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    \topmatter
\title First steps of local symplectic algebra \endtitle
\author V.I.Arnold  \endauthor
\address Steklov Mathematical Institute, 8 Gubkina street,
117966 MOSCOW, GSP--1, RUSSIA \endaddress
%\translator\nofrills{xxxxxxx} x \endtranslator
    \endtopmatter
\document

$$
\alignat{2}
  & \hbox{\bf Dedicated to D.B.Fuchs} \\
\endalignat
$$

\bigskip

    \head 1. Introduction \endhead

The goal of this paper is the solution of a very special problem of symplectic
singularity theory -- that of the classification of the symplest singularities
of curves in a symplectic manifold. The interest of this problem consists in
the following observation: there exist nonobvious discrete symplectic invariants
of such singularities. These
 invariants should be expressed in terms of the local
algebra's interaction with symplectic structure. The study of this interaction,
which I propose to call {\it local symplectic algebra}, is necessary for the
understanding of the results of the present paper,
 which at present seem to me to be rather misterious.

 The classical Darboux--Givental theorem claims, that the germ of a smooth
 submanifold of a symplectic manifold is defined (up to a symplectomorphism)
 by the restriction of the symplectic form to the tangent space of
  the submanifold.

 In the case of a smooth curve this restriction vanishes. The results of
 the present paper suggest that something nontrivial
  remains from the symplectic structure
 at the singular points of the curve. It would be interesting to
 describe this ghost of the symplectic structure in terms
  of the local algebra of
 the singularirty. In this paper such a formula is missing: I just
 provide the classification of the curves with simplest singularities in a
 symplectic manifold.

 A {\it singularity} of a curve is a germ of a smooth
 (holomorphic, formal, ...)
mapping of a line into a smooth (holomorphic, formal, ...) manifold at a
singular point (where the derivative of the mapping vanishes). The
singularities are considered up to the diffeomorphisms of the images in the
target
manifold (that is, up to the right-left equivalence in terms of the
singularity theory).

A singularity is callled {\it simple}, if all the singularites of the
neighbouring
mappings belong to a finite set of equivalence classes. The classification
of simple singularities of curves is described in \cite{1}.

The codimension of the set of nonsimple singularities of mappings of
a line to $\C^N$ equals to $6N-6$ (nonsimle singularites first occur at
some points of some curves of generic families, if the number of the
 parameters reaches $6N-7$).

 The list of the simple singularities of curves starts with the series

 $$
  A_2\leftarrow A_4\leftarrow \dots ,
  \ \ \ \ A_{2k} = \( t^2,\ t^{2k+1},\ 0,\dots ,\ 0\).
  $$

  These are all the singularitis, whose Taylor series contains a
   nonzero quadratic term (neglecting the infinitely degenerate ones).
   The list of simple singularities of curves contains also all the
   (finitely degenerate) singularities, whose Taylor expansion starts
   from the nonzero cubic term (they form two 3-indices series)
   and seven (one-index) series of finitely degenerate singularities,
   whose $6$--jet has the form $\(t^4,\ t^6,\ 0,\dots ,\ 0\).$

   The Taylor series of the remaining "sporadic" simple singularities
   of curves start with $\(t^4,\ t^5,\ 0,\dots ,\ 0\)$
(7 curves), $\(t^4,\ t^7,\ 0,\dots ,\ 0\)$ (15 curves),
$\(t^5,\ 0,\dots ,\ 0\)$  (6 curves) and $\(t^6,\ 0,\dots ,\ 0\)$ (3 curves).
 They live
in spaces of dimension at least 2 (5 curves), at least 3 (13 curves),
at least 4 (9 curves), at least 5 (4 curves) and at least 6 (one curve).


 The principal result of the present paper
 is the classification of the singularities
 $A_{2k}$ in the symplectic space (up to the symplectomorphisms). All such
 singularities are simple. It means, that each curve singularity $A_{2k}$
 has only a finite number of symplectic forms.


 The fact, that these singularities should have several symplectic forms is
 evident. Indeed, the flag, formed by the tangent line and by osculating
 two-plane of the curve at the singular point is intrinsically defined as a flag
 in the tangent space to the ambient manifold. The symplectic 2-form vanish
 or not vanish at the 2-plane of the flag, providing at least two
 symplectically different $A_{2k}$ singularities.

