# The multiplicative structure on polynomial continuous valuations.

Semyon Alesker
Department of Mathematics, Tel Aviv University, Ramat Aviv
69978 Tel Aviv, Israel
e-mail<EMAIL_ADDRESS>
###### Abstract

We introduce a canonical structure of a commutative associative filtered
algebra with the unit on polynomial smooth valuations and study its
properties. The induced structure on the subalgebra of translation invariant
smooth valuations has especially nice properties (it is the structure of the
Frobenius algebra). We also present some applications.

##  0 Introduction.

The purpose of this paper is to introduce a canonical structure of a
commutative associative filtered algebra with the unit on polynomial smooth
valuations and study its properties. The induced structure on the subalgebra
of translation invariant smooth valuations has especially nice properties (it
is the structure of the Frobenius algebra). We will also present some
applications. Part of the results of this paper was announced in [3].

Let us describe our main results in more detail. Let us remind first of all
some notation and definitions. Let $V$ be a real vector space of finite
dimension $n$. Let ${\cal K}(V)$ denote the class of all convex compact
subsets of $V$. If we fix on $V$ a Euclidean metric then we can define the
Hausdorff metric $d_{H}$ on ${\cal K}(V)$ as follows:

$d_{H}(A,B):=\inf\\{\varepsilon>0|A\subset(B)_{\varepsilon}\mbox{ and
}B\subset(A)_{\varepsilon}\\},$

where $(U)_{\varepsilon}$ denotes the $\varepsilon$-neighborhood of a set $U$.
However the topology on ${\cal K}(V)$ does not depend on the choice of a
Euclidean metric on $V$. Moreover, equipped with this topology, ${\cal K}(V)$
becomes a locally compact topological space (Blaschke’s selection theorem).

######  0.1 Definition.

a) A function $\phi:{\cal K}(V)\longrightarrow\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}$ is called a valuation if for
any $K_{1},\,K_{2}\in{\cal K}(V)$ such that their union is also convex one has

$\phi(K_{1}\cup K_{2})=\phi(K_{1})+\phi(K_{2})-\phi(K_{1}\cap K_{2}).$

b) A valuation $\phi$ is called continuous if it is continuous with respect
the Hausdorff metric on ${\cal K}(V)$.

For the classical theory of valuations we refer to the surveys [16] and [15].
Let us remind the definition of a polynomial valuation introduced by
Khovanskii and Pukhlikov [9], [10].

######  0.2 Definition.

A valuation $\phi$ is called polynomial of degree at most $d$ if for every
$K\in{\cal K}(V)$ the function $x\mapsto\phi(K+x)$ is a polynomial on $V$ of
degree at most $d$.

Note that valuations polynomial of degree 0 are called translation invariant
valuations. Polynomial valuations have many nice combinatorial-algebraic
properties ([9], [10]).

######  0.3 Example.

(1) The Euler characteristic $\chi$ is a continuous translation invariant
valuation (remind that $\chi(K)=1$ for any convex compact set $K$).

(2) Let $\mu$ be a measure on $V$ with a polynomial density with respect to a
Lebesgue measure. Fix $A\in{\cal K}(V)$. Then

$\phi(K):=\mu(K+A)$

is a continuous polynomial valuation (here $K+A:=\\{k+a|k\in K,\,a\in A\\}$).

######  0.4 Remark.

In [3] we have announced a result that polynomial continuous valuations are
dense in the space of all continuous valuations. However during the
preparation of this paper we found a gap in our original argument. So we do
not know if this fact is true.

Let us remind a basic definition from the representation theory. Let $\rho$ be
a continuous representation of a Lie group $G$ in a Fréchet space $F$. A
vector $\xi\in F$ is called $G$-smooth if the map $g\mapsto\rho(g)\xi$ is
infinitely differentiable map from $G$ to $F$. It is well known that the
subset $F^{sm}$ of smooth vectors is a $G$-invariant linear subspace dense in
$F$. Moreover it has a natural topology of a Fréchet space (which is stronger
than that induced from $F$), and the representation of $G$ is $F^{sm}$ is
continuous. Moreover all vectors in $F^{sm}$ are $G$-smooth.

We will denote by $GL(V)$ the group of all linear transformations of $V$, and
by $Aff(V)$ the group of all affine transformations of $V$.

We will especially be interested in polynomial valuations which are
$GL(V)$-smooth. The space of $GL(V)$-smooth valuations polynomial of degree at
most $d$ will be denoted by $PVal_{d}^{sm}(V)$. This is a Fréchet space. Let
$PVal^{sm}(V)$ denote the inductive limit of the Fréchet spaces
$PVal_{d}^{sm}(V)$ (with the topology of inductive limit).

In Section 1 we define a canonical structure of commutative associative
algebra with unit on $PVal^{sm}(V)$ where the unit is the Euler
characteristic. Let us give the idea of this construction. Let us denote by
${\cal G}^{\prime}(V)$ the linear space of valuations on $V$ which are finite
linear combinations of valuations from Example 0.3 (2). It turns out that
${\cal G}^{\prime}(V)\cap PVal^{sm}(V)$ is dense in $PVal^{sm}(V)$ (see the
proof of Lemma 1.1). Let $W$ be another linear real vector space. Let us
define the exterior product $\phi\boxtimes\psi\in{\cal G}^{\prime}(V\times W)$
of two valuations $\phi\in{\cal G}^{\prime}(V),\,\psi\in{\cal G}^{\prime}(W)$.
Let $\phi(K)=\sum_{i}\mu_{i}(K+A_{i}),\,\psi(L)=\sum_{j}\nu_{j}(L+A_{j})$.
Define

$(\phi\boxtimes\psi)(M):=\sum_{i,j}(\mu_{i}\boxtimes\nu_{j})(M+(A_{i}\times\\{0\\})+(\\{0\\}\times
B_{j})),$

where $\mu_{i}\boxtimes\nu_{j}$ denotes the usual product measure.

Now let us define a product on ${\cal G}^{\prime}(V)$. Let
$\Delta:V\hookrightarrow V\times V$ denote the diagonal imbedding. For
$\phi,\,\psi\in{\cal G}^{\prime}(V)$ let

$\phi\cdot\psi:=\Delta^{*}(\phi\boxtimes\psi),$

where $\Delta^{*}$ denotes the restriction of a valuation on $V\times V$ to
the diagonal. By Proposition 1.4 this product is associative and commutative,
the Euler characteristic is the identity element in this algebra, and this
product extends (uniquely) to a continuous product on $PVal^{sm}(V)$
(Proposition 1.9).

We have the following natural filtration on $PVal^{sm}(V)$:

$\gamma_{i}=\\{\phi\in PVal^{sm}(V)|\,\phi(K)=0\,\forall K\mbox{ such that
}\dim K<i\\}.$

Then
$PVal^{sm}(V)=\gamma_{0}\supset\gamma_{1}\supset\dots\supset\gamma_{n}\supset\gamma_{n+1}=0$.
However this filtration is not compatible with the multiplicative structure.
Nevertheless there exists another filtration $W_{i}$ which can be
characterized as follows.

######  0.5 Theorem.

There exists a unique filtration $W_{i}$ on $PVal^{sm}(V)$ such that

(1) $\\{W_{i}\\}$ is compatible with the multiplicative structure, i.e.
$W_{i}\cdot W_{j}\subset W_{i+j}$;

(2) $\gamma_{i+1}\subset W_{i}\subset\gamma_{i}\mbox{ for all }i.$

(3) $W_{0}=\gamma_{0},\,W_{1}=\gamma_{1}$;

(4) $W_{i}$ is a closed subspace of $PVal^{sm}(V)$.

(5) $W_{i}$ is $Aff(V)$-invariant.

This theorem is a reformulation of Theorem 3.8 of Section 3. The explicit
construction of this filtration is given in Section 3. One of the ways to
describe this filtration explicitly is as follows (see Proposition 3.4):

$\phi\in W_{i}\mbox{ iff
}\lim_{r\longrightarrow+0}r^{-i+1}\phi(rK+x)=0\,\forall K\in{\cal K}(V),\,x\in
V.$

It turns out that one can easily describe the associated graded algebra
$gr_{W}(PVal^{sm}(V)$ in terms of the algebra of translation invariant smooth
valuations. In Section 3 we prove the following result (Theorem 3.9).

######  0.6 Theorem.

There exists a canonical isomorphism of graded algebras

$gr_{W}(PVal^{sm}(V))\simeq Val^{sm}(V)\otimes\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}[V]$

where the $i$-th graded term in the right hand side is equal to
$Val_{i}^{sm}(V)\otimes\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}[V]$ where $\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}[V]$ denote the algebra of
polynomial functions on $V$.

In the proof of this theorem we construct this isomorphism explicitly.

Now let us discuss in more detail the case of translation invariant
valuations. We will denote the space of all translation invariant continuous
valuations on $V$ by $Val(V)$. Remind that a valuation $\phi$ is called
$i$-homogeneous if $\phi(\lambda K)=\lambda^{i}\phi(K)$ for all $\lambda\geq
0,\,K\in{\cal K}(V)$. We will denote by $Val_{i}(V)$ the subspace
$i$-homogeneous valuations. By a result of McMullen [13] one has a
decomposition:

$Val(V)=\oplus_{i=0}^{n}Val_{i}(V).$ (1)

Also one has

$Val_{0}(V)=\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}\cdot\chi,\,Val_{n}(V)=\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}\cdot vol$ (2)

where the first equality is trivial, and the second one is due to Hadwiger
[8]. We have also further decomposition of these space with respect to parity.
Namely we say that a valuation $\phi$ is even if $\phi(-K)=\phi(K),\,\forall
K\in{\cal K}(V)$. Similarly $\phi$ is called odd if
$\phi(-K)=-\phi(K),\,\forall K\in{\cal K}(V)$. The subspace of even
translation invariant valuations will be denoted by $Val^{0}(V)$, and the
subspace of odd translation invariant valuations will be denoted by
$Val^{1}(V)$. Similarly $Val_{i}^{0}(V)$ and $Val_{i}^{1}$ will denote their
subspaces of $i$-homogeneous valuations. We obviously have

$Val_{i}(V)=Val_{i}^{0}(V)\oplus Val_{i}^{1}(V).$ (3)

One of the main results on translation invariant valuations we will use in
this paper is as follows (it was proved in [2]).

######  0.7 Theorem (Irreducibility Theorem).

The natural representations of the group $GL(V)$ in $Val_{i}^{0}(V)$ and
$Val_{i}^{1}(V)$ are irreducible.

Below we will denote by
$Val^{sm}(V),\,Val_{i}^{sm}(V),\,(Val_{i}^{0}(V))^{sm},\,(Val_{i}^{1}(V))^{sm}$
the subspaces of $GL(V)$-smooth vectors of the corresponding spaces. The space
$Val^{sm}(V)$ is a subalgebra of $PVal^{sm}(V)$. Moreover the degree of
homogeneity and parity are (obviously) compatible with the multiplication and
define the structure of the bigraded algebra on $Val^{sm}(V)$ (when the first
grading is by  Z and the second grading is by $\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$Z}/2\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$Z}$). The non-trivial property of
it is the following version of the Poincaré duality (remind that $Val_{n}(V)$
is one dimensional).

######  0.8 Theorem.

The maps

$(Val_{i}^{0}(V))^{sm}\longrightarrow((Val_{n-i}^{0})^{*})^{sm}\otimes
Val_{n}(V),\,\mbox{ and }$
$(Val_{i}^{1}(V))^{sm}\longrightarrow((Val_{n-i}^{1})^{*})^{sm}\otimes
Val_{n}(V)$

induced by the multiplication
$(Val_{i}^{0}(V))^{sm}\otimes(Val_{n-i}^{0}(V))^{sm}\longrightarrow
Val_{n}(V)$ and
$(Val_{i}^{1}(V))^{sm}\otimes(Val_{n-i}^{1}(V))^{sm}\longrightarrow
Val_{n}(V)$ are isomorphisms.

This result is proved in Section 2 (Theorem 2.1). The proof is based on
Irreducibility Theorem 0.7. In Section 2 we also compute explicitly some
examples of product of valuations (Proposition 2.2). Note that the
isomorphisms in Theorem 0.8 commute with the natural action on the group
$GL(V)$. An attempt to understand these isomorphisms from purely
representation theoretical point of view was done in [5].

If we forget the grading by parity then we obtain a version of the Poincaré
duality on the graded algebra $Val^{sm}(V)=\oplus_{i=0}^{n}Val_{i}^{sm}(V).$

We also describe the algebra structure on the space of isometry invariant
continuous valuation on an $n$-dimensional Euclidean space (as a vector space
it is described by the classical Hadwiger characterization theorem [8]). This
graded algebra is isomorphic to the graded algebra of truncated polynomials
$\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}[x]/(x^{n+1})$ (see Theorem 2.6
for more details).

Now let us discuss some applications of the above results to the spaces of
valuations invariant under a group. Let $G$ be a compact subgroup of $GL(V)$.
Let us denote by $Val^{G}(V)$ the space of $G$-invariant translation invariant
continuous valuations. Again we have the decomposition

$Val^{G}(V)=\oplus_{i=0}^{n}Val_{i}^{G}(V).$

Let us denote $h_{i}:=\dim Val_{i}^{G}(V).$

######  0.9 Theorem.

Assume that $G$ acts transitively on the projective space $\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$P}(V)$.

(i)Then $Val^{G}(V)$ is finite dimensional.

(ii) $Val^{G}(V)\subset Val^{sm}(V).$

(iii) $Val^{G}(V)$ is finite dimensional graded subalgebra of $Val^{sm}(V)$
satisfying the Poincaré duality:

$Val^{G}_{i}(V)\otimes Val_{n-i}^{G}(V)\longrightarrow
Val_{n}^{G}(V)=\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}\cdot vol$

is a perfect pairing. In particular $h_{i}=h_{n-i}$.

