $\Dh(i)=2$, then
$P(n_i)\subset P(n_{i-1})$ and
$v_2(n_i)\geq v_2(n_{i-1})+2$;
\item[(+1)]
if $\Dh(i)=1$, then
the difference $P(n_i)\setminus P(n_{i-1})$ is either empty
or consists of one element $p$, in which case both integers $p-1$ and
$p+1$ are in $P(n_{i-1})$;
\item[(0)]
if $\Dh(i)=0$, then
the difference $P(n_i)\setminus P(n_{i-1})$
consists of one element $p$ and either $p-1$ or $p+1$ is in $P(n_{i-1})$;
\item[(-1)]
if $\Dh(i)=-1$, then
the difference $P(n_i)\setminus P(n_{i-1})$
consists either
of two elements $p-1$ and $p+1$ for some $p\in P(n_{i-1})$,
or the difference consists of one element;
\item[(-2)]
if $\Dh(i)=-2$, then
the difference $P(n_i)\setminus P(n_{i-1})$
consists
of two elements
(that is, $P(n_i)\supset P(n_{i-1})$);
moreover, one of these two elements is equal to $p+1$ for some $p\in
P(n_{i-1})$.
\end{enumerate}
\end{enumerate}
\end{lemma}


\begin{proof}
We write $p_0,p_1,\dots,p_r$ for the elements of $P(n_{i-1})$ in
descending order.
We have $n_i=n_{i-1}-2\iw_i$.
On the other hand,
by Theorem \ref{i1}, there exists a nonnegative integer $m$ such that
$2^m<n_{i-1}$, $\iw_i\equiv n_{i-1}\pmod{2^m}$, and $1\leq\iw_i\leq2^m$.
The condition $2^m<n_{i-1}$ means that $m< p_0$.
Let us take the element $p_s$ with maximal even $s$ such that
$m<p_s$.

If $m=p_s-1$, then $\iw_i=2^{p_s-1}-2^{p_{s+1}}+2^{p_{s+2}}-\dots$ and,
therefore,
$$
n_i=2^{p_0}-2^{p_1}+\dots-2^{p_{s-1}}+
2^{p_{s+1}}-2^{p_{s+2}}+\dots+(-1)^{r-1}2^{p_r}\;.
$$
If $s=r$ and $p_{r-1}+1=p_{r-2}$,
we get that $P(n_i)$ is $P(n_{i-1})$ without $p_r$ and $p_{r-2}$.
Otherwise, $P(n_i)$ is $P(n_{i-1})$ without $p_s$.

Below in this proof, we are assuming that $m<p_s-1$.

If $s=r$  (and $m<p_r-1$), then $\iw_i=2^m$ and
$n_i=n_{i-1}-2^{m+1}$.
In the case when $m=p_r-2$ we get that $P(n_i)$ is $P(n_{i-1})$ with
$p_r$ transformed to $p_r-1$.
In the case when $m<p_r-2$ we get that $P(n_i)$ is $P(n_{i-1})$ with
$m+1$ added.

Below in this proof, we are assuming that $s<r$.

If $p_s-1>m> p_{s+1}$, then $\iw_i=2^m-2^{p_{s+1}}+2^{p_{s+2}}-\dots$
%(with the exception of the case when $s=r-2$ and $m=p_r$ which we consider a bit later)
and,
therefore,
%\begin{multline*}
$$
n_i=2^{p_0}-2^{p_1}+\dots-2^{p_{s-1}}+2^{p_s}-
2^{m+1}+\\
2^{p_{s+1}}-2^{p_{s+2}}+\dots+
(-1)^{r+1}2^{p_r}\;.
$$
%\end{multline*}
This is the right representation of $n_i$ (and, therefore, $P(n_i)$ is
$P(n_{i-1})$ with $m+1$ added).