 The calculations below show that the number of the symplectically different
 versions of the curve singularity    $A_{2k}$ equals $2k+1.$ This shows that
 the restriction of the symplectic form to the curve is highly nontrivial at
 the singular points of the curve (vanishing outside the singular point).


    \head 2. Classification of the symplectic singularities of the curves
    of the series $A_{2k}$ \endhead

Consider the symplectic space $\C^{2n}$ equipped with the standard Darboux
coordinates $(p_i,\ q_i)$
(the symplectic structure being $\sum dp_i\wedge  dq_i$).


 The following curves provide all the symplectic versions of the $A_{2k}$
 singularities :
 $$
 A_{2k,0}:\ \  \( q_1 = t^2,\  p_1 = t^a,\ p_i =q_i = 0
 \ \hbox{for}\  i > 1 \);
 $$
 $$
 A_{2k,r}:\ \ \( q_1 = t^2,\  q_2 = t^a,\   p_1 = t^b,\  p_2 = 0,\
 p_i =q_i = 0\ \  \hbox{for}\  i > 2 \),
 $$
 where $a = 2k+1,\ b = a +2r,\ \ r > 0.$

 \bigskip

{\bf Theorem 1.} {\it Almost every curve in a symplectic space of dimension
$2n\geq 4,$ whose Taylor series at the singular point
 has a nonzero quadratic term,
is simple and symplectomorphic
 to one of the curves $ A_{2k,r}\ ,\ \  0\leq r\leq 2k.$ }

 \medskip

The exceptional curves are nonsimple and form a subset of
 codimension infinity in the space
of curves. In the holomorphic case such an infinitely degenerate curve
"covers itself more than once" (the mapping of the complex line to the complex
space being not a topological immersion). In the formal or real smooth case
the infinitely degenerate germs verify the same infinite system of the
equations on the coefficients of the Taylor series, as in the holomorphic
case, but their topological description is not so simple.

{\bf Remark.} In the case $r =2k$ one might replace the monomial $t^b$ in
the normal form by zero. The resulting "plane" curve is symplectically
equivalent to the original one. It belongs to the Lagrange plane $p =0.$
The other curves do not belong to any smooth Lagrange manifold.
The invariant $b$ mesures the maximal order of tangency of the curve with a
 smooth Lagrange (or isotropic) surface at the singular point.

\medskip

 Denote the symplectic curves $ A_{2k,0}$ and $ A_{2k,r}$ by the symbols
 $(2;a)$ and $(2,a;b),$ where $ a =2k+1,\ \ b =2k+r,\ \ 0< r <2k.$
 Denote by $(2,a)$ the curve
 $$
 \( q_1 = t^2,\  q_2 = t^a,\  q_i = 0 \ \hbox{for}\  i > 2,\ p_j =0 \).
  $$
 This curve is symplectomorphic (as we sall see later) to $A_{2k,2k}.$

 The symplectic classification of the singular curves in $\C^4$ starts from
 $$
 \alignat{12}
  (2;3)^0\  & \leftarrow &\ (2,3;5)^1\ & \leftarrow &\ (2,3)^2 \ & &  &   &
&  &  & \\
 &    & \uparrow\qquad &  & \uparrow \qquad &\ &  &  &  &  &  & \\
 & & (2;5)^3\ &\ \leftarrow &\ (2,5;7)^4\ &\ \leftarrow &\ (2,5;9)^5\
  &\ \leftarrow &\ (2,5;11)^6\  &\ \leftarrow  &\ (2,5)^7
   \qquad\qquad\qquad & \\
&  &  &  &\uparrow\qquad &
 &\uparrow\qquad & & \uparrow\qquad & & \uparrow \qquad\qquad\qquad  & \\
  & & & & (2;7)^6 & \leftarrow & (2,7;9)^7 & \leftarrow
  &\ (2,7;11)^8 & \leftarrow &\ (2,7;13)^9 \  \leftarrow \dots \qquad
  &\\
   & & & & & & \uparrow\qquad & & \uparrow\qquad & &\uparrow\qquad\qquad
    \qquad  & \\
  & & & & & &\ (2;9)^9\ &\ \leftarrow &\ (2,9;11)^{10}\ &\ \leftarrow
  &\ (2,9;13)^{11} \leftarrow \dots \qquad &\\
  &  & & & & & & &  \uparrow\qquad & & \uparrow\qquad\qquad\qquad  & \\
  & & & & &  &  &  & \dots\qquad & & \dots  ,\qquad\qquad\qquad
   & \\
 \endalignat
 $$
where the upper index is the codimension in the space of singular curves in
$\C^4.$  I hope there are no further adjacencies. I have verified that
there is no adjacency from $(2;5)^3$ to $(2,3)^2$
(permitted by the codimension).