(iv) $Val_{1}^{G}(V)$ is spanned by the intrinsic volume $V_{1}$, and
$Val^{G}_{n-1}(V)$ is spanned by the intrinsic volume $V_{n-1}$. Thus
$h_{1}=h_{n-1}=1$.

(v) Assume in addition that $-Id\in G$. Then the numbers $h_{i}$ satisfy the
Lefschetz inequalities:

$h_{i}\leq h_{i+1}\mbox{ for }i<n/2.$

For the definition of the intrinsic volumes we refer to [17]. Note that part
(i) of this theorem was proved in [1] (Theorem 8.1), and part (ii) in [4] (see
the proof of Corollary 1.1.3). Part (iii) is a direct consequence of Theorem
0.8. Note that the only proof we know of the equality $h_{i}=h_{n-i}$ for a
general compact group $G$ is based on the existence of the multiplicative
structure on a much larger (infinite dimensional) space $Val^{sm}(V)$ and a
version of the Poincaré duality for this larger algebra which is in turn is
based on the Irreducibility Theorem 0.7. Part (iv) will be proved in Section 2
of this paper. Part (v) of Theorem 0.9 was proved in [4] and it is based on a
version of the hard Lefschetz theorem for even valuations. We expect that this
result (i.e. hard Lefschetz theorem) should be true also for odd valuations;
in that case the condition $-Id\in G$ in part (v) of Theorem 0.9 could be
omitted. However we do not know this.

The paper is organized as follows. In Section 1 we define the multiplicative
structure on polynomial valuations and study its properties. In Section 2 we
study the properties of the subalgebra of translation invariant smooth
valuations. In Section 3 we introduce and study filtrations on polynomial
valuations. In Section 4 we have some further remarks and discuss some
examples.

Acknowledgements. We would like to thank J. Bernstein for very useful
discussions.

##  1 The product on valuations.

Let us agree on a notation. In the rest of the paper we will denote by
$Pol_{d}(V)$ the space of homogeneous polynomials of degree $d$. Let us denote
by ${\cal G}(V)$ the linear space of valuations on $V$ which are finite linear
combinations of valuations of the form $K\mapsto\mu(i(K)+A)$ where
$i:V\hookrightarrow V^{\prime}$ is a linear imbedding of $V$ into a larger
linear space $V^{\prime}$, and $A\in{\cal K}(V^{\prime})$ is a fixed convex
compact set, and $\mu$ is a smooth measure on $V^{\prime}$.

######  1.1 Lemma.

${\cal G}(V)\cap PVal^{sm}(V)$ is dense in $PVal^{sm}(V)$.

Proof. Let $\phi$ be a valuation of the above form, i.e. $\phi(K)=\mu(i(K)+A)$
where $i:V\hookrightarrow V^{\prime}$, $\mu$ is a smooth measure on
$V^{\prime}$, and $A\in{\cal K}(V^{\prime})$. We have to prove that for any
$d$ the space ${\cal G}(V)\cap PVal^{sm}_{d}(V)$ is dense in $PVal_{d}(V)$.
Let us prove is by induction in $d$. For $d=0$ this is precisely McMullen’s
conjecture which was proved in [2] (as an easy consequence of the
Irreducibility Theorem 0.7). Let us assume that $d>0$. Then we have a
continuous map $PVal^{sm}_{d}(V)\longrightarrow Val^{sm}\otimes Pol_{d}(V)$
which is defined as follows. For any $\phi\in PVal^{sm}_{d}(V)$ we have

$\phi(K+x)=P_{d}(K)(x)+\mbox{ lower order terms }$

where $P_{d}\in(Val(V)\otimes Pol_{d}(V))^{sm}$. However by Lemma 1.5 below
$(Val(V)\otimes Pol_{d}(V))^{sm}=Val^{sm}(V)\otimes Pol_{d}(V)$ So the map
$\phi\mapsto P_{d}$ is the desired map. By the assumption of induction ${\cal
G}(V)\cap PVal_{d-1}(V)$ is dense in $PVal_{d-1}(V)$. It follows from the case
$d=0$ that the image of ${\cal G}(V)\cap PVal_{d}(V)$ in $Val^{sm}(V)\otimes
Pol_{d}(V)$ has a dense image. Hence the lemma follows. Q.E.D.

We need the following technical lemma.

######  1.2 Lemma.

Let $\phi_{1},\dots,\phi_{s}\in{\cal G}(V)$. Then there exists a linear space
$V^{\prime}$, a linear imbedding $f:V\hookrightarrow V^{\prime}$, smooth
measures $\mu_{i}$ on $V^{\prime}$, and $A_{1},\dots,A_{s}\in{\cal
K}(V^{\prime})$ such that $\phi_{i}(K)=\mu_{i}(f(K)+A_{i}),\,i=1,\dots,s$.

Proof. By definition $\phi_{i}$ has the form

$\phi_{i}(K)=\nu_{i}(g_{i}(K)+B_{i})$

where $g_{i}:V\hookrightarrow V_{i}^{\prime},\,B_{i}\in{\cal
K}(V_{i}^{\prime})$, and $\nu_{i}$ are smooth measures on $V_{i}^{\prime}$.
Let $V^{\prime}$ be the inductive limit of the system
$\\{V\overset{g_{i}}{\hookrightarrow}V_{i}^{\prime}\\}_{i=1}^{s}$. It would be
convenient to have an explicit construction for it. Let $T_{i}$ be a
complement of $g_{i}(V)$ in $V_{i}^{\prime}$. Then $V^{\prime}$ is isomorphic
to $V\oplus T_{1}\oplus\dots T_{s}$. With this decomposition $f$ is the
identity imbedding. Let us construct $A_{i},\,\nu_{i}$ say for $i=1$. Note
that $V_{1}^{\prime}$ is isomorphic to $V\oplus T_{1}$. Let us denote also by
$B_{1}$ the image of $B_{1}$ in $V^{\prime}$. Let $S:=V_{2}\oplus\dots\oplus
V_{s}$. Let us fix a Lebesgue measure $dvol_{S}$ on $S$. Fix $Q\in{\cal K}(S)$
any set of volume 1. Set $A_{1}:=B_{1}\times Q$. Let
$\mu_{1}:=\nu_{1}\boxtimes dvol_{S}$. These choices satisfy the lemma. Q.E.D.

Let $W$ be another linear real vector space. Let us define the exterior
product $\phi\boxtimes\psi\in{\cal G}(V\times W)$ of two valuations
$\phi\in{\cal G}(V),\,\psi\in{\cal G}(W)$. By Lemma 1.2 we may assume that
these valuations have the form
$\phi(K)=\sum_{i}\mu_{i}(f(K)+A_{i}),\,\psi(L)=\sum_{j}\nu_{j}(g(L)+B_{j})$
where $f:V\hookrightarrow V^{\prime},\,g:W\hookrightarrow W^{\prime}$ are
imbeddings, $A_{i}\in{\cal K}(V^{\prime}),\,B_{j}\in{\cal K}(W^{\prime})$.
Define

$(\phi\boxtimes\psi)(M):=\sum_{i,j}(\mu_{i}\boxtimes\nu_{j})((f\times
g)(M)+(A_{i}\times\\{0\\})+(\\{0\\}\times B_{j})),$

where $\mu_{i}\boxtimes\nu_{j}$ denotes the usual product measure on
$V^{\prime}\times W^{\prime}$.

Let us denote by $CVal(V)$ the closure of ${\cal G}(V)$ in the Fréchet space
of all continuous valuations on $V$ with the topology of uniform convergence
on compact subsets on ${\cal K}(V)$.

######  1.3 Proposition.

(i) For $\phi\in{\cal G}(V),\,\psi\in{\cal G}(W)$ their exterior product
$\phi\boxtimes\psi\in{\cal G}(V\times W)$ is well defined.

(ii) The exterior product is bilinear with respect to each argument.

(iii) Fix $\phi\in{\cal G}(V)$. Then the map ${\cal G}(W)\longrightarrow{\cal
G}(V\times W)$ given by $\psi\mapsto\phi\boxtimes\psi$ extends (uniquely) by
continuity to the map $CVal(V)\longrightarrow CVal(V\times W)$.

(iv)

$(\phi\boxtimes\psi)\boxtimes\eta=\phi\boxtimes(\psi\boxtimes\eta).$

(v) Let $f:V\hookrightarrow V_{1},\,g:W\hookrightarrow W_{1}$ be two
imbeddings. Let $\phi\in{\cal G}(V_{1}),\,\psi\in{\cal G}(W_{1})$. Then

$(f\times g)^{*}(\phi\boxtimes\psi)=f^{*}\phi\boxtimes g^{*}\psi.$

Proof. Note that the parts (ii), (iv), and (v) are obvious. Now let us fix a
valuation $\phi\in{\cal G}(V)$ of the form $\phi(K)=\mu(K+A),\,A\in
V^{\prime}$. From the definition of the exterior product one easily gets the
following formula:

$(\phi\boxtimes\psi)(M)=\int_{x\in
V^{\prime}}\psi\left((M+(A\times\\{0\\}))\cap(\\{x\\}\times W)\right)d\mu(x).$

Now the parts (i) and (iii) follow easily. Q.E.D.

Now let us define a product on ${\cal G}(V)$. Let $\Delta:V\hookrightarrow
V\times V$ denote the diagonal imbedding. For $\phi,\,\psi\in{\cal G}(V)$ let

$\phi\cdot\psi:=\Delta^{*}(\phi\boxtimes\psi),$

where $\Delta^{*}$ denotes the restriction of a valuation on $V\times V$ to
the diagonal.

######  1.4 Proposition.

The above defined multiplication uniquely extends by continuity to ${\cal
G}(V)$. Then ${\cal G}(V)$ becomes an associative commutative unital algebra
where the unit is the Euler characteristic $\chi$.

Proof. The associativity follows from Proposition 1.3 (iv). The commutativity
is obvious. Let us prove that the Euler characteristic $\chi$ is the unit. Let
a valuation $\psi$ has the form $\psi(K)=\mu(K+A)$ where $A$ is a fixed set
from ${\cal K}(V)$. Let $\Delta:V\longrightarrow V\times V$ be the diagonal
imbedding. Then by the definition of the product we have

$(\chi\cdot\psi)(K)=\int_{x\in V}\chi\left((\Delta
K+(A\times\\{0\\}))\cap(\\{x\\}\times V)\right)d\mu(x)=\mu(K+A)=\psi(K).$

Q.E.D.

Next let us prove the following technical lemma which is well known.

######  1.5 Lemma.

Let ${\cal F}$ be a Fréchet $G$-module which is admissible and of finite
length. Let $S$ be a finite dimensional $G$-module. Then $({\cal F}\otimes
S)^{sm}={\cal F}^{sm}\otimes S$.

Proof. Clearly ${\cal F}^{sm}\otimes S$ is a dense subspace of $({\cal
F}\otimes S)^{sm}$ (both spaces are admissible $G$-modules of finite length
and they have the same underlying Harish-Chandra modules). Since $S$ is finite
dimensional and ${\cal F}^{sm}$ is a Fréchet space then ${\cal F}^{sm}\otimes
S$ is also a Fréchet space. Now the statement follows from the Casselman-
Wallach theorem [6] which says that if $f:U\longrightarrow V$ is a morphism of
admissible $G$-modules of finite length which induces a surjection on the
underlying Harish-Chandra modules then $f:U^{sm}\longrightarrow V^{sm}$ is
also surjection (we should apply it to the identity map $id:{\cal
F}^{sm}\otimes S\longrightarrow({\cal F}\otimes S)^{sm}$). Q.E.D.

Let $PVal_{d}(V)$ denote the space of continuous valuations on $V$ polynomial
of degree at most $d$. Let $\Omega^{n}_{d}$ denote the (finite dimensional)
space of $n$-densities on $V$ with polynomial coefficients of degree at most
$d$ (clearly $\Omega^{n}_{d}$ is isomorphic to
$(\oplus_{i=0}^{d}Sym^{i}V^{*})\otimes|\wedge^{n}V^{*}|$).

Let us denote by $\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$P}_{+}(V^{*})$ the manifold of
oriented lines passing through the origin in $V^{*}$. Let $L$ denote the line
bundle over $\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$P}_{+}(V^{*})$ such that its fiber
over an oriented line $l$ consists of linear functionals on $l$.

We are going to construct a natural map

$\Theta_{k,d}:\Omega^{n}_{d}\otimes C^{\infty}((\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$P}_{+}(V^{*}))^{k},L^{\boxtimes
k})\longrightarrow PVal_{d}(V)$

which commutes with the natural action of the group $GL(V)$ on both spaces and
induces epimorphism on the subspaces of smooth vectors.

The construction is as follows. Let
$\mu\in\Omega^{n}_{d},\,A_{1},\dots,A_{s}\in{\cal K}(V)$. Then
$\int_{\sum_{j=1}^{s}\lambda_{j}A_{j}}\mu$ is a polynomial in $\lambda_{j}\geq
0$ of degree at most $n+d$. This can be easily seen directly, but it was also
proved in general for polynomial valuations by Khovanskii and Pukhlikov [9].
Also it easily follows that the coefficients of this polynomial depend
continuously on $(A_{1},\dots,A_{s})\in{\cal K}(V)^{s}$ with respect to the
Hausdorff metric. Hence we can define a continuous map
$\Theta_{s,d}^{\prime}:\Omega^{n}_{d}\times{\cal K}(V)^{s}\longrightarrow
PVal_{d}(V)$ given by

$(\Theta_{s,d}^{\prime}(\mu;A_{1},\dots,A_{s}))(K):=\frac{\partial^{s}}{\partial\lambda_{1}\dots\partial\lambda_{s}}\big{|}_{\lambda_{j}=0}\int_{K+\sum_{j=1}^{s}\lambda_{j}A_{j}}\mu.$

It is clear that $\Theta_{s,d}^{\prime}$ is Minkowski additive with respect to
each $A_{j}$. Namely, say for $j=1$, one has
$\Theta_{s,d}^{\prime}(\mu;aA_{1}^{\prime}+bA_{1}^{\prime\prime},A_{2},\dots,A_{s})=a\Theta_{s,d}^{\prime}(\mu;A_{1}^{\prime},A_{2},\dots,A_{s})+b\Theta_{s,d}^{\prime}(\mu;A_{1}^{\prime\prime},A_{2},\dots,A_{s})$
for $a,\,b\geq 0$.