%In the excluded case
%of $s=r-1$ and $m=p_r$, we have $\iw_i=2^{p_r}$ and
%$$
%n_i=\dots+2^{p_{r-1}}-2^{p_r+1}-2^{p_r}=\dots+2^{p_{r-1}}-2^{p_r+2}+2^{p_r}\;.
%$$
%If $p_{r-1}=p_r+2$, then $P(n_i)$ is $P(n_{i-1})$ without $p_{r-1}$;
%otherwise $P(n_i)$ is $P(n_{i-1})$ with $p_r+2$ added.

It remains to consider the case with $m\leq p_{s+1}$ (while $s<r$).
For this case, let us first assume that $s=r-1$.
Then $\iw_i=2^m$ and $n_i=n_{i-1}-2^{m+1}$.
If $m<p_r-2$, then $P(n_i)$ is $P(n_{i-1})$ with $p_r+1$ and $m+1$ added.
If $m=p_r-2$, then $P(n_i)$ is $P(n_{i-1})$ with $p_r$ removed and
$p_r+1$ and $p_r-1$ added.
In the case with $m=p_r-1$, one has:
if $p_{r-1}>p_r+2$, then $P(n_i)$ is $P(n_{i-1})$ with $p_r$ removed and
$p_r+1$ added;
if $p_{r-1}=p_r+2$, then $P(n_i)$ is $P(n_{i-1})$ with $p_r$ and $p_{r-1}$
removed while $p_r+1$ added.
Finally, in the case with
$m=p_r$, we have:
if $p_{r-1}=p_r+2$, then $P(n_i)$ is $P(n_{i-1})$ without $p_{r-1}$,
otherwise $P(n_i)$ is $P(n_{i-1})$ with $p_r+2$ added.

We finish the proof considering the case with $m\leq p_{s+1}$ and $s<r-1$.
We have: $\iw_i=2^{p_{s+2}}-2^{p_{s+3}}+\dots+(-1)^r2^{p_r}$ and
\begin{multline*}
n_i=2^{p_0}-2^{p_1}+\dots+2^{p_s}-2^{p_{s+1}+1}+2^{p_{s+1}}-2^{p_{s+2}}
+\dots+(-1)^{r+1}2^{p_r}\;.
\end{multline*}
So, if $p_s>p_{s+1}+1$, then $P(n_i)$ is $P(n_{i-1})$ with
$p_{s+1}+1$ added;
otherwise $P(n_i)$ is $P(n_{i-1})$ with
$p_s$ removed.
\end{proof}

\begin{cor}
\label{P for odd}
For odd-dimensional $\phi$,
and for any $i\in[1,\;\Height]$,
one has
$$
\Height(n_{i-1})-\Height(n_i)\leq1\;.
$$
\qed
\end{cor}

\begin{rem} In fact,
%under the above condition,
for odd-dimensional $\phi$,
we have
$\Height(n_i) = \Height(n_{i-1}) \pm 1,$ which is true as well for
an even-dimensional $\phi$, if
%the relative index at level $i-1$
$\iw_i$
is a
proper binary suffix (and not a proper 2-power divisor) of
%$\dim  {\phi_{K_{i-1}}}_{\rm an} $,
$n_{i-1}$ (see Remark \ref{interpretation}),
but we will not use this fact explicitly during our proof.

\end{rem}

\begin{cor}
\label{HR}
For the height $\Height(\phi)$ of an arbitrary anisotropic quadratic form
$\phi$ of odd dimension $n$, one has
$\Height(\phi)\geq\Height(n)$
(that is, Theorem \ref{main} holds in odd dimension).
\end{cor}

\begin{proof}
We have: $\Height(n_\Height)=0$ (simply because $n_\Height=1$) and
(by Corollary \ref{P for odd}) $\Height(n_{i-1})-\Height(n_i)\leq1$
for every $i\in[1,\;\Height]$.
Therefore, $\Height(n_0)\leq\Height$.
Since $\phi$ is assumed to be anisotropic, $n=n_0$, and we are done.
\end{proof}



\section
{One more property of splitting patterns}
\label{One more property of splitting patterns}