\head  3. Reduction to four-dimensional problem \endhead

A finitely degenerated curve, defined by a mapping of a line, whose
quadratic term at the critical point is not vanishing, belongs to a smoooth
2-surface. The image curve is defined on this surface by an equation of
the form $x^a =y^2,\ a = 2k+1$ (in some local coordinates on the surface, see
\cite{1}).

Consider the restriction of the symplectic form of
 the ambient space to this surface's
tangent space at the singulat point of the curve. This 2-dimensional
subspace is defined unambiguously (does not depend on the choice of the
surface). The restriction of the symplectic form to this 2-space might be a
nonzero or zero bilinear form.

\bigskip

{\bf1$^o$.} {\it Let the restriction be a nonzero bilinear form.}

\bigskip

{\bf Lemma 1.} {\it In this case the curve is symplectomorphic to
$A_{2k,0}:\  q_1 = t^2,\ p_1 = t^a\ (a = 2k+1).$}

{\bf Proof.} By the Darboux-Givental theorem the 2-surface is
symplectomorphic to the coordinate plane $(q_1,p_1).$ Lemma 1 follows now
from the 2-dimensional version of it :

\medskip

{\bf Lemma 2.} {\it The curve $x^a = y^2$ on a symplectic surface is
 symplectomorphic to the curve $q_1^a =p_1^2,$ where $(p_1,q_1)$ are
 some Darboux coordinates.}

{\bf Proof.} Let $\w =dp_1\wedge dq_1 = f dx\wedge dy,\  f(0)>0.$
Consider the quasihomogeneous Euler vector field
$v =2x\partial_x +ay\partial_y.$ This field is tangent to our curve.

For any function $g$ the field $gv$ is also tangent to it. To reduce $f$
to 1 (using the  homotopy method) it suffices to solve (regularly) the
homological equation $L_{gv}\w = h\w$ with respect to the unknown
function $g.$ Since $d\w =0$ this equation takes the form
$$
\hbox{div}(gv)=h.
$$

Denote by $g_{l,m}$ and $h_{l,m}$ the Taylor coefficients of $g$ and $h.$
The solution of the homological equation in the holomorphic (or formal)
case is provided by the formula $g_{l,m} = h_{l,m}/\(2(b+1) + a(m+1)\).$
In the smooth case one solves the linear inhomogeneous first order partial
derivatives equation for $g$ integrating along the characteristics (starting
at the origin with $g=0$).

Lemma 2 and hence Lemma 1 is thus proved.

\bigskip

{\bf2$^o.$} {\it Let the restriction of the
 symplectic form to the tangent plane
of the surface vanish.}

\bigskip


{\bf Lemma 3.} {\it In this case the mapping of the line to the
 symplectic space can be written in the form

 $$
 \( q_1 = t^3,\ \ q_2 = t^a,\ \ p_1 = f(t),\
  \ p_2 = g(t),\ \ p_{>2} = q_{>2} =0\),
 $$
 where $f,g = o(t^a)$ and $p,q$ are some
  Darboux coordinates, $\w =\sum dp_i\wedge dq_i.$}

{\bf Proof.} The tangent plane of the surface at the singular
point of the curve is symplectically reducible to the coordinate plane
$(q_1, q_2).$ Hence in some Darboux coordinates  the mapping takes the form
$ \( q_1 = t^2,\ \ q_2 = t^a,\ \ q_{>2} =0,\ p = 0\),$
up to a perturbation, which is $o(t^a)$.

The projection to the $q$-plane along the $p$-plane can be
therefore reduced to
the form
$ \( q_1 = t^3,\ \ q_2 = t^a,\ \ q_{>2} =0\)$ by
 a diffeomorphism of the q-plane.
This diffeomorphism induces a symplectomorphism,
 preserving the Lagrangian plane $p =0.$
Therefore in the new coordinates $p = o(t^a).$

But every smooth function of $t,$ which is $o(t^a),$
can be represented as the restriction of a smooth function of
$(q_1,\ q_2)$ to the curve $q_1 = t^2,\ \ q_2 = t^a$ according to the
preparation theorem.