Remind that for any $A\in{\cal K}(V)$ one defines the supporting functional
$h_{A}(y):=\sup_{x\in A}(y,x)$ for any $y\in V^{*}$. Thus $h_{A}\in
C(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$P}_{+}(V^{*}),L)$. Moreover it is
well known (and easy to see) that $A_{N}\longrightarrow A$ is the Hausdorff
metric if and only if $h_{A_{N}}\longrightarrow h_{A}$ in
$C(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$P}_{+}(V^{*}),L)$. Also any section
$F\in C^{2}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$P}_{+}(V^{*}),L)$ can be presented
as a difference $F=G-H$ where $F,\,H\in C^{2}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$P}_{+}(V^{*}),L)$ are supporting
functionals of some convex compact sets and
$\max\\{||G||_{2},||H||_{2}\\}\leq||F||_{2}$. (Indeed one can choose
$G=F+R\cdot h_{D},\,H=R\cdot h_{D}$ where $D$ is the unit Euclidean ball, and
$R$ is a large enough constant depending on $||F||_{2}$.) Hence we can
uniquely extend $\Theta_{s,d}^{\prime}$ to a multilinear continuous map (which
we will denote by the same letter):

$\Theta_{s,d}^{\prime}:\Omega^{n}_{d}\times(C^{2}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$P}_{+}(V^{*}),L))^{s}\longrightarrow
PVal_{d}(V).$

By the L. Schwartz kernel theorem (or, better, using the explicit formulas) it
follows that this map gives rise to a linear map

$\Theta_{s,d}:\Omega^{n}_{d}\otimes C^{\infty}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$P}_{+}(V^{*})^{s},L^{\boxtimes
s})\longrightarrow PVal_{d}^{sm}(V)$

which we wanted to construct.

Moreover the map $\Theta_{k,d}$ can be factorized as follows. Let
$\Delta:\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$P}_{+}(V^{*})\hookrightarrow(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$P}_{+}(V^{*}))^{k}$ denotes the
diagonal imbedding. Let ${\cal O}$ denote the sheaf of infinitely smooth
functions on $(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$P}_{+}(V^{*}))^{k}$. Let ${\cal J}$
denote the subsheaf of ideals of ${\cal O}$ of the diagonal
$\Delta(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$P}_{+}(V^{*}))$ in
$(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$P}_{+}(V^{*}))^{k}$. Then one has
the canonical map

$C^{\infty}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$P}_{+}(V^{*})^{k},L^{\boxtimes
k})\longrightarrow C^{\infty}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$P}_{+}(V^{*})^{k},L^{\boxtimes
k}\otimes_{\cal O}{\cal J}^{2k+1})$

which is the $2k$-jet of a section of $L^{\boxtimes k}$ in the transversal
direction to the diagonal. By writing down an explicit formula one can easily
prove

######  1.6 Lemma.

The map $\Theta_{k,d}$ factorizes as

$\Psi_{k,d}:\Omega^{n}_{d}\otimes C^{\infty}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$P}_{+}(V^{*})^{k},L^{\boxtimes
k}\otimes_{\cal O}{\cal J}^{2k+1})\longrightarrow PVal_{d}(V).$

We will study this map $\Psi_{k,d}$. Note that it depends on $k$ and $d$ which
will be fixed from now on. Let us denote by $\Psi_{d}$ the sum of the maps
$\bigoplus_{k=0}^{n-1}\Psi_{k,d}$. Thus
$\Psi_{d}:\Omega^{n}_{d}\otimes(\bigoplus_{k=0}^{n-1}C^{\infty}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$P}_{+}(V^{*})^{k},L^{\boxtimes
k}\otimes_{\cal O}{\cal J}^{2k+1}))\longrightarrow PVal^{sm}_{d}(V).$

######  1.7 Lemma.

The map $\Psi_{d}$ has a dense image.

######  1.8 Corollary.

The map $\Psi_{d}$ is an epimorphism.

This corollary follows immediately from Lemma 1.7 and the Casselman-Wallach
theorem. So let us prove Lemma 1.7.

Proof of Lemma 1.7. We will prove it by induction in $d$. For $d=0$ this is
just McMullen’s conjecture proved in [2]. Assume that the statement of Lemma
1.7 holds for $d-1$. It is sufficient to show that the map
$Sym^{d}(V)\otimes|\wedge^{n-1}V^{*}|\otimes(\bigoplus_{k=0}^{n-1}C^{\infty}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$P}_{+}(V^{*})^{k},L^{\boxtimes
k}\otimes_{\cal O}{\cal
J}^{2k+1}))\longrightarrow(PVal_{d}(V)/PVal_{d-1}(V))^{sm}$ is onto. However
the last space is equal to $(Sym^{d}(V)\otimes Val(V))^{sm}=Sym^{d}(V)\otimes
Val^{sm}(V)$ (using Lemma 1.5). Thus we have reduced the claim again to the
case $d=0$. Q.E.D.

######  1.9 Proposition.

The multiplication

$(PVal_{i}(V))^{sm}\otimes(PVal_{j}(V))^{sm}\longrightarrow(PVal_{i+j}(V))^{sm}$

is continuous.

Proof. Since this map commutes with the action of $GL(V)$ it is sufficient to
prove that the map

$(PVal_{i}(V))^{sm}\otimes(PVal_{j}(V))^{sm}\longrightarrow PVal_{i+j}(V)$

is continuous. In fact we will prove a bit more. We will prove that the
exterior product of valuations is a continuous bilinear map

$(PVal_{i}(V))^{sm}\times(PVal_{j}(W))^{sm}\longrightarrow PVal_{i+j}(V\times
W).$

For the simplicity of the notation we will assume that $W=V$.

Let us fix a large natural number $N$. Let
$\phi\in(PVal_{i}(V))^{sm},\,\psi\in(PVal_{j}(V))^{sm}$ such that
$\phi=\sum_{p}\omega_{p}\otimes
l_{p},\,\psi=\sum_{q}\omega^{\prime}_{q}\otimes l^{\prime}_{q}$ where
$\omega_{p}\in\Omega^{n}_{i},\,\omega^{\prime}_{q}\in\Omega^{n}_{j},\,l_{p},\,l^{\prime}_{q}\in\bigoplus_{k=0}^{n-1}C^{\infty}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$P}_{+}(V^{*})^{k},L^{\boxtimes
k}\otimes_{\cal O}{\cal J}^{2k+1})$ and

$max\\{\sum_{p}||\omega_{p}||,\,\sum_{q}||\omega^{\prime}_{q}||,\,\sum_{p}||l_{p}||_{2N},\,\sum_{q}||l^{\prime}_{q}||_{2N}\\}\leq
1$

where $||\cdot||$ is a norm on $\Omega^{n}_{i}$, and $||\cdot||_{N}$ is the
norm on $\bigoplus_{k=0}^{n-1}C^{2N}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$P}_{+}(V^{*})^{k},L^{\boxtimes
k}\otimes_{\cal O}{\cal J}^{2k+1}).$

Then it is easy to see that for a constant $C$ (depending on $n$ and $k$ only)
there exist $L_{p},\,L^{\prime}_{q}\in C^{\infty}((\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$P}_{+}(V^{*}))^{k},L^{\boxtimes
k})$ which extend $l_{p}$ and $l^{\prime}_{q}$ respectively and such that
$max\\{\sum_{p}||L_{p}||_{2N},\,\sum_{q}||L^{\prime}_{q}||_{2N}\\}<C$. Then
one can write

$L_{p}=\sum_{s=1}^{\infty}\gamma_{s,p,1}\otimes\dots\otimes\gamma_{s,p,k},\,L^{\prime}_{q}=\sum_{s=1}^{\infty}\gamma^{\prime}_{s,q,1}\otimes\dots\otimes\gamma^{\prime}_{s,q,k}$

where $\gamma_{s,p,i},\,\gamma_{s,q,i}\in C^{\infty}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$P}_{+}(V^{*}),L)$ and

$max\\{\sum_{s}\sum_{k=0}^{n-1}\sum_{t=1}^{k}||\gamma_{s,p,t}||_{N},\,\sum_{s}\sum_{k=0}^{n-1}\sum_{t=1}^{k}||\gamma^{\prime}_{s,q,t}||_{N}\\}<C$

where $C$ is a constant which might be different from the previous $C$. We may
assume that $N\geq 2$. Then every $\gamma\in C^{\infty}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$P}_{+}(V^{*}),L)$ can be written as
$\gamma=\gamma_{1}-\gamma_{2}$ such that $\gamma_{1},\,\gamma_{2}$ are
supporting functionals of convex sets and
$max\\{||\gamma_{1}||_{N},||\gamma_{2}||_{N}\\}\leq C||\gamma||_{N}$ (using
the same argument as in the construction of the maps $\Theta_{k,d}$). Hence we
may assume that all $\gamma_{s,p,t},\,\gamma^{\prime}_{s,q,t}$ are supporting
functionals of convex sets $A_{s,p,t},\,A^{\prime}_{s,q,t}$ respectively. Let
$\Omega_{p,q}:=d\omega_{p}\wedge d\omega^{\prime}_{q}$; it is a top-degree
form on $V\times V$. Then by the definition of the exterior product of
valuations

$(\phi\boxtimes\psi)(K)=$
$\sum_{s=1}^{\infty}\sum_{k,k^{\prime}=0}^{n-1}\sum_{p,q}\frac{\partial^{k}}{\partial\lambda_{1}\dots\partial\lambda_{k}}\frac{\partial^{k^{\prime}}}{\partial\mu_{1}\dots\partial\mu_{k^{\prime}}}|_{0}\int_{K+\sum_{m=1}^{k}\lambda_{m}(A_{s,p,m}\times
0)+\sum_{m^{\prime}=1}^{k^{\prime}}\lambda_{m^{\prime}}(0\times
A^{\prime}_{s,p,m^{\prime}})}\Omega_{p,q}.$

Note that

$\frac{\partial^{k}}{\partial\lambda_{1}\dots\partial\lambda_{k}}\frac{\partial^{k^{\prime}}}{\partial\mu_{1}\dots\partial\mu_{k^{\prime}}}|_{0}\int_{K+\sum_{m=1}^{k}\lambda_{m}(A_{s,p,m}\times
0)+\sum_{m^{\prime}=1}^{k^{\prime}}\lambda_{m^{\prime}}(0\times
A^{\prime}_{s,q,m^{\prime}})}\Omega_{p,q}=$
$\prod_{m}||\gamma_{s,p,m}||_{0}\prod_{m^{\prime}}||\gamma_{s,q,m^{\prime}}||_{0}\cdot$
$\frac{\partial^{k}}{\partial\lambda_{1}\dots\partial\lambda_{k}}\frac{\partial^{k^{\prime}}}{\partial\mu_{1}\dots\partial\mu_{k^{\prime}}}|_{0}\int_{K+\sum_{m=1}^{k}\lambda_{m}(\frac{A_{s,p,m}}{||\gamma_{s,p,m}||_{0}}\times
0)+\sum_{m^{\prime}=1}^{k^{\prime}}\lambda_{m^{\prime}}(0\times\frac{A^{\prime}_{s,q,m^{\prime}}}{||\gamma_{s,q,m^{\prime}}||_{0}})}\Omega_{p,q}.$

The last integral is a polynomial in $\lambda_{m},\,\mu_{m^{\prime}}$ of
degree at most $i+j+2n$. Hence in order to get an estimate on its derivatives
it is sufficient to estimate the polynomial itself for all
$\lambda_{m},\,\mu_{m^{\prime}}$ of absolute value at most 1. If the set $K$
is uniformly bounded then the whole Minkowski sum of sets under the integral
is uniformly bounded. Hence all the integrals are uniformly bounded, and we
get

$|(\phi\boxtimes\psi)(K)|\leq
C\sum_{s=1}^{\infty}\sum_{k,k^{\prime}=0}^{n-1}\sum_{p,q}\left(\prod_{m}||\gamma_{s,p,m}||_{0}\prod_{m^{\prime}}||\gamma_{s,q,m^{\prime}}||_{0}\right).$

The last sum is estimated by a constant. Q.E.D.

##  2 Translation invariant valuations.

In this section we discuss the subalgebra $(Val(V))^{sm}$ of smooth
translation invariant valuations. It has a structure of Frobenius algebra,
namely it satisfies a version of the Poincaré duality with respect to the
natural grading.

Remind that by a result of Hadwiger [8] the space $Val_{n}(V)$ is one
dimensional and it is spanned by a Lebesgue measure. First observe that the
multiplication
$(Val_{i}^{0}(V))^{sm}\otimes(Val_{n-i}^{1}(V))^{sm}\longrightarrow
Val_{n}(V)$ is trivial since the product of even and odd valuations must be
odd, and all valuations of the maximal degree of homogeneity are even. The
main result of this section is

######  2.1 Theorem.

(i) The map
$(Val_{i}^{0}(V))^{sm}\otimes(Val_{n-i}^{even}(V))^{sm}\longrightarrow
Val_{n}(V)$ is a perfect pairing. More precisely the induced map

$(Val_{i}^{0}(V))^{sm}\longrightarrow(Val_{n-i}^{even}(V)^{*})^{sm}\otimes
Val_{n}(V)$

is an isomorphism.