In this section, $\{n_0>n_1>\dots>n_\Height\}$ with $\Height\geq1$
is the splitting pattern of
an anisotropic quadratic form $\phi$
of {\em even} dimension $n=n_0$.
We write $X$ for the projective quadric given by $\phi$ (see
\cite{neravenstva}
for the definition of the reduced modulo $2$ Chow group $\BCH$
of a variety
and for the terminology
concerning algebraic cycles on $X^2=X\times X$).

\begin{prop}[{\cite[th.~3.3 and th.~5.1]{neravenstva}}]
\label{jakoby}
Let $\alpha\in\BCH(X^2)$ be the minimal cycle containing $h^0\times l_0$.
Assume that $\alpha$ also contains $h^q\times l_q$ for some integer
$q\in[1,\;[n/2])$
and take
the minimal $q$ with this property.
Let $i\in[1,\;\Height)$ be the maximal integer such that $\jw_i\leq q$
(then in fact $q=\jw_i$ by \cite[prop. 3.1]{neravenstva}).
Then
$v_2(\iw_{i+1})\geq v_2(\iw_1)$.
Moreover, if
$v_2(\iw_2+\dots+\iw_i)\geq v_2(\iw_1)+2$
(in particular, if $i=1$, in which case the sum $\iw_2+\dots+\iw_i$
is understood to be $0$, while
by definition $v_2(0)=\infty$),
then $v_2(\iw_{i+1})\leq v_2(\iw_1)+1$ so that
$v_2(\iw_{i+1})$ is equal to $v_2(\iw_1)$ or $v_2(\iw_1)+1$
in this case.
\end{prop}

A situation in which
the condition on $\alpha$ in Proposition \ref{jakoby}
is always satisfied,
is described in:

\begin{thm}[{\cite{Izhboldin-Vishik}, for a more elementary proof see \cite{KM2}}]
\label{power of 2}
If $n_1+\iw_1$ ($=n_0-\iw_1$) is not a $2$ power, then the
minimal cycle on $X^2$ containing $h^0\times l_0$
also contains $h^q\times l_q$ for some $q\in[1,\;[n/2])$.
\end{thm}

The main result of the current section, which is the basement of the proof
of Theorem \ref{main} for even dimension, is:

\begin{thm}
\label{vot tak}
Assume that $v_2(n_i)\geq v_2(n_{i-1})+2$ for some $i\in [1,\;\Height)$.
Then the open interval $(i,\;\Height)$,
contains an integer $i'\in (i,\;\Height)$ such that
$
|v_2(n_{i'})-v_2(n_{i-1})|\leq 1
$.
\end{thm}

\begin{proof}
It suffices to consider the case of $i=1$
(see Remark \ref{knebuschevshchina}).
Note that $\Height\geq2$
(otherwise $[1,\;\Height)=\emptyset$).
We set $p=v_2(n_0)$.
By assumption, we have $v_2(n_1)\geq p+2$.
Therefore $v_2(\iw_1)=p-1$.
Clearly, $\iw_1+n_1$ is not a power of $2$;
therefore, by Theorem \ref{power of 2},
the minimal cycle on $X^2$ containing $h^0\times l_0$
also contains $h^q\times l_q$ for some $q\in[1,\;[n/2])$.
Let $j$ be the maximal integer
satisfying $\jw_j\leq q$.
We are going to show that
$v_2(n_j)$ or $v_2(n_{j+1})$ is in $[p-1,\;p+1]$ for this $j$.
Then we can take $i'=j$ in the first case and $i'=j+1$ in the second case.
Note that $i'\ne1,\Height$ (because of $v_2(n_1)\geq p+2$, while
$v_2(n_\Height)=\infty$; we recall that $\dim\phi$ is even in this section).