Hence our curve belongs to some 4-submanifold, defined by the equations
$q_{>2} = 0$, $p_{>2} = F(q_1,\ q_2)$  and parametrised by the
coordinates $(q_1, q_2, p_1, p_2).$

The restriction of the symplectic structure to this
 submanifold takes the form
 $dp_1\wedge dq_1 +dp_2\wedge dq_2 $ (since  $dq_{>2} =0$).

 According to Darboux-Givental theorem, all the symplectic
 4-submanifolds are locally symplectomorphic. Hence
 our manifold can be symplectically sent to the coordinate 4-space
 $(p_1,q_1,p_2,q_2),\  p_{>2} =q_{>2} = 0.$
 Reducing the curve inside this space to the above form by its
 symplectomorphism, we get Lemma 3.

 \head 4. Lagrange approximating surface. \endhead

 Now we study curves in symplectic 4-space, equipped with
 Darboux coordinates,
  $\w = dp_1\wedge dq_1 +dp_2\wedge dq_2. $

  {\bf Lemma 4.} {\it The mapping of a line to the symplectic space
  $$
q_1 =t^2,\ \ q_2 = t^a,\ \ p_1 = f(t),\ \ p_2 = g(t),
$$
where $f,g = o(t^a),\  a =2k+1,$ can be transformed to the form
$$
Q_1 =t^2,\ \ Q_2 = t^a,\ \ P_1 = \sum_{ r= 1}^{ 2k-1} c_r t^{a+2r} \ ,\ \
P_2 = 0
$$
by a symplectomorphism of the form
 $P = p -\frac{\partial S}{\partial q},\ Q =q.$
}

{\bf Proof .} For simplicity denote $q_1 =Q_1 = x,\ \ q_2 =Q_2 = y.$ Choose
$$
S(x,y) =A(x) +y B(x) +y^2 C(x) +y^3 D(x)
$$
(we shall see in section 6 that the higher degree terms in $y$ are
not helpful.)

We can represent $f$ and $g$ along our curve as
$$
f = u(x) + y z(x),\ \ \  g = v(x) + y w(x),
$$
where $u$ and $v$ are divisible by $x^{k+1},$ while $z$ and $w $ are
divisible by $x$.

We get the following shifts in the $p$-direction :

$$
\frac{\partial S}{\partial x} =
 A^\prime +y B^\prime +y^2 C^\prime + y^3 D^\prime,
$$
$$
\frac{\partial S}{\partial y} =
 B + 2y C +3 y^2 D.
 $$
 Along our curve $ y^2 = x^a.$ Therefore along it we have

$$
\frac{\partial S}{\partial x} =
(A^\prime +x^a C^\prime) +y (B^\prime + x^a D^\prime),
$$
$$
\frac{\partial S}{\partial y} =
 (B  +3 x^a D) + 2y (2C).
 $$

 Choose now $C,\ A$ and $B$ from the equations $ 2C = w, \ \
 A^\prime + x^a C^\prime = u,\ \ B+3 x^a D = v,$ where $D$ will be choosen
 later. Along our curve we get
 $ g- \frac{\partial S}{\partial y} = 0$ (hence  $P_2 = 0$)
 and
  $ f - \frac{\partial S}{\partial x} = y\( z- \(B^\prime + x^aD^\prime\)\) =
   y\(z - v^\prime + 2 x^a D^\prime + 3 a D x^{a-1}\).$
Choose now $D$ to kill all the terms of degree $a-1$ and higher in
$z-v^\prime.$  We thus reduce the bracket to the polynomial in $x$
 of degree smaller than $a-1=2k$
$$
z-v^\prime +  2 x^a D^\prime + 3 a D x^{a-1} =\sum^{2k-1}_{r=1} c_r x^r.
$$
Here $r>0,$ since $z(0) = v^\prime (0) = 0.$  Along our curve
$P_1 = y \sum c_r x^r = \sum c_r t^{a+2r},$ and Lemma 4  is proved.