(ii) The map
$(Val_{i}^{1}(V))^{sm}\otimes(Val_{n-i}^{1}(V))^{sm}\longrightarrow
Val_{n}(V)$ is a perfect pairing in the above sense.

Before we prove the theorem we will need a proposition which is of independent
interest.

######  2.2 Proposition.

Let $V$ be an $n$-dimensional Euclidean space. Let

$\phi(K)=V(K[i],A_{1},\dots,A_{n-i}),\,\psi(K)=V(K[n-i],B_{1},\dots,B_{i})$

where $A_{p},\,B_{q}\in{\cal K}(V)$ are fixed. Then

$(\phi\cdot\psi)(K)={n\choose
i}^{-1}V(A_{1},\dots,A_{n-i},-B_{1},\dots,-B_{i})vol(K).$

First we have the following simple identity.

######  2.3 Claim.

$V(K[i],A_{1},\dots,A_{n-i})=(n(n-1)\dots(i+1))^{-1}\frac{\partial^{n-i}}{\partial\lambda_{1}\dots\partial\lambda_{n-i}}\big{|}_{0}vol(K+\sum_{j=1}^{n-i}\lambda_{j}A_{j}).$

Proof of Proposition 2.2. Using Claim 2.3 we have

$(\phi\cdot\psi)(K)=(n(n-1)\dots(i+1))^{-1}(n(n-1)\dots(n-i+1))^{-1}\cdot$
$\frac{\partial^{n-i}}{\partial\lambda_{1}\dots\partial\lambda_{n-i}}\big{|}_{0}\frac{\partial^{i}}{\partial\mu_{1}\dots\partial\mu_{i}}\big{|}_{0}vol_{2n}(\Delta(K)+\sum_{j=1}^{n-i}\lambda_{j}(A_{j}\times
0)+\sum_{l=1}^{i}\mu_{l}(0\times B_{l}))$

where $\Delta:V\hookrightarrow V\times V$ is the diagonal imbedding. Again by
Claim 2.3

$\frac{\partial^{n-i}}{\partial\lambda_{1}\dots\partial\lambda_{n-i}}\big{|}_{0}\frac{\partial^{i}}{\partial\mu_{1}\dots\partial\mu_{i}}\big{|}_{0}vol_{2n}(\Delta(K)+\sum_{j=1}^{n-i}\lambda_{j}(A_{j}\times
0)+\sum_{l=1}^{i}\mu_{l}(0\times B_{l}))=$ $(2n\dots(n+1))\cdot
V_{2n}(\Delta(K)[n];A_{1}\times 0,\dots,A_{n-i}\times 0;0\times
B_{1},\dots,0\times B_{i}).$

Hence we obtain

$(\phi\cdot\psi)(K)={2n\choose n}{n\choose
i}^{-1}V_{2n}(\Delta(K)[n];A_{1}\times 0,\dots,A_{n-i}\times 0;0\times
B_{1},\dots,0\times B_{i}).$

We will need a lemma.

######  2.4 Lemma.

Let $X=Y\oplus Z$ be an orthogonal decomposition of the Euclidean space $X$.
Let $\dim X=N,\,\dim Y=n$. Let $M\in{\cal K}(Y),\,A_{1},\dots A_{N-n}\in{\cal
K}(X).$ Then

$V_{N}(M[n];A_{1},\dots A_{N-n})={N\choose
n}^{-1}vol_{n}(M)V_{N-n}(Pr_{Z}A_{1},\dots,Pr_{Z}A_{N-n})$

where $Pr_{Z}$ denotes the orthogonal projection onto $Z$.

Let us postpone the proof of this lemma and continue proving Proposition 2.2.
In our situation $X=V\times V,\,Y=\Delta(V)=\\{(x,x)\\},\,Z=\\{(x,-x)\\}.$
Note that

$Pr_{Z}((x,0))=\frac{1}{2}(x,-x),\,Pr_{Z}((0,x))=\frac{1}{2}(-x,x).$

Let us denote by $\Delta^{\prime}$ the imbedding $V\hookrightarrow V\times V$
given by $\Delta^{\prime}(x)=(x,-x)$. Using Lemma 2.4 we get

$(\phi\cdot\psi)(K)={n\choose
i}^{-1}vol_{n}(\Delta(K))V_{n}(\frac{\Delta^{\prime}A_{1}}{2},\dots,\frac{\Delta^{\prime}A_{n-i}}{2},\frac{-\Delta^{\prime}B_{1}}{2},\dots,\frac{-\Delta^{\prime}B_{i}}{2})=$
$(\phi\cdot\psi)(K)={n\choose
i}^{-1}V(A_{1},\dots,A_{n-i},-B_{1},\dots,-B_{i})vol(K).$

Q.E.D.

Proof of Lemma 2.4. We may assume that $A_{1}=\dots=A_{N-n}=A$. We have

$V_{N}(M[n],A[N-n])=\frac{(N-n)!}{N!}\frac{d^{n}}{d\varepsilon^{n}}\big{|}_{0}vol_{N}(A+\varepsilon
M).$

We have

$vol_{N}(A+\varepsilon M)=\int_{z\in Z}vol_{n}((A+\varepsilon M)\cap(z+Y))dz=$
$\int_{z\in Z}vol_{n}((A\cap(z+Y))+\varepsilon M)dz=\int_{z\in
Pr_{Z}A}\varepsilon^{n}vol_{n}M+O(\varepsilon^{n-1})=$
$\varepsilon^{n}vol_{N-n}(Pr_{Z}A)vol_{n}M+O(\varepsilon^{n-1}).$

This proves Lemma 2.4. Q.E.D.

Proof of Theorem 2.1. The kernel of the map
$(Val_{i}^{0}(V))^{sm}\longrightarrow(Val_{n-i}^{0}(V)^{*})^{sm}\otimes
Val_{n}(V)$ is $GL(V)$-invariant subspace (and similarly in the odd case).
Hence by the Irreducibility Theorem 0.7 it must be either trivial or coincide
with the whole space. We will show that the multiplication in non-trivial.
Assume that it is proved. Then by the Irreducibility Theorem 0.7 the image the
above map is a dense subspace, and hence it is equal to the whole space by the
Casselman-Wallach theorem. Thus it remains to check in both cases that the
product is non-zero.

First let us check it in the even case. By Proposition 2.2 the product of the
intrinsic volumes $V_{i}\cdot V_{n-i}\neq 0$.

Now let us consider the odd case. We want to show that the multiplication
$(Val_{i}^{1}(V))^{sm}\otimes(Val_{n-i}^{1}(V))^{sm}\longrightarrow
Val_{n}(V)$ is non-zero. We have

$(Val^{\varepsilon}_{i}(V))^{sm}\otimes(Val^{\delta}_{j}(V))^{sm}\longrightarrow(Val^{\varepsilon+\delta}_{i+j}(V))^{sm}$

where the addition of upper indexes is understood modulo 2.

######  2.5 Lemma.

The multiplication

$(Val^{\varepsilon}_{1}(V))^{sm}\otimes(Val^{\varepsilon}_{1}(V))^{sm}\longrightarrow(Val^{0}_{2}(V))^{sm}$

is non-zero for $\varepsilon\in\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$Z}/2\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$Z}$ and hence has a dense image.

First let us finish proving Theorem 2.1 assuming this lemma. Consider the
multiplication maps (we will omit for brevity the upper sign $sm$):

$Val^{1}_{1}(V)\otimes(Val^{0}_{1}(V))^{\otimes(i-1)}\longrightarrow
Val_{i}^{1}(V);$
$Val^{1}_{1}\otimes(Val^{0}_{1})^{\otimes(n-1-i)}\longrightarrow
Val_{n-i}^{1}.$

It is sufficient to show that the composition

$\left(Val^{1}_{1}(V)\otimes(Val^{0}_{1}(V))^{\otimes(i-1)}\right)\otimes\left(Val^{1}_{1}(V)\otimes(Val^{0}_{1}(V))^{\otimes(n-1-i)}\right)\longrightarrow
Val_{n}(V)$

is non-zero. This is equivalent to show that

$Val^{1}_{1}(V)\otimes
Val^{0}_{1}(V)\otimes(Val^{0}_{1}(V))^{\otimes(n-2)}\longrightarrow
Val_{n}(V)$

is non-zero. Note that the image
$(Val^{0}_{1}(V))^{\otimes(n-2)}\longrightarrow Val_{n-2}^{0}(V)$ is non-zero
since the power of the intrinsic volume $(V_{1})^{n-2}$ does not vanish. Hence
this image is a dense subspace in $Val_{n-2}^{0}(V)$. Hence it is sufficient
to show that the map

$Val^{1}_{1}(V)\otimes Val^{1}_{1}(V)\otimes Val_{n-2}^{0}(V)\longrightarrow
Val_{n}(V)$

is non-zero. By Lemma 2.5 the map $Val^{1}_{1}(V)\otimes
Val^{1}_{1}(V)\longrightarrow Val^{0}_{2}(V)$ has a dense image. As we have
previously proved the map $Val^{0}_{2}(V)\otimes
Val_{n-2}^{0}(V)\longrightarrow Val_{n}(V)$ is non-zero. This immediately
implies Theorem 2.1. Q.E.D.

Proof of Lemma 2.5. First let reduce the statement to the case $n=2$. Let us
fix a 2-dimensional subspace $W\subset V$. It is easy to see that the
restriction map $Val(V)\longrightarrow Val(W)$ is non-zero and its image is
$GL(W)$-invariant. Hence by the Irreducibility Theorem the image of this
restriction is a dense subspace. Moreover the multiplication commutes with the
restriction. Hence it is sufficient to show that

$Val^{1}_{1}(W)\otimes Val^{1}_{1}(W)\longrightarrow Val_{2}(W)$

is non-zero. By Proposition 2.2 we have for $A,B\in{\cal K}(W)$:

$V(\cdot,A)\cdot V(\cdot,B)=\kappa V(A,-B)\cdot vol(\cdot),$

where $\kappa$ is a non-zero normalizing constant.

Now let us choose valuations
$\phi(\cdot):=V(\cdot,A)-V(\cdot,-A),\,\psi(\cdot):=V(\cdot,B)-V(\cdot,-B)$
with $A$ and $B$ to be chosen later. Then we get

$(\phi\cdot\psi)(K)=2\kappa(V(A,B)-V(A,-B))\cdot vol(K).$

If we choose $A$ and $B$ so that $V(A,B)\neq V(A,-B)$ then the above product
does not vanish. Q.E.D.

For any subgroup $G$ of the group of linear transformations of the space $V$
let us denote by $Val^{G}(V)$ the space of translation invariant continuous
valuations invariant with respect to $G$. Let now $V$ be a Euclidean space of
dimension $n$. Also $Val^{G}_{i}(V)$ denote the subspace of $Val^{G}(V)$ of
$i$-homogeneous valuations. Then it immediately follows from McMullen’s
theorem [13] that $Val^{G}(V)=\oplus_{i=0}^{n}Val_{i}^{G}(V)$. Let $O(n)$
denote the full orthogonal group, and $SO(n)$ denote the special orthogonal
group. Let $V_{i}$ denote the $i$-th intrinsic volume (see e.g. [17]). It was
proved by Hadwiger that for any $i$
$Val_{i}^{O(n)}=Val_{i}^{SO(n)}=\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}\cdot V_{i}$. Now we describe the
algebra structure on the space $Val^{O(n)}(V)$.

######  2.6 Theorem.

There exists an isomorphism of graded algebras $\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}[x]/(x^{n+1})\simeq
Val^{O(n)}(V)$ given by $x\mapsto V_{1}$.

Proof. For any $i$ the valuation $(V_{1})^{i}\in Val_{i}^{O(n)}$. Hence it
must be proportional to $V_{i}$. By Proposition 2.2 the constant of
proportionality does not vanish. This implies the result. Q.E.D.

Now let us prove part (iv) of Theorem 0.9. Namely let us prove that if $G$ is
a compact subgroup of $GL(V)$ acting transitively on the projective space
$\mbox{\rm$~{}\vrule height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$P}_{+}(V)$
then $Val_{1}^{G}(V)=\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}\cdot V_{1}$ and
$Val_{n-1}^{G}=\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}\cdot V_{n-1}$. Here we assume
that we consider the intrinsic volumes $V_{i}$ with respect to a Euclidean
metric on $V$ invariant under $G$ (which is unique up to a proportionality).
It was proved by McMullen [14] that every translation invariant continuous
$(n-1)$-homogeneous valuation $\phi$ has the form

$\phi(K)=\int_{S^{n-1}}f(x)dS_{n-1}(K,x)$

where $dS_{n-1}(K,\cdot)$ denotes the $(n-1)$-area measure of $K$ (see [17]),
and $f$ is a continuous function on the sphere $S^{n-1}$. Moreover $f$ can be
chosen to be orthogonal (with respect to the Lebesgue measure on $S^{n-1}$) to
any linear functional on the sphere, and under this restriction $f$ is defined
uniquely by $\phi$. Hence in our situation $f$ can be chosen $G$-invariant. If
$G$ acts transitively on the projective space then it acts transitively on the
sphere. Hence $f$ must be constant. Hence $\phi\in\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}\cdot V_{n-1}$.