By the first part of Proposition \ref{jakoby}, $v_2(\iw_{j+1})\geq p-1$.
Consequently, by Theorem \ref{i1}, $v_2(n_j)\geq p-1$ as well.
Since $n_1=2(\iw_2+\dots+\iw_j)+n_j$,
it follows that
$v_2(\iw_2+\dots+\iw_j)+1\geq p-1$.
If $v_2(\iw_2+\dots+\iw_j)< p+1$, then
$v_2(n_j)=v_2(\iw_2+\dots+\iw_j)+1\in[p-1,\;p+1]$.
So, it remains to consider the case when
$v_2(\iw_2+\dots+\iw_j)\geq p+1$, where we may apply
the second part of Proposition \ref{jakoby} as well, stating that
$v_2(\iw_{j+1})\in\{p-1,p\}$.
%If $v_2(n_j)>v_2(\iw_{j+1})+1$, then
%$v_2(n_{j+1})=v_2(\iw_{j+1})+1\in\{p,p+1\}$.
%If $v_2(n_j)=v_2(\iw_{j+1})+1$, then
%$v_2(n_j)\in\{p,p+1\}$.
%Finally, if $v_2(n_j)<v_2(\iw_{j+1})+1$, then
%$v_2(n_j)=v_2(\iw_{j+1})$ and therefore
%$v_2(n_j)\in\{p-1,p\}$.
Since now $v_2(n_j)\geq p+2$ while $n_j=2\iw_{j+1}+n_{j+1}$,
it follows that $v_2(n_{j+1})=v_2(\iw_{j+1})+1\in\{p,p+1\}$.
\end{proof}

\begin{cor}
\label{tak vot}
Under the condition of Theorem \ref{vot tak}, we set $p=v_2(n_{i-1})$
(note that $p\in P(n_{i-1})$, while $\min P(n_i)\geq p+2$).
Then there exists $i'\in (i,\;\Height)$ such that
the set $P(n_{i'})$ contains an element $p'$
with $|p'-p|\leq1$.
\end{cor}

\begin{proof}
Take $i'$ as in Theorem \ref{vot tak} and set $p'=v_2(n_{i'})$.
\end{proof}


\section
{Proof of the Main Theorem}
\label{Proof for even dimensions}




\begin{proof}[Proof of Theorem \ref{main}]
We only need to prove Theorem \ref{main} for even-dimensional forms
(see Corollary \ref{HR}).
So, let $\{n_0>n_1>\dots>n_\Height\}$ with $\Height\geq1$
be the splitting pattern of
an anisotropic quadratic form $\phi$
of even dimension $n=n_0$.


Let $H$ be the set $\{1,2,\dots,\Height\}$ of $\Height=\Height(\phi)$ elements.
For any $i\in H$, we let $\Dh(i)=\Height(n_{i-1})-\Height(n_i)$.
Let $C$ be the subset of $H$ consisting of all $i\in H$ such that
$\Dh(i)=2$ (we recall that $\Dh(i)\leq2$ for any $i\in H$ by Lemma
\ref{what happens}).
We prove Theorem \ref{main} by constructing
a map $f\!:C\to H$ such that
$\Dh(j)\leq 1-|f^{-1}(j)|$ for any $j\in f(C)$
(in particular, $f(C)\subset H\setminus C$).
Since the subsets $f^{-1}(j)\cup\{j\}$, where $j$ runs over $H\setminus C$,
are disjoint and cover $H$,
while  the average value of $\Dh$ on each such subset is $\leq1$,
the average value of $\Dh$ on $H$ is $\leq1$, and
the statement of Theorem \ref{main} follows by the same argument as
the proof
of the
odd-dimensional case (see Corollary \ref{HR}).

To define $f$, let us consider any $i\in C$.
By Lemma \ref{what happens}, $v_2(n_i)\geq v_2(n_{i-1})+2$.
Therefore, by Corollary \ref{tak vot}, there exists $i'\in(i,\;\Height)$
such that the set $P(n_{i'})$ contains an element $p'$ satisfying
$|p'-p|\leq1$ for $p=v_2(n_{i-1})$.
Taking the minimal $i'$ with this property, we set $f(i)=i'$.
Also we define $g(i)$ to be
the minimal element of $P(n_{f(i)})$ such that
$|g(i)-p|\leq1$.

The map $f$  is constructed.
It only remains to check the stated properties of $f$.