{\bf Remark.} The meaning of these calculation is the choice of the
 Lagrange surface  $p = \frac{\partial S}{\partial q},$ which have the
 maximal tangency order with our curve at the singular point.
  The distance to this surface is of order $t^b,$ where
  $b = a +2\hbox{min}\{r|\ c_r \neq 0\}.$
  If all the coefficients $c_r$ are vanishing, the distance is zero
  (the curve belong to a smooth Lagrange surface).

  \head  5. Proof of the normal form theorem. \endhead

  Suppose that $c_r$ is the first nonzero coefficient
  of the polynomial of Lemma 4.
  Taking into account Lemmata 1--4, we see, that it remains
  to prove the following

 {\bf Lemma 5.}{\it  The curve, defined in terms of the Darboux
 coordinates in symplectic $\C^4$ as the image of the mapping
 $$
 (q_1 = t^2,\ \ q_2 = t^a,\ \ p_1 = t^b,\ \ p_2 = 0)
 $$
$ (a =2k+1,\ \ b =a +2r,\ \ 0 < r < 2k),$ is symplectomorphic to the
curve, defined as the image of the mapping
$$
\(Q_1 = T^2,\ \ \ Q_2 = T^a,\ \  P_1 = T^b F(Q_1),\ \ P_2 = 0 \)
$$
whatever be the function $F,$ provided that $F(0)\neq 0.$}

{\bf Proof.} We use the symplectomorphism, induced by the change of
the $q$-variables $q_1 =E^2 Q_1,\ \ q_2 =E^a Q_2,$ where
$E = E(Q_1),\ \ E(0)\neq 0.$

The curve $q_1^a = q_2^2$ is sent to itself: $Q_1^a = Q_2^2.$
Since $dq_1 = \(2E E^\prime Q_1 + E^2\)dQ_1,$
$dq_2 = aE^{a-1} Q_2 dQ_1 + E^a dQ_2,$ we find from
$p_1dq_1 + p_2dq_2 =  P_1dQ_1 + P_2 dQ_2$ that
$$
P_1 = p_1\(2E E^\prime Q_1 + E^2\) + p_2aE^{a-1}Q_2,\ \ P_2 = p_2E^a.
$$

Along our curve $p_1 = q_2 q_1^r = E^{a+2r}Q_2Q_1^r,\ \ p_2 =0.$
Therefore along this curve
$$
P_2 = 0,\ \
P_1 = Q_2Q_1^rE^{a+2r}\(2E E^\prime Q_1 + E^2\) =
 T^b E^{a+2r}\(2E E^\prime Q_1 + E^2\).
 $$
 To get the required value of $P_1$ it suffices to solve (with respect to the
unknown smooth function $E(Q_1)$) the singular differential equation
$2E^\prime Q_1 = F E^c -E ,$ where $c= - a  - 2r -1.$


Consider the system defined by the regular vector field on the
$(E,Q)$-plane (we denote $Q_1 = Q$):
$$
\frac{dE}{ds} = F E^c - E,\ \  \frac{dQ}{ds} = 2Q.
$$
The vector field has a stationnary point $(E_0, 0):\qquad$
$Q = 0,\ \ E_0^{c-1} = {F(0)}^{-1} \neq 0.$
This point is a saddle point with eigenvalues $\lambda_1 = 2$ and
$\lambda _2 = F(0)cE_0^{c-1} - 1 = c - 1 = -a -2r-2.$
The separatrice, corresponding to $\lambda_1 = 2,$ is analytic
(smooth if $F$ is smooth) at the  origin. It is the graph of the required
regular solution $E = E(Q),\ \ E(0) = E_0\neq 0.$
Lemma 5 (and therefore the theorem) is proved.

\head 6. Symplectic invariants of singularities. \endhead

We shall now prove that all the curves $A_{2k,r}$ of
 Theorem 1 are symplectically different. The difference between
 $A_{2k,0}$ and al the others is obvious, since $k$ is the invariant of the
 diffeomorphisms and $A_{2k,0}$ is the only case, where the corresponding
 tangent two-space is not isotropic.