Now let us consider the case of $Val_{1}^{G}(V)$. The result follows from the
previous case and the Poincaré duality. A more elementary way to see it is as
follows. Let us construct a canonical imbedding $Val_{1}(V)\hookrightarrow
C^{-\infty}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$P}_{+}(V),L^{*}\otimes|\omega|))$
where the target space is the space of generalized sections of the line bundle
$L^{*}\otimes|\omega|$ over $\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$P}_{+}(V)$, where $L$ was defined
in Section 1, and $|\omega|$ is the line bundle of densities over
$\mbox{\rm$~{}\vrule height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$P}_{+}(V)$.
This imbedding (in fact in a more general situation) was first considered by
Goodey and Weil [7]; now we essentially reproduce their argument. It is well
known that any valuation $\phi\in Val_{1}(V)$ is Minkowski additive, i.e.
$\phi(\lambda A+\mu B)=\lambda\phi(A)+\mu\phi(B)$ for all $A,\,B\in{\cal
K}(V)$ and $\lambda,\,\mu\geq 0$. Note that the supporting functional of any
convex compact set is a continuous section of the line bundle $L$ over
$\mbox{\rm$~{}\vrule height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$P}_{+}(V)$
defined in Section 1. However any $C^{\infty}$-section $F$ of $L$ can
presented as a difference of two smooth supporting functionals of convex
compact sets, $F=G-H$ such that $\max\\{||G||_{2},||H||_{2}\\}\leq||F||_{2}$.
Hence using the Minkowski additivity of $\phi$ we can extend it uniquely to a
continuous linear functional on $C^{\infty}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$P}_{+}(V),L)$. Clearly we get a
continuous imbedding $Val_{1}(V)\hookrightarrow
C^{-\infty}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$P}_{+}(V),L^{*}\otimes|\omega|)).$
However if the group $G$ acts transitively on the sphere then the space of
$G$-invariant (generalized) sections is at most one dimensional. It follows
that $Val_{1}^{G}(V)=\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}\cdot V_{1}$. Q.E.D.

##  3 Filtrations on polynomial valuations.

Let us define the following filtration on $PVal^{sm}(V)$:

$\gamma_{i}=\\{\phi\in PVal^{sm}(V)|\,\phi(K)=0\,\forall K\mbox{ such that
}\dim K<i\\}.$

Then
$PVal(V)=\gamma_{0}\supset\gamma_{1}\supset\dots\supset\gamma_{n}\supset\gamma_{n+1}=0$.
Note that

$\gamma_{i}\cap Val^{sm}(V)=(Val^{1}_{i-1}(V))^{sm}\oplus
Val_{i}^{sm}(V)\oplus\dots\oplus Val_{n}^{sm}(V).$

Let us introduce another filtration on $(PVal)^{sm}$. Set

$W_{i}(PVal_{d}^{sm}):=\sum_{k=0}^{n-i}Im(\Psi_{k,d}).$

We have

$PVal_{d}^{sm}=W_{0}(PVal_{d}^{sm})\supset
W_{1}(PVal_{d}^{sm})\supset\dots\supset W_{n}(PVal_{d}^{sm}).$

Moreover $W_{n}(PVal_{d}^{sm})$ coincides with densities on $V$ polynomial of
degree at most $d$.

######  3.1 Lemma.

For any $d$

$W_{i}(PVal_{d}^{sm})\cap Val^{sm}(V)=Val_{i}^{sm}(V)\oplus
Val_{i+1}^{sm}(V)\oplus\dots\oplus Val_{n}^{sm}(V).$

Proof. Obviously we have an inclusion

$Val_{i}^{sm}(V)\oplus Val_{i+1}^{sm}(V)\oplus\dots\oplus
Val_{n}^{sm}(V)\subset W_{i}(PVal_{d}^{sm})\cap Val^{sm}(V).$

Since $W_{i}(PVal_{d}^{sm})\subset\gamma_{i}$ (see Proposition 3.7(i) below)
we have

$W_{i}(PVal_{d}^{sm})\cap Val^{sm}(V)\subset\left((Val_{i}^{sm}(V)\oplus
Val_{i+1}^{sm}(V)\oplus\dots\oplus Val_{n}^{sm}(V)\right)\oplus
Val_{i-1}^{1}(V).$

It is sufficient to show that $W_{i}(PVal_{d}^{sm})\cap Val_{i-1}^{1}=0$.
Assume that $\phi\in
W_{i}(PVal_{d}^{sm})\cap(Val_{i-1}^{1}(V))^{sm},\,\phi\neq 0$. By the Poincaré
duality there exists $\psi\in(Val_{n-i+1}(V))^{sm}$ such that
$\phi\cdot\psi\neq 0$. We may also assume that $\psi(K)=vol(K[n-i+1],A[i-1])$
where $A\in{\cal K}(V)$ is fixed. By Proposition 1.3(iii) for fixed $\psi$ the
map $CVal(V)\longrightarrow CVal(V)$ given by $\xi\mapsto\xi\cdot\psi$ is a
continuous map (with the topology of uniform convergence on compact subsets of
${\cal K}(V)$).

######  3.2 Claim.

$\psi\cdot W_{i}(PVal_{d}^{sm})=0.$

Hence by continuity it follows that $\psi\cdot W_{i}(PVal_{d}^{sm})=0$. We get
a contradiction. Q.E.D.

######  3.3 Proposition.

$W_{i}(PVal_{d}^{sm})$ is a closed subspace of $PVal_{d}^{sm}$.

Proof. The statement follows from the Casselman-Wallach theorem and the fact
that the source and the target of the map $\Psi_{k,d}$ are admissible
$GL(V)$-modules of finite length. Q.E.D.

######  3.4 Proposition.

Let $\phi\in PVal_{d}^{sm}$. If $\phi\in W_{i}(PVal_{d^{\prime}}^{sm})$ for
some $d^{\prime}\geq d$ then

$\lim_{r\longrightarrow+0}\frac{1}{r^{i-1}}\phi(rK+x)=0\,\forall x\in
X,\,\forall K\in{\cal K}(V).$ $None$

Oppositely if the condition $(\star)$ holds then $\phi\in
W_{i}(PVal_{d}^{sm})$.

Proof. Let us assume that $\phi\in PVal_{d^{\prime}}^{sm}$. Then
$\int_{rK+x_{0}+\sum_{j=1}^{n-i}\lambda_{j}A_{j}}F(x)dx=F(x_{0})vol(rK+\sum_{j=1}^{n-i}\lambda_{j}A_{j})+\mbox{
higher order terms }$. The condition $(\star)$ follows.

Let us prove the other direction. Let $\phi\in PVal_{d}^{sm}$ and satisfies
$(\star)$. We will prove the result by induction in $d$. For $d=0$ the
statement is clear. Let us assume now that the statement is true for
valuations of degree of polynomiality less than $d$. We have

$\phi(K+x)=P_{d}(K)(x)+\mbox{ lower order terms }.$

Thus $P_{d}\in Val^{sm}\otimes Pol_{d}$. From the condition $(\star)$ we get:

$0=\lim_{r\longrightarrow+0}\frac{1}{r^{i-1}}\phi(rK+x)=\lim_{r\longrightarrow+0}\frac{1}{r^{i-1}}P_{d}(rK)(x)+\dots.$

Hence $P_{d}\in(Val_{i}^{sm}\oplus\dots\oplus Val_{n}^{sm})\otimes Pol_{d}$.
It follows from the Casselman-Wallach theorem that there exists $\psi\in
W_{i}(PVal_{d}^{sm})$ such that $\psi(K+x)=P_{d}(K)(x)+\dots$. Applying the
assumption of induction to the valuation $\phi-\psi$ we prove the statement.
Q.E.D.

We get immediately the following corollary.

######  3.5 Corollary.

(i) For $d^{\prime}>d$ $W_{i}(PVal_{d^{\prime}}^{sm})\cap
PVal_{d}^{sm}=W_{i}(PVal_{d}^{sm}).$

(ii) Let $f:U\longrightarrow V$ be an injective imbedding of linear spaces.
Then $f^{*}(W_{i}(PVal_{d}^{sm}(V)))\subset W_{i}(PVal_{d}^{sm}(U))$.

Using part (i) of this corollary let us define the filtration $W_{i}$ on all
smooth polynomial valuations $PVal^{sm}$ so that $W_{i}\cap
PVal_{d}^{sm}=W_{i}(PVal_{d}^{sm}).$

######  3.6 Theorem.

$W_{i_{1}}(PVal_{d_{1}}^{sm})\otimes W_{i_{2}}(PVal_{d_{2}}^{sm})\subset
W_{i_{1}+i_{2}}(PVal_{d_{1}+d_{2}}^{sm}).$

Proof. It is clear from the definitions that

$W_{i_{1}}(PVal_{d_{1}}^{sm}(V))\boxtimes
W_{i_{2}}(PVal_{d_{2}}^{sm}(W))\subset
W_{i_{1}+i_{2}}(PVal_{d_{1}+d_{2}}(V\times W)).$

Now the statement follows from Corollary 3.5(ii). Q.E.D.

######  3.7 Proposition.

(i) For any $i$

$\gamma_{i+1}\subset W_{i}\subset\gamma_{i}.$

(ii) $W_{1}=\gamma_{1}.$

Proof. Let us prove part (i). Let us prove first the inclusion
$\gamma_{i+1}\subset W_{i}$. Let $\phi\in\gamma_{i+1}(PVal_{d}^{sm})$. We will
prove the statement by the induction in $d$. Assume first that $d=0$. Then
$\gamma_{i+1}(Val^{sm})=(Val_{i}^{1})^{sm}\oplus
Val_{i+1}^{sm}\oplus\dots\subset W_{i}(Val^{sm})=Val_{i}^{sm}\oplus
Val_{i+1}^{sm}\oplus\dots.$

Now let us assume that the statement holds for valuations polynomial of degree
less than $d$. Let us prove it for $d$. We have

$\phi(K+x)=P_{d}(K)(x)+\mbox{ lower order terms }.$

Thus $P_{d}\in\gamma_{i+1}(Val^{sm})\otimes Pol_{d}\subset
W_{i}(Val^{sm})\otimes Pol_{d}$. It follows from the case $d=0$ and the
Casselman-Wallach theorem that there exists $\psi\in W_{i}(PVal_{d}^{sm})$
such that

$\psi(K+x)=P_{d}(K)(x)+\mbox{ lower order terms }.$

Applying the assumption of induction to the valuation $\phi-\psi$ we obtain
the result.

The second inclusion $W_{i}\subset\gamma_{i}$ follows from the fact that if
$\dim K<i$ then
$\int_{K+\sum_{j=1}^{n-i}\lambda_{j}A_{j}}\mu=O\left((\sqrt{\sum_{j=1}^{n-i}\lambda_{j}^{2}})^{n-i+1}\right)$
when $\lambda\longrightarrow 0$.

Let us prove part (ii). It remains to prove the inclusion $\gamma_{1}\subset
W_{1}$. Assume that $\phi\in\gamma_{1}\cap PVal_{d}^{sm}$. We will prove by
induction in $d$ that $\phi\in W_{1}$. For $d=0$ the statement is clear. As
previously we can write

$\phi(K+x)=P_{d}(K)(x)+\mbox{ lower order terms }$

with $P_{d}\in\gamma_{1}(Val^{sm})\otimes Pol_{d}=W_{1}(Val^{sm})\otimes
Pol_{d}$. Again the Casselman-Wallach theorem implies that there exists
$\psi\in W_{1}(PVal_{d}^{sm})$ such that

$\psi(K+x)=P_{d}(K)(x)+\mbox{ lower order terms }.$

Applying the assumption of induction to the valuation $\phi-\psi$ we prove the
result. Q.E.D.

The next theorem gives an axiomatic characterization of the filtration
$W_{i}$.

######  3.8 Theorem.

Assume that we have another filtration $\\{\tilde{W}_{i}\\}$ on $PVal^{sm}$
such that

(1) $\\{\tilde{W}_{i}\\}$ is compatible with the multiplicative structure,
i.e. $\tilde{W}_{i}\cdot\tilde{W}_{j}\subset\tilde{W}_{i+j}$;

(2) $\gamma_{i+1}\subset\tilde{W}_{i}\subset\gamma_{i}\mbox{ for all }i.$

(3) $\tilde{W}_{0}=\gamma_{0},\,\tilde{W}_{1}=\gamma_{1}$;

(4) $\tilde{W}_{i}\cap PVal_{d}^{sm}$ is a closed subspace of $PVal_{d}^{sm}$
for all $i,\,d$.

(5) $\tilde{W}_{i}$ is $Aff(V)$-invariant.

Then $\tilde{W}_{i}=W_{i}$.

Proof. One has to prove that $\tilde{W}_{i}\cap
PVal_{d}^{sm}=W_{i}(PVal_{d}^{sm})$ for all $i,\,d$. As previously the general
case reduces to the case $d=0$. Let us prove the statement in this case. We
may assume that $i>1$. Remind that

$\gamma_{i+1}\cap Val^{sm}=(Val_{i}^{1})^{sm}\oplus
Val_{i+1}^{sm}\oplus\dots\oplus Val_{n}^{sm};$ $\gamma_{i}\cap
Val^{sm}=(Val_{i-1}^{1})^{sm}\oplus Val_{i}^{sm}\oplus\dots\oplus
Val_{n}^{sm}.$

First let us show that $\tilde{W}_{i}\cap(Val_{i-1}^{1})^{sm}=0.$ Otherwise
due to $Aff(V)$-invariance of $\tilde{W}_{i}$ and $GL(V)$-irreducibility of
$Val_{i-1}^{1}$ this intersection will be equal to $(Val_{i-1}^{1})^{sm}$.
Note that it follows from Proposition 2.2 and Irreducibility Theorem 0.7 that
$(Val_{i-1}^{1})^{sm}\cdot(Val_{1}^{1})^{sm}$ is dense in
$(Val_{i}^{0})^{sm}$. But $(Val_{1}^{1})^{sm}\subset\gamma_{1}=\tilde{W}_{1}$.
Also $\tilde{W}_{i}\cdot\tilde{W}_{1}\subset\tilde{W}_{i+1}$. Hence
$\tilde{W}_{i+1}\supset(Val_{i}^{0})^{sm}$. Hence
$\gamma_{i+1}\supset(Val_{i}^{0})^{sm}$. This is a contradiction.