First of all we note that by the very definition
of $f$, for any $j\in f(C)$,
the difference of sets $P(n_{j})\setminus P(n_{j-1})$ is nonempty
(and so, $\Dh(j)\ne2$ by Item II+2 of Lemma \ref{what happens}).
Moreover, this difference contains an element $p$ such that
$\{p-1,\;p+1\}\not\subset P(n_{j-1})$
(and so, $\Dh(j)\ne1$ by Item II+1 of Lemma \ref{what happens}).
Therefore, by Lemma \ref{what happens}, $\Dh(j)\leq0$.


Now let $j$ be an element of $f(C)$ with $|f^{-1}(j)|\geq2$.
Let $i_1<i_2$ be two different elements of $f^{-1}(j)$.
Note that $i_1<i_2<j$ and $|p_2-p_1|>1$ (where $p_1=v_2(n_{i_1-1})$,
$p_2=v_2(n_{i_2-1})$) by definition of $f(i_1)$.
We are going to show that $\Dh(j)\leq-1$.
As we already know, $\Dh(j)\leq0$.
If $\Dh(j)=0$, then by Item II-0 of Lemma \ref{what happens}
the difference $P(n_{j})\setminus P(n_{j-1})$
consists of one element $p'$ and either $p'-1$ or $p'+1$ is in $P(n_{j-1})$.
Since the difference $P(n_{j})\setminus P(n_{j-1})$
consists of one element $p'$,
we have $p'=g(i_1)=g(i_2)$, and it follows that $\{p_1,p_2\}=\{p'-1,p'+1\}$
(in particular, $|p_2-p_1|=2$).
Consequently, the set $P(n_{j-1})$ contains neither $p'-1$ nor $p'+1$,
a contradiction.

Now let $j$ be an element of $f(C)$ with $|f^{-1}(j)|\geq3$.
Let $i_1,i_2,i_3$ be three different elements of $f^{-1}(j)$.
The equalities $g(i_1)=g(i_2)=g(i_3)$ do not hold
simultaneously (simply because the three conditions
$|p_2-p_1|=2$, $|p_3-p_2|=2$, and $|p_1-p_3|=2$ can not
be satisfied simultaneously).
On the other hand, the difference $P(n_j)\setminus P(n_{j-1})$
can have at most two elements.
Therefore we may assume that $g(i_1)=g(i_2)$ and that
$g(i_3)$ is different from $g(i_1)=g(i_2)$.
We set $p'=g(i_1)=g(i_2)$.
We are going to show that $\Dh(j)=-2$.
As we already know, $\Dh(j)\leq-1$.
If $\Dh(j)=-1$, then by Item II-1 of Lemma \ref{what happens}
the difference $P(n_{j})\setminus P(n_{j-1})$
consists of $\tilde{p}-1$ and $\tilde{p}+1$ for some $\tilde{p}\in P(n_{j-1})$.
However $p'$ is neither $\tilde{p}-1$ nor $\tilde{p}+1$, a
contradiction.

We finish the proof by showing that $|f^{-1}(j)|$ is never $\geq4$.
Indeed, if $|f^{-1}(j)|\geq4$, then the difference
$P(n_j)\setminus P(n_{j-1})$ contains two elements $p'$ and $p''$ such
that none of $p'\pm1$ and of $p''\pm1$ is in $P(n_{j-1})$, contradicting
Lemma \ref{what happens}.
\end{proof}

\medskip

\section{Final Example}
We close with an illustration:
Let $\phi$ be an anisotropic quadratic form of dimension $n$ over a
field of characteristic not 2. Above we proved that
$\Height(\phi) \in [\Height(n),[n/2]]$.
For $n$ odd one can be more specific. It
is a nice exercise to deduce that $\Height(\phi) \equiv \Height(n)
\mod 2$ for forms of odd dimension $n$. So,
$$\Height(\phi) \in \{\Height(n),\Height(n)+2,\dots,(n-1)/2\}.$$

\smallskip
{\bf Example:} For $n=21$ we have $\Height(n) = 4$, hence
  $\Height(\phi) \in \{4,6,8,10\}$ and, in fact, each of the four
  values is the height $h(\phi)$ of some 21-dimensional anisotropic
  form $\phi$, cf. \cite{V2}.
\smallskip

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\end{document}