 We shall now show, that $r$ is a symplectic invariant of the singularity
 $A_{2k,r},\ \ 0 < r < 2k $  in $\C^4.$

 {\bf Theorem 2.} {\it The curve
  $(q_1 = t^2,\ q_2 = t^a,\ p_1 = t^b,\ p_2 =0),$
 $a =2k+1, \ b =a +2r,\ 0 < r < 2k $ has no smooth Lagrange surface, whose
 tangency with the curve at the origin
  is higher than that of the plane $p = 0.$}

 {\bf Proof.} Suppose there exists such a better approximating surface.
 First we prove, that its tangent plane at the singular point should
 be the plane $p=0.$

 {\bf Lemma 6.} {\it If a smooth surface has the property that the distance
  from the point of our curve to this surface
   is vanishing for $t=0$ with a higher order
  than $t^a$, then  the tangent plan of the surface at the singular
   point of the curve is the plane $ p=0.$}

 {\bf Proof.} Write the regular equation of the surface in the form
 $$
 Aq_1 + Bq_2 + Cp_1 + D p_2 + R(p,q) = 0,
 $$
 where $R$ is the remaining part of the Taylor series (containing quadratic
 and higher order terms). Substituing for $p$ and $q$ their expressions, we get
 $$
 At^2 + Bt^a + Ct^b + R_2t^4 +R_3t^6 +\dots + O(t^{a+2}) + ... \ \ .
 $$
 Since $b > a > 2,$ we get $A = 0$ (otherwise the vanishing order would be
 $2 < a$). Since $a < b$  is odd, we get $B = 0$ (otherwise the vanishing
 order would be equal to $a$ or would be even smaller).
 Hence $ A = B = 0$ and hence the tangent plane is the plane $p=0.$
 Lemma 6 is thus proved.

 According to Lemma 6, the better approximating Lagrange surface would have the
 equation $ p = \frac{\partial S}{\partial q}.$

 We shall denote $q_1 = x,\ \ q_2 = y.$ The terms, containing $y^4$ will give
 the contribution to $p,$ which is $O(y^3) = O(t^{3a})$ along our curve.
 For $r < 2k$ we have $ 3a > b$
 (since $b = a + 2r < a + 4k = 6k+1 < 6k+3$).
 Therefore these terms are small with respect to $t^b$ and can't change the
 tangency order. Thus it suffices to consider
  the Lagrange surface, defined by
 $$
 S = A + y B + y^2 C + y^3 D.
 $$
 Along the curve $ x = t^2,\  y = t^a$ we get
 $$
 \frac{\partial S}{\partial x} = \(A^\prime + x^a C^\prime\)
  +  y\(B^\prime + x^a D^\prime\),
 $$
  $$
 \frac{\partial S}{\partial y} = \(B + 3 x^a D\) +  y\(2C\).
 $$
 To have a better tangency we need  have  along the curve
   $$
 \frac{\partial S}{\partial y} = o(t^b),\ \
   \frac{\partial S}{\partial x} - t^b = o(t^b).
   $$

We shall see that these requirements are contradictory.

The only term with $t^b$ in $ \frac{\partial S}{\partial x}$
is proportional to the term containing $x^r$ in $B^\prime.$
Indeed, $x^ay = t^{3a},\ \ 3a > b,$ and therefore $D^\prime$
contributes only $o(t^b).$ Therefore $B$ contains $x^{r+1}$
 with a nonzero coefficient. Since $ r+1 < 2k+1 = a,$
 $\frac{\partial S}{\partial y}$ contains the term  $x^{r+1} = t^{2r+2}$
  with nonzero
 coefficient.
 The condition $\frac{\partial S}{\partial y} = o(t^b)$
 therefore requires the inequality $2r+2 > b = a +2r, $
 which is impossible since $ a\geq 3.$

 The contradiction shows that the required Lagrange surface
(approximating the curve better than the plane $p=0$)
does not exist.

Theorem 2 is thus proved. The quantity $r$ is therefore a symplectic
invariant of the curve $A_{2k,r}$ in $\C^4.$ It remains an
invariant in higher-dimensional symplectic manifold (it follows from
calculations similar to the calculations above).

\bigskip

\head Acknowledgements \endhead

  Partially supported by the
Russian Basic Research Foundation, project 96-0101104.
The author is grateful to the Max Plank Institute f\"ur
Mathematic, Bonn, where this article had been written
in June 1998.

    \Refs\nofrills{References}
\widestnumber\no1
    \ref\no1
\by V.I.Arnold
\paper Simple singularities of curves.
\jour Proc.of the Steklov  Mathematical Institute,
 volume
\vol226\yr1999.
%  \moreref\jour Math. Revs.
%\vol20\yr1959\pages103
\endref
\endRefs
\enddocument
\end