Thus it remains to prove that $\tilde{W}_{i}\supset(Val_{i}^{0})^{sm}$. Remind
that since $\tilde{W}_{1}=\gamma_{1}$ we have
$(Val_{1}^{0})^{sm}\subset\tilde{W}_{1}$. Hence
$((Val_{1}^{0})^{sm})^{i}\subset\tilde{W}_{i}$. Again Proposition 2.2 and
Irreducibility Theorem 0.7 imply that $((Val_{1}^{0})^{sm})^{i}$ is dense in
$(Val_{i}^{0})^{sm}$. So the result follows. Q.E.D.

Let us consider the associated graded algebra
$gr_{W}(PVal^{sm}):=\bigoplus_{i=0}^{n}W_{i}/W_{i+1}.$ The next result gives a
description of it.

######  3.9 Theorem.

There exists a canonical isomorphism of graded algebras

$gr_{W}(PVal^{sm}(V))\simeq Val^{sm}(V)\otimes\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}[V]$

where the $i$-th graded term in the right hand side is equal to
$Val_{i}^{sm}(V)\otimes\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}[V]$ where $\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}[V]$ denote the algebra of
polynomial functions on $V$.

Proof. Let us construct the isomorphism explicitly. Let us define a map

$Q:W_{i}\longrightarrow Val_{i}^{sm}\otimes\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}[V]$

by $Q(\phi)(K)(x)=\lim_{r\longrightarrow 0}r^{-i}\phi(rK+x).$ First notice
that $Q(\phi)$ is indeed a translation invariant valuation. Let us check it
say for $x=0$. We have:

$\phi(r(K+a))=r^{i}Q(\phi)(K+a)(0)+o(r^{i}).$

On the other hand

$\phi(r(K+a))=\phi(rK+ra)=r^{i}Q(\phi)(K)(ra)+o(r^{i})=r^{i}Q(\phi)(K)(0)+o(r^{i}).$

Hence $Q(\phi)(K+a)(0)=Q(\phi)(K)(0)$.

Next the kernel of $Q$ is equal to $W_{i+1}$ by Proposition 3.4. Let us show
that $Q$ is surjective. Let us check that for any $d$ the map

$Q:W_{i}(PVal_{d}^{sm})\longrightarrow Val_{i}^{sm}\otimes\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}[V]_{\leq d}$

is surjective (where $\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}[V]_{\leq d}$ denote the space of
polynomials of degree at most $d$). Both spaces are admissible Fréchet
representations of $GL(V)$ of finite length. Hence by the Casselman-Wallach
theorem it is sufficient to show that $Q$ has a dense image. Let $\phi\in
W_{i}(PVal_{d}^{sm})$ be a valuation of the form

$\phi(K)=\frac{\partial^{n-i}}{\partial\lambda_{1}\dots\partial\lambda_{n-i}}\big{|}_{\lambda_{j}=0}\int_{K+\sum_{j=1}^{n-i}\lambda_{j}A_{j}}f(x)dx$
(4)

where $A_{1},\dots A_{n-i}\in{\cal K}(V)$ are fixed, and $f$ is a polynomial
of degree at most $d$. Then it is easy to see that
$Q(\phi)(K)(x)=f(x)\frac{\partial^{n-i}}{\partial\lambda_{1}\dots\partial\lambda_{n-i}}\big{|}_{\lambda_{j}=0}vol(K+\sum_{j}\lambda_{j}A_{j}).$
Since by the (proved) McMullen’s conjecture the valuations of the form
$\frac{\partial^{n-i}}{\partial\lambda_{1}\dots\partial\lambda_{n-i}}\big{|}_{\lambda_{j}=0}vol(K+\sum_{j}\lambda_{j}A_{j})$
are dense in $Val_{i}$ we deduce that $Q$ is an epimorphism.

It remains to prove that $Q$ is a homomorphism of algebras. Let $\phi\in
W_{i}(PVal_{d}^{sm})$ be a representative of an element from $W_{i}/W_{i+1}$
of the form (4), and $\psi\in W_{j}(PVal_{d^{\prime}}^{sm})$ be a
representative of an element of $W_{j}/W_{j+1}$ of the form

$\frac{\partial^{n-j}}{\partial\mu_{1}\dots\partial\mu_{n-j}}\big{|}_{\mu_{l}=0}\int_{K+\sum_{l=1}^{n-j}\mu_{l}B_{l}}g(x)dx$

where $B_{1},\dots B_{n-j}\in{\cal K}(V)$ are fixed, and $g$ is a polynomial
of degree at most $d^{\prime}$. Then

$(\phi\boxtimes\psi)(K)=$
$\frac{\partial^{n-i}}{\partial\lambda_{1}\dots\partial\lambda_{n-i}}\frac{\partial^{n-j}}{\partial\mu_{1}\dots\partial\mu_{n-j}}\big{|}_{0}\int_{K+\sum_{m}\lambda_{m}(A_{m}\times\\{0\\})+\sum_{l}\mu_{l}(\\{0\\}\times
B_{l})}f(x)g(y)dxdy.$

Then clearly

$Q(\phi\times\psi)(K)((x,y))=$
$f(x)g(y)\frac{\partial^{n-i}}{\partial\lambda_{1}\dots\partial\lambda_{n-i}}\frac{\partial^{n-j}}{\partial\mu_{1}\dots\partial\mu_{n-j}}\big{|}_{0}vol(K+\sum_{m}\lambda_{m}(A_{m}\times\\{0\\})+\sum_{l}\mu_{l}(\\{0\\}\times
B_{l}))=$ $(Q(\phi)\times Q(\psi))(K)((x,y)).$

The result follows. Q.E.D.

##  4 Further remarks.

In Theorem 2.6 of this paper we have described the algebra
$Val^{O(n)}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$R}^{n})=Val^{SO(n)}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$R}^{n})$ of isometry invariant
continuous valuations on an $n$-dimensional Euclidean space
$\mbox{\rm$~{}\vrule height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$R}^{n}$. We
would like to study the space of isometry invariant continuous valuations in
all dimensions simultaneously. More precisely assume that
$i_{n}:\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$R}^{n}\hookrightarrow\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$R}^{n+1}$ is the standard isometric
imbedding when the last coordinate vanishes. It follows from the Hadwiger
characterization theorem that for any $k\leq n$ the restriction map
$i_{n}^{*}:Val_{k}^{O(n+1)}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$R}^{n+1})\longrightarrow
Val_{k}^{O(n)}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$R}^{n})$ is an isomorphism. In
other words for a fixed $k$ the sequence
$Val_{k}^{O(n)}\overset{i_{n}^{*}}{\leftarrow}Val_{k}^{O(n+1)}\overset{i_{n+1}^{*}}{\leftarrow}\dots$
stabilizes. Let us denote this limit vector space by $Val_{k}^{O(\infty)}$.
Consider the stable algebra of isometry invariant valuations

$Val^{O(\infty)}:=\oplus_{k=0}^{\infty}Val_{k}^{O(\infty)}.$

From Theorem 2.6 we easily deduce the following statement:

######  4.1 Proposition.

The graded algebra $Val^{O(\infty)}=Val^{SO(\infty)}$ is isomorphic to the
graded algebra of polynomials in one variable $\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}[x]$ with the grading given by
the degree of a polynomial.

In [4] we have described in geometric terms the vector space
$Val^{U(m)}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}^{m})$ of translation invariant
unitarily invariant continuous valuations on a Hermitian space
$\mbox{\rm$~{}\vrule height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}^{m}$. It
would be of interest to describe the algebra structure of this space (compare
with Theorem 0.9). However it might be of interest as well to describe the
stable algebra of translation invariant unitarily invariant continuous
valuations $Val^{U(\infty)}$ which is defined similarly to the previous case
using the following lemma.

######  4.2 Lemma.

Let $i_{n}:\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}^{n}\hookrightarrow\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}^{n+1}$ be the standard Hermitian
imbedding such that the last coordinate vanishes. For any $k$ the restriction
map $i_{n}^{*}:Val_{k}^{U(n+1)}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}^{n+1})\longrightarrow
Val_{k}^{U(n)}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}^{n})$ is an isomorphism provided
$n\geq k$.

Proof. In [2] we have shown that $\dim Val_{k}^{U(n)}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}^{n})=1+\left[\frac{min\\{k,2n-k\\}}{2}\right]$.
Hence if $n\geq k$ we have $\dim Val_{k}^{U(n)}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}^{n})=\dim
Val_{k}^{U(n+1)}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}^{n+1})=1+[k/2]$. Hence it
suffices to prove that $i_{n}^{*}$ is injective. Let
$\\!{}^{\textbf{R}}Gr_{k}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}^{n})$ denote the Grassmannian of
real $k$-dimensional subspaces of $\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}^{n}$. In [1], [2] we have used
an imbedding of $Val_{k}^{0}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}^{n})$ to the space
$C(\\!{}^{\textbf{R}}Gr_{k}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}^{n}))$ of continuous functions
on the Grassmannian as follows. Let $\phi\in Val_{k}^{0}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}^{n})$. For any
$E\in\\!{}^{\textbf{R}}Gr_{k}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}^{n})$ consider the restriction
$\phi|_{E}\in Val_{k}(E)$. By a result by Hadwiger [8] $Val_{k}(E)$ coincides
with the space of Lebesgue measures on $E$. Hence $\phi|_{E}=f(E)vol_{E}$
where $vol_{E}$ is the Lebesgue measure induced by the Hermitian metric. Thus
$\phi\mapsto[E\mapsto f(E)]$ defines a map

$Val_{k}^{0}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}^{n})\longrightarrow
C(\\!{}^{\textbf{R}}Gr_{k}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}^{n})).$

The non-trivial fact essentially due to D. Klain [11], [12] is that this map
is injective. Let us consider the restriction of this map to $U(n)$-invariant
valuations. Then its image is contained in the space of $U(n)$-invariant
continuous functions on $\\!{}^{\textbf{R}}Gr_{k}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}^{n})$. Hence it is enough to
prove that the restriction map $C(\\!{}^{\textbf{R}}Gr_{k}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}^{n+1}))^{U(n+1)}\longrightarrow
C(\\!{}^{\textbf{R}}Gr_{k}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}^{n}))^{U(n)}$ is injective if
$n\geq k$ where the restriction is considered under the imbedding
$\\!{}^{\textbf{R}}Gr_{k}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}^{n})\hookrightarrow\\!{}^{\textbf{R}}Gr_{k}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}^{n+1})$. It is enough to check
that each $U(n+1)$-orbit in $\\!{}^{\textbf{R}}Gr_{k}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}^{n+1})$ intersects non-trivially
with $\\!{}^{\textbf{R}}Gr_{k}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}^{n})$. This fact follows
immediately from the explicit description of $U(n)$-orbits on
$\\!{}^{\textbf{R}}Gr_{k}(\mbox{\rm$~{}\vrule
height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$C}^{n})$ in terms of Kähler angles
due to H. Tasaki [18]. Q.E.D.

We would like to notice that the next interesting case to classify is the
algebra of translation invariant continuous valuations on the quaternionic
space $\mbox{\rm$~{}\vrule height=6.5pt,width=0.5pt,depth=0.3pt\\!\\!$H}^{n}$
invariant under the group $Sp(n)Sp(1)$.

## References

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Adding this rescaling factor to the training eases the burden on the
programmable wave propagation, as it no longer has to funnel all of the powers
into the discrete and incomplete basis of output modes.

The results are shown in Fig. A20. We observe that the 2D-programmable
waveguide is able to execute the unitary matrix operations with high fidelity
for the $N=100$ case. For $N=150$, the fidelity is lower, but the matrix
elements of the target unitary matrix and the matrix performed by the
2D-programmable waveguide still agree to within 25%.


# Isotypic equivalence of Abelian $p$-groups with separable reduced parts

Elena Bunina  Department of Mathematics, Bar–Ilan University, 5290002 Ramat
Gan, ISRAEL<EMAIL_ADDRESS>
###### Abstract.

We prove that two elementarily equivalent Abelian $p$-groups with separable
reduced parts are isotypically equivalent. Also as the corollary we prive that
any Abelian $p$-group with a separable reduced part if $\omega$-strongly
homogeneous.

###### Key words and phrases:

Types, isotypic equivalence, elementary equivalence, Abelian $p$-groups

###### 2020 Mathematics Subject Classification:

03C52, 20K10

##  1\. Introduction

In this paper we study Abelian $p$-groups $A$ with a separable reduced part.
Our goal is to figure out how close are such groups in case they have the same
sets of types realized in $A$. We show that these groups have the same types
(are _isotypically equivalent_) if and only if they are elementarily
equivalent.

In this paper (Abelian) groups are our main subject, so we do not consider
here rings, semigroups, etc., even thought most of definitions below make
sense for arbitrary algebraic structures.

###### Definition 1.1.

Let $G$ be a group and $(g_{1},\dots,g_{n})$ a tuple of its elements. The type
of this tuple in $G$, denoted $\mathrm{tp}^{G}(g_{1},\dots,g_{n})$, is the set
of all first order formulas in free variables $x_{1},\dots,x_{n}$ in the
standard group theory language which are true on $(g_{1},\dots,g_{n})$ in $G$
(see [6] or [8] for details).

###### Definition 1.2.

The set of all types of tuples of elements of $G$ is denoted by
$\mathrm{tp}(G)$. Following [12], we say that two groups $G$ and $H$ are
isotypic if $\mathrm{tp}(G)=\mathrm{tp}(H)$, i. e., if any type realized in
$G$ is realized in $H$, and vice versa.

Isotypic groups appear naturally in logical (algebraic) geometry over groups
which was developed in the works of B. I. Plotkin and his co-authors (see [10,
11, 12] for details), they play an important part in this subject. In
particular, it turns out that two groups are logically equivalent if and only
if they have the same sets of realizable types. So there arise two fundamental
algebraic questions which are interesting in its own right: what are possible
types of elements in a given group $G$ and how much of the algebraic structure
of $G$ is determined by the types of its elements?

Isotypicity property of groups is related to the elementary equivalence
property, though it is stronger. Indeed, two isotypic groups are elementarily
equivalent, but the converse does not hold. For example, if we denote by
$F_{n}$ a free group of finite rank $n$, then groups $F_{n}$ and $F_{m}$ for
$2\leqslant m<n$ are elementarily equivalent [7, 15], but not isotypic [9].
Furthermore, Theorem 1 from [9] shows that two finitely generated isotypic
nilpotent groups are isomorphic, but there are examples, due to Zilber, of two
elementarily equivalent non-isomorphic finitely generated nilpotent of class
$2$ groups [16].

Isotypicity is a very strong relation on groups, which quite often implies
their isomorphism. This explains the need of the following definition.

###### Definition 1.3.

We say that a group $G$ is _defined by its types_ if every group isotypic to
$G$ is isomorphic to $G$.

It was noticed in [9] that every finitely generated group $G$ which is defined
by its types satisfies a (formally) stronger property. Namely, we say that

###### Definition 1.4.

A finitely generated group $G$ is _strongly defined by types_ if for any
isotypic to $G$ group $H$ every elementary embedding $G\to H$ is an
isomorphism.

Miasnikov and Romanovsky in the paper [9] proved that

1) every virtually polycyclic group is strongly defined by its types;

2) every finitely generated metabelian group is strongly defined by its types;

3) every finitely generated rigid group is strongly defined by its types. In
particular, every free solvable group of finite rank is strongly defined by
its types.

R. Sclinos (unpublished) proved that finitely generated homogeneous groups are
defined by types. Moreover, finitely generated co-hopfian and finitely
presented hopfian groups are defined by types. Nevertheless, the main problem
in the area remains widely open:

######  Problem 1.5 ([11]).

Is it true that every finitely generated group is defined by types?

In the recent paper [2] Gvozdevsky proved that any field of finite
transcendence degree over a prime subfield is defined by types. Also he gave
several interesting examples of certain countable isotypic but not isomorphic
structures: totally ordered sets, rings, and groups.

Although in the most of classes of groups isotypicity is a more stronger
property than elementary equivalence, for the class of Abelian $p$-groups ($p$
is a fixed prime number) with separable reduced parts classifications up to
isotypicity and up to elementary equivalence coincide.

This paper is devoted to the following theorem:

######  Theorem A.

For any prime $p$ two Abelian $p$-groups $A_{1}$ and $A_{2}$ with separable
reduced parts are isotypic if and only if they are elementarily equivalent.

We hope that it is the first step towards the description of all groups
isotypic to a given Abelian group. Of course it is not reasonable to hope that
on the whole class of Abelian groups isotypicity would coincide with
elementary equivalence.

##  2\. Elementary equivalence of Abelian groups

###### Definition 2.1.

Two groups are called _elementarily equivalent_ if their first order theories
coincide.

Elementary equivalent Abelian groups were completely described in 1955 by
Wanda Szmielew in [14] (see also Eclof and Fisher, [3]).

To formulate her theorem we need to introduce a set of special invariants of
Abelian groups.

Let $A$ be an Abelian group, $p$ a prime number, $A[p]$ be the subgroup of
$A$, containing all elements of $A$ of the orders $p$ or $1$ (it is so-called
$p$-_socle_ of the group $A$), $kA$ be the subgroup of $A$, containing all
elements of the form $ka$, $a\in A$.

The first invariant is

$D(p;A):=\lim\limits_{n\to\infty}\dim((p^{n}A)[p])\text{ for every prime }p.$

Note that for every $n\in\mathbb{N}$ the subgroup $(p^{n}A)[p]$ consists of
elements which are annulated being multiplied by $p$, therefore it is a vector
space over the field $\mathbb{Z}_{p}$. So $\dim(p^{n}A)[p]$ is for every
$n\in\mathbb{N}$ a well-defined cardinal number.

Since $p^{n+1}A\subset p^{n}A$, then $(p^{n+1}A)[p]\subset(p^{n}A)[p]$,
therefore

$\dim(p^{n+1}A)[p]\leqslant\dim(p^{n}A)[p]\text{ for all }n\in\mathbb{N}.$

Consequently we have a non-increasing sequence of cardinal numbers, which has
the smallest element.

Thus for any Abelian group $A$ and any prime $p$ the invariant $D(p;A)$ is
well-defined.

The second invariant is

$Tf(p;A):=\lim\limits_{n\to\infty}\dim(p^{n}A/p^{n+1}A).$

This invariant is also well-defined for any prime $p$ and any Abelian group
$A$.

The third invariant  is

$U(p,n-1;A):=\dim((p^{n-1}A)[p]/(p^{n}A)[p]),$

which is called the _Ulm_ invariant, it defines the number of copies of
$\mathbb{Z}_{p^{n}}$ in $A$.

The last invariant is $\,\mathrm{Exp}\,(A)$ which is an _exponent_ of $A$ (the
smallest natural number $n$ such that $\forall a\in A\,na=0$).

######  Theorem 2.2 (Szmielew theorem on elementary classification of Abelian
groups).

Two Abelian groups $A_{1}$ and $A_{2}$ are elementarily equivalent if and only
if their elementary invariants $D(p;\cdot)$, $Tf(p;\cdot)$, $U(p,n-1;\cdot)$
and $\,\mathrm{Exp}\,(\cdot)$ pairwise coincide _(_ more precisely, they are
either finite and coincide or simultaneously are equal to infinity _)_.

##  3\. Abelian $p$-groups, their structure and elementary equivalence

Let $p$ be some prime number, $A$ be an Abelian $p$-group.

It is said that an element $a\in A$ is _divisible by_ a positive integer $n$
(denoted as $n\mid a$) if there is an element $x\in A$ such that $nx=a$. A
group $D$ is called _divisible_ if $n\mid a$ for all $a\in D$ and all natural
$n$. The groups $\mathbb{Q}$ and $\mathbb{Z}(p^{\infty})$ are examples of
divisible groups. Any divisible subgroup is a direct summand of a group. A
group $A$ is called _reduced_ if it has no nonzero divisible subgroups.

A subgroup $G$ of a group $A$ is called _pure_ if the equation $nx=g\in G$ is
solvable in $G$ whenever it is solvable in the entire group $A$. In other
words, $G$ is pure if and only if

$\forall n\in\mathbb{Z}\quad nG=G\cap nA.$

A subgroup $B$ of a group $A$ is called a $p$-_basic subgroup_ if it satisfies
the following conditions:

  1. (1)

$B$ is a direct sum of cyclic $p$-groups and infinite cyclic groups;

  2. (2)

$B$ is pure in $A$;

  3. (3)

$A/B$ is $p$-divisible.

Every group, for every prime $p$, contains $p$-basic subgroups [4].

Now we focus on $p$-groups, where $p$-basic subgroups are particularly
important. If $A$ is a $p$-group and $q$ is a prime different from $p$, then
evidently $A$ has only one $q$-basic subgroup, namely $0$. Therefore, in
$p$-groups we may refer to the $p$-basic subgroups simply as _basic_
subgroups, without confusion.

We need the following facts about basic subgroups.

######  Theorem 3.1 ([13]).

Assume that $B$ is a subgroup of a $p$-group $A$,
$B=\bigoplus\limits_{n=1}^{\infty}B_{n}$, and $B_{n}$ is a direct sum of
groups $\mathbb{Z}(p^{n})$. Then $B$ is a basic subgroup of $A$ if and only if
for every integer $n>0$, the subgroup $B_{1}\oplus\dots\oplus B_{n}$ is a
maximal $p^{n}$-bounded direct summand of $A$.

Any Abelian $p$-group $A$ is a direct sum of its divisible part $D$
(isomorphic to $\bigoplus\limits_{\varkappa_{0}}\mathbb{Z}(p^{\infty})$) and
its reduced part $\overline{A}$ with a basic subgroup

$B=\bigoplus\limits_{n=1}^{\infty}\left(\bigoplus\limits_{\varkappa_{n}}\mathbb{Z}(p^{n})\right).$

The basic subgroup $B$ is dense in $\overline{A}$ in its $p$-adic topology.

An infinite system $L=\\{a_{i}\\}_{i\in I}$ of elements of the group $A$ is
called _independent_ if every finite subsystem of $L$ is independent. An
independent system $M$ of $A$ is _maximal_ if there is no independent system
in $A$ containing $M$ properly. By the _rank_ $r(A)$ of a group $A$ we mean
the cardinality of a maximal independent system containing only elements of
infinite and prime power orders. The _final rank_ of a basic subgroup $B$ of a
$p$-group $A$ is the infimum of the cardinals $r(p^{n}B)$.

###### Definition 3.2.

Given $a\in A$, the greatest nonnegative integer $r$ for which $p^{r}x=a$, is
solvable for some $x\in A$, is called the _$p$ -height_ $h_{p}(a)$ of $a$. If
$p^{r}x=a$ is solvable whatever $r$ is, $a$ is of _infinite $p$-height_,
$h_{p}(a)=\infty$. If it is completely clear from the context which prime $p$
is meant, we call $h_{p}(a)$ simply the _height_ of $a$ and write $h(a)$.

###### Definition 3.3.

A reduced Abelian $p$-group $A$ is called _separable_ , if it does not contain
any non-zero elements of infinite height.

For a reduced $p$-group $A$ _the first Ulm subgroup_ of $A$ is

$A^{\mathbf{1}}=\bigcap_{n=1}^{\infty}p^{n}A,$

it is the subgroup of $A$ consisting of all elements of $A$ of infinite
height. Therefore a reduced $A$ is separable if and only if
$A^{\mathbf{1}}=0$.

######  Proposition 3.4 ([4]).

An element $a$ of prime power order belongs to a finite direct summand of $A$
if and only if $\langle a\rangle$ contains no elements of infinite height.

######  Theorem 3.5 ([1]).

Suppose that $B$ is a subgroup of $p$-group $A$,

$B=B_{1}\oplus B_{2}\oplus\dots\oplus B_{n}\oplus\dots,$

where

$B_{n}\cong\bigoplus_{\mu_{n}}\mathbb{Z}(p^{n}).$

The subgroup $B$ is a basic subgroup of $A$ if and only if

$A=B_{1}\oplus B_{2}\oplus\dots\oplus B_{n}\oplus(B_{n}^{*}+p^{n}A),$

where $n\in\mathbb{N}$,

$B_{n}^{*}=B_{n+1}\oplus B_{n+2}\oplus\dots$

Since the group $B$ has a basis, and the quotient group $A/B$ is a direct sum
of groups isomorphic to $\mathbb{Z}(p^{\infty})$ (i. e. $A/B$ also has a
generating system which can be easily described) then it is natural to combine
these generating systems and to obtain a one for $A$.

We write

$B=\bigoplus_{i\in I}\langle a_{i}\rangle\text{ and }A/B=\bigoplus_{j\in
J}C_{j}^{*},\text{ where }C_{j}^{*}=\mathbb{Z}(p^{\infty}).$

If the direct summand $C_{j}^{*}$ is generated by cosets
$c_{j1}^{*},\dots,c_{jn}^{*},\dots$ modulo $B$ with $pc_{j1}^{*}=0$,
$pc_{j,n+1}^{*}=c_{jn}^{*}$ ($n=1,2,\dots$), then, by the purity of $B$ in
$A$, in the group $A$ we can pick out $c_{jn}\in c_{jn}^{*}$ of the same order
as $c_{jn}^{*}$. Then we get the following set of relations:

$pc_{j1}=0,\ pc_{j,n+1}=c_{jn}=b_{jn}\quad(n\geq 1,b_{jn}\in B),$

where $b_{jn}$ must be of order $\leq p^{n}$, since $o(c_{jn})=p^{n}$.

The system $\\{a_{i},c_{jn}\\}_{i\in I,j\in J,n\in\omega}$ will be called a
_quasibasis_ of $A$.

######  Proposition 3.6.

_([5])_ If $\\{a_{i},c_{jn}\\}$ is a quasibasis of the $p$-group $A$, then
every $a\in A$ can be written in the following form:

(3.1)
$a=s_{1}a_{i_{1}}+\dots+s_{m}a_{i_{m}}+t_{1}a_{j_{1}n_{1}}+\dots+t_{r}a_{j_{r}n_{r}},$

where $s_{i}$ and $t_{j}$ are integers, no $t_{j}$ is divisible by $p$, and
the indices $i_{1},\dots,i_{m}$ as well as $j_{1},\dots,j_{r}$ are distinct.
The condition (3.1) is unique in the sence that $a$ uniquely defines the terms
$sa_{i}$ and $tc_{jn}$.

Let us now concentrate on elementary equivalence of two Abelian $p$-groups
with separable reduced parts.

Suppose that $A=D\oplus\overline{A}$, where $\overline{A}$ is reduced and
separable (i. e., does not contain any elements of infinite height) and has
$B$ as its basis subgroup.

Assume also that

$D\cong\bigoplus_{\varkappa_{0}}\mathbb{Z}(p^{\infty}),\quad
B_{k}\cong\bigoplus_{\varkappa_{k}}\mathbb{Z}(p^{k}),\quad k=1,\dots,n,\dots$

1. For the first invariant $D(p;A)$ let us fix some $n\in\mathbb{N}$ and represent $A$ as

$A=D\oplus B_{1}\oplus B_{2}\oplus\dots\oplus
B_{n}\oplus(B_{n}^{*}+p^{n}\overline{A})$

as in Theorem 3.5.

Then

$p^{n}A=(p^{n}D)\oplus p^{n}(B_{n}^{*}+p^{n}\overline{A})=D\oplus
p^{n}(B_{n}^{*}+p^{n}\overline{A}),$

therefore

$\dim p^{n}A[p]=\varkappa_{0}+\dim p^{n}(B_{n}^{*}+p^{n}\overline{A})[p]$

and

$\lim\limits_{n\to\infty}\dim p^{n}A[p]=\varkappa_{0},$

since $\bigcap\limits_{n=1}^{\infty}B_{n}^{*}=0$ and
$\bigcap\limits_{n=1}^{\infty}p^{n}\overline{A}=0$.

2. The second invariant is

$\lim\limits_{n\to\infty}\dim(p^{n}A/p^{n+1}A)=\lim\limits_{n\to\infty}\dim(p^{n}D/p^{n+1}D\oplus
p^{n}\overline{A}/p^{n+1}\overline{A})=\lim\limits_{n\to\infty}\dim(p^{n}\overline{A}/p^{n+1}\overline{A})=\\\
=\lim\limits_{n\to\infty}\dim(p^{n}B_{1}/p^{n+1}B_{1}+\dots+p^{n}B_{n+1}/p^{n+1}B_{n+1}+p^{n}(B_{n+1}^{*}+p^{n+1}\overline{A})/p^{n+1}(B_{n+1}^{*}+p^{n+1}\overline{A}))=\\\
=\lim\limits_{n\to\infty}(\varkappa_{n+1}+\varkappa_{n+2}+\dots)=0.$

3. The third invariant is

$U(p,n-1;A):=\dim((p^{n-1}A)[p]/(p^{n}A)[p])=\varkappa_{n}.$

Therefore two Abelian $p$-groups are elementarily equivalent if and only if
their divisible parts are elementarily equivalent and their basic subgroups
are elementarily equivalent. In its turn elementary equivalence of divisible
parts means that either both of them contain finite number of
$\mathbb{Z}(p^{\infty})$ in the direct sum and are isomorphic, or both of them
contain infinite number of $\mathbb{Z}(p^{\infty})$ in the direct sum. The
same is for any direct summand $B_{i}$ of the basic subgroups.

##  4\. Types of elements

Suppose that the decomposition $A=D\oplus\overline{A}$ is fixed. Suppose also
that we have a tuple $(g_{1},\dots,g_{m})$ of elements of $A$ and their type
$\mathrm{tp}(g_{1},\dots,g_{m})$. If $g_{i}=d_{i}+a_{i}$, $i=1,\dots,m$, is a
decomposition of these elements with respect to the direct summands $D$ and
$\overline{A}$, then $\mathrm{tp}(d_{1},\dots,d_{m},a_{1},\dots a_{m})$
contains
$\mathrm{tp}(g_{1},\dots,g_{m})=\mathrm{tp}(d_{1}+a_{1},\dots,d_{m}+a_{m})$.
Therefore we always can assume that we study only types
$\mathrm{tp}(d_{1},\dots,d_{\ell},a_{1},\dots,a_{m})$,
$d_{1},\dots,d_{\ell}\in D$, $a_{1},\dots,a_{m}\in\overline{A}$.

Let us for some fixed decomposition $A=D\oplus\overline{A}$ have
$d_{1},\dots,d_{\ell}\in D$ and $a_{1},\dots,a_{m}\in\overline{A}$. The
elements $d_{1},\dots,d_{\ell}$ generate a direct summand

$\langle
d_{1},\dots,d_{\ell}\rangle=D_{1}\cong\bigoplus\limits_{n_{0}}\mathbb{Z}(p^{\infty})\text{
in }D.$

Knowing $\mathrm{tp}(d_{1},\dots,d_{\ell})$ we can easily define $n_{0}$: it
is $k-1$, where $k$ is the minimal natural number such that there exist a
subset $\\{m_{1},\dots,m_{k}\\}\subset\\{1,\dots,\ell\\}$,
$\alpha_{1},\dots,\alpha_{k}\in\mathbb{Z}$,
$0\leqslant\alpha_{i}<\mathrm{ord}(d_{m_{i}})$, $\alpha_{1},\dots,\alpha_{k}$
are not all zeros, such that

$\alpha_{1}d_{m_{1}}+\dots+\alpha_{k}d_{m_{k}}=0.$

Now let us study the elements $a_{1},\dots,a_{m}$. All these elements have
finite heights. Suppose that

$\mathrm{ord}(a_{i})=p^{t_{i}},\quad h(a_{i})=p^{s_{i}},\quad i=1,\dots,\ell.$

Let us take instead of every $a_{i}$ an element $b_{i}$ such that
$p^{s_{i}}b_{i}=a_{i}$. Then every $b_{i}$ has the height $0$ and therefore
generates a direct summand

$a_{i}\in\langle b_{i}\rangle\cong\mathbb{Z}(p^{t_{i}+s_{i}}).$

If we take the subgroup $\overline{B}=\langle b_{1},\dots,b_{m}\rangle$, it is
a finite subgroup of $\overline{A}$. Since $\overline{A}$ does not contain any
elements of infinite height and $\overline{B}$ is finite, then the heights of
all nonzero elements of $\overline{B}$ in $\overline{A}$ are bounded,
therefore $\overline{B}$ is embedded in a direct summand of $\overline{A}$.
This summand is finite and is isomorphic to

$\overline{B}_{1}\oplus\dots\oplus\overline{B}_{q},\text{ where
}\overline{B}_{i}\cong\bigoplus\limits_{n_{i}}\mathbb{Z}(p^{i}).$

Of course all $n_{i}$ are defined by formulas with $a_{1},\dots,a_{m}$ as
parameters.

Also, we can find a basic subgroup $B$ of $\overline{A}$ such that
$\overline{B}$ is a direct summand of $B$.

Now we are ready to prove the main theorem (Theorem A).

Proof. Let $A_{1}$ and $A_{2}$ be two elementarily equivalent Abelian
$p$-groups, $A_{1}=D_{1}\oplus\overline{A}_{1}$,
$A_{2}=D_{2}\oplus\overline{A}_{2}$ be their decompositions in the direct sum
of divisible and reduced separable subgroups,
$(d_{1},\dots,d_{\ell},a_{1},\dots,a_{m})$ be a tuple of element of $A_{1}$,
where $d_{1},\dots,d_{\ell}\in D_{1}$ and
$a_{1},\dots,a_{m}\in\overline{A}_{1}$. These elements generate (in the sense
above) a direct summand of $A_{1}$, isomorphic to

$C=C_{d}\oplus
C_{r}\cong\bigoplus\limits_{n_{0}}\mathbb{Z}(p^{\infty})\oplus\bigoplus\limits_{t=1}^{m}\left(\bigoplus\limits_{n_{t}}\mathbb{Z}(p^{t})\right).$

Therefore if

$D_{1}=\bigoplus\limits_{\varkappa_{0}}\mathbb{Z}(p^{\infty}),\quad
D_{2}=\bigoplus\limits_{\varkappa_{0}^{\prime}}\mathbb{Z}(p^{\infty})$

and

$B_{1}=\bigoplus\limits_{t=1}^{\infty}\left(\bigoplus\limits_{\varkappa_{t}}\mathbb{Z}(p^{t})\right),\quad
B_{2}=\bigoplus\limits_{t=1}^{\infty}\left(\bigoplus\limits_{\varkappa_{t}^{\prime}}\mathbb{Z}(p^{t})\right),$

then for all $i=0,1,\dots,m$

$n_{i}\leqslant\varkappa_{i}=\varkappa_{i}^{\prime}\text{ or
}\varkappa_{i},\varkappa_{i}^{\prime}\text{ are both infinite}.$

So we can denote

$A_{1}=D_{1}\oplus\overline{A}_{1}=(C_{d}\oplus\widetilde{C}_{d})\oplus(C_{r}\oplus\widetilde{B}_{1}^{(1)}\oplus\dots\oplus\widetilde{B}_{1}^{(m)})\oplus(\overline{B}_{1}^{(m+1)}+p^{m}\overline{A}_{1})=C\oplus\widetilde{C}_{1}.$

Similarly the group $A_{2}$ can be decomposed as

$A_{2}=D_{2}\oplus\overline{A}_{2}\cong(C_{d}\oplus\widetilde{C}_{d}^{\prime})\oplus(C_{r}\oplus\widetilde{B}_{2}^{(1)}\oplus\dots\oplus\widetilde{B}_{2}^{(m)})\oplus(\overline{B}_{2}^{(m+1)}+p^{m}\overline{A}_{2})=C\oplus\widetilde{C}_{2}$

and

$\widetilde{C}_{1}\equiv\widetilde{C}_{2}.$

Therefore we have
$(d_{1}^{\prime},\dots,d_{\ell}^{\prime},a_{1}^{\prime},\dots,a_{m}^{\prime})\in
C\in A_{2}$ (the same as $(d_{1},\dots,d_{\ell},a_{1},\dots,a_{m})\in C\in
A_{1}$ such that

$\mathrm{tp}(d_{1}^{\prime},\dots,d_{\ell}^{\prime},a_{1}^{\prime},\dots,a_{m}^{\prime})=\mathrm{tp}(d_{1},\dots,d_{\ell},a_{1},\dots,a_{m}).$

Therefore $A_{1}$ and $A_{2}$ are isotypic. $\blacksquare$

###### Remark 4.1.

From [4], Theorem 77.3, it follows that any countable reduced separable
Abelian $p$-group is a direct sum of cyclic groups. Therefore for countable
groups with separable reduced part elementary equivalence coincides not only
with isotypicity, but also with isomorphism.

For noncountable Abelian $p$-groups with rseparable reduced parts there are
many examples of elementary equivalence (and isotypical) non-isomorphic
groups.

For example we can take any unbounded basic $p$-group $B$ of the power
$\varkappa>\aleph_{0}$ and the $p$-part of it’s closure in $p$-adic topology
$A$. Then both $A$ and $B$ are reduced separable $p$-groups with the basic
subgroup $B$, so they are elementary equivalent and isotypical, but not
isomorphic.

##  5\. Strong homogeneity

###### Definition 5.1.

A model $\mathcal{M}$ is called _$\omega$ -strongly homogeneous_, if for any
$a_{1},\dots,a_{n},b_{1},\dots,b_{n}\in M$ if
$\mathrm{tp}\,(a_{1},\dots,a_{n})=\mathrm{tp}\,(b_{1},\dots,b_{n})$, then
there exists an automorphism $\varphi\in\,\mathrm{Aut}\,\mathcal{M}$ such that
$\varphi(a_{i})=b_{i}$, $i=1,\dots,n$.

From the proof of Theorem A we can derive the following corollary:

######  Corollary 5.2.

For any prime $p$ any Abelian $p$-group $A$ with a separable reduced part is
$\omega$-strongly homogeneous.

Proof. Let us take our group $A$ and $a_{1},\dots,a_{n},b_{1},\dots,b_{n}\in
A$ with $\mathrm{tp}\,(a_{1},\dots,a_{n})=\mathrm{tp}\,(b_{1},\dots,b_{n})$.
According to the previous considerations we can (without loss of generality)
assume, that $a_{1},\dots,a_{k},b_{1},\dots,b_{k}\in D$, where $D$ is the
divisible part of $A$, and $a_{k+1},\dots,a_{n}\in A_{1}$,
$b_{k+1},\dots,b_{n}\in A_{2}$, where $A_{1}$ and $A_{2}$ are reduced and
$A=D\oplus A_{1}=D\oplus A_{2}$.

Since $\mathrm{tp}\,(a_{1},\dots,a_{k})=\mathrm{tp}\,(b_{1},\dots,b_{k})$,
these elements are contained in isomorphic minimal direct summands of $D$:

$D_{1}\cong D_{2}\cong\bigoplus\limits_{\ell}\mathbb{Z}(p^{\infty}).$

In these groups $a_{1},\dots,a_{k}$ and $b_{1},\dots,b_{k}$ are such elements
that for any $\alpha_{1},\dots,\alpha_{k}\in\mathbb{Z}$ the linear
combinations $\alpha_{1}a_{1}+\dots+\alpha_{k}a_{k}$ and
$\alpha_{1}b_{1}+\dots+\alpha_{k}b_{k}$ have the same orders. Therefore there
exists an isomorphism $\varphi_{1}:D_{1}\to D_{2}$ such that
$\varphi_{1}(a_{i})=b_{i}$ for all $i=1,\dots,k$.

Since $D=D_{1}\oplus D_{1}^{\prime}=D_{2}\oplus D_{2}^{\prime}$, where
$D_{1}^{\prime}\cong D_{2}^{\prime}$, the isomorphism $\varphi$ can be
extended up to an automorphism $\varphi_{2}\in\,\mathrm{Aut}\,D$.

Now let us consider the elements $a_{k+1},\dots,a_{n}\in A_{1}$ and
$b_{k+1},\dots,b_{n}\in A_{2}$. Since
$\mathrm{tp}\,(a_{k+1},\dots,a_{n})=\mathrm{tp}\,(b_{k+1},\dots,b_{n})$, these
elements are contained in isomorphic minimal direct summands $B_{1}$ of
$A_{1}$ and $B_{2}$ $A_{2}$, respectively.

By the same reasons, since for any
$\alpha_{k+1},\dots,\alpha_{n}\in\mathbb{Z}$ the linear combinations
$\alpha_{k+1}a_{k+1}+\dots+\alpha_{n}a_{n}$ and
$\alpha_{1}b_{1}+\dots+\alpha_{k}b_{k}$ have the same orders and heights,
there exists an isomorphism $\varphi_{3}:B_{1}\to B_{2}$ such that
$\varphi_{3}(a_{i})=b_{i}$ for all $i=k+1,\dots,n.$

Since $A_{1}=B_{1}\oplus C_{1}$ and $A_{2}=B_{2}\oplus C_{2}$, where
$C_{1}\cong C_{2}$, we can extend $\varphi_{3}$ up to an isomorphism
$\varphi_{4}:A_{1}\to A_{2}$.

Since $A=D\oplus A_{1}=D\oplus A_{2}$, the mappings $\varphi_{2}$ and
$\varphi_{4}$ give us the required automorphism $\varphi$. $\blacksquare$

Acknowledgements. Our sincere thanks go to Alexey Miasnikov and Eugene Plotkin
for very useful discussions regarding this work and permanent attention to it.

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