Table Of ContentTHE RADICAL OF THE KERNEL OF A CERTAIN
DIFFERENTIAL OPERATOR AND APPLICATIONS TO
LOCALLY ALGEBRAIC DERIVATIONS
7
1
0
2 WENHUA ZHAO
n
a Abstract. Let R be a commutative ring, A an R-algebra (not
J
necessary commutative) and V an R-subspace or R-submodule of
2 A. By the radical of V we mean the set of allelements a∈A such
2
that am ∈V for all m≫0. We derive (and show) some necessary
] conditions satisfied by the elements in the radicals of the kernels
A
ofsome(partial)differentialoperatorssuchasalldifferentialoper-
R ators of commutative algebras; the differential operators P(D) of
. (noncommutative)Awithcertainconditions,whereP(·)isapoly-
h
t nomial in n commutative free variables and D =(D1,D2,...,Dn)
a are either n commutatinglocally finite R-derivationsorn commu-
m
tating R-derivations of A such that for each 1 ≤ i ≤ n, A can be
[ decomposed as a direct sum of the generalized eigen-subspaces of
1 Di; etc. We then apply the results mentioned above to study R-
v derivationsofAthatarelocallyalgebraicorlocallyintegrableover
4 R. In particular, we show that if R is an integral domain of char-
2 acteristic zero and A is reduced and torsion-free as an R-module,
1
then A has no nonzero locally algebraic R-derivations. We also
6
0 show a formula for the determinant of a differential vandemonde
. matrixovercommutativealgebras. Thisformulanotonlyprovides
1
some information for the radicals of the kernels of ordinary differ-
0
7 entialoperatorsofcommutativealgebras,butalsoisinterestingon
1 its own right.
:
v
i
X
r
a 1. Background and Motivation
Let R be a commutative ring and A an R-algebra (not necessary
commutative). A derivation D of A is a map from A to A such that
D(ab) = D(a)b+aD(b) for all a,b ∈ A. If D is also R-linear, we call
it an R-derivation of A.
Date: January 24, 2017.
2000 Mathematics Subject Classification. 47F05, 47E05,16W25, 16D99.
Key words and phrases. Theradical,the kernelofadifferentialoperator,locally
algebraic or integral derivations, a differential vandemonde determinant, Mathieu
subspaces (Mathieu-Zhao spaces).
TheauthorhasbeenpartiallysupportedbytheSimonsFoundationgrant278638.
1
2 WENHUAZHAO
For each a ∈ A, we denote by ℓ the map from A to A that maps
a
b ∈ Atoab. Wecalltheassociativealgebrageneratedbyℓ (a ∈ A)and
a
all derivations of A the Weyl algebra of A, and denote it by W(A). The
subalgebra of W(A) generated by ℓ (a ∈ R) and all R-derivations of
a
A will be denoted by W (A). Elements of W(A) are called differential
R
operators of A.
ForeachΦ ∈ W(A),itiswell-knownandalsoeasytocheckthatthere
exist some derivations D = (D ,D ,...,D ) of A and a polynomial
1 2 n
P(ξ) ∈ A[ξ] in n noncommutative free variables ξ = (ξ ,ξ ,...,ξ )
1 2 n
such that Φ = P(D), where P(D) throughout this paper is defined by
first writing all the coefficients of P(ξ) on the left and then replacing ξ
i
by D for all 1 ≤ i ≤ n. Furthermore, if Φ ∈ W (A), the same is true
i R
with D (1 ≤ i ≤ n) being R-derivations of A and P(ξ) ∈ R[ξ]. We call
i
the differential operator Φ = P(D) an ordinary differential operator of
A, if P(ξ) is univariate, and a partial differential operator of A if P(ξ)
is multivariate.
Next, we recall the following two notions of associative algebras that
were first introduced in [Z2, Z3].
Definition 1.1. An R-subspace V of an R-algebra A is said to be a
Mathieu subspace (MS) of A if for all a,b,c ∈ A with am ∈ V for all
m ≥ 1, we have bamc ∈ V for all m ≫ 0.
Note that a MS is also called a Mathieu-Zhao space in the literature
(e.g., see [DEZ, EN, EH], etc.), as suggested by A. van den Essen [E2].
The introduction of this notion is mainly motivated by the study in
[M, Z1] of the well-known Jacobian conjecture (see [Ke, BCW, E1]).
See also [DEZ]. But, a more interesting aspect of the notion is that it
provides a natural but highly non-trivial generalization of the notion
of ideals.
Definition 1.2. [Z3, p.247] Let V be an R-subspace (or a subset) of
an R-algebra A. We define the radical r(V) of V to be
(1.1) r(V):= {a ∈ A|am ∈ V for all m ≫ 0}.
When A is commutative and V is an ideal of A, r(V) coincides with
the radical of V. So this new notion is also interesting on its own right.
It is also crucial for the study of MSs. For example, it is easy to see
that every R-subspace V of an R-algebra A with r(V) = nil(A) is a
MS of A, where nil(A) denotes the set of all nilpotent elements of A.
We will frequently use this fact (implicitly) throughout this paper.
Recent studies show that many MSs arise from the images of dif-
ferential operators, especially, from the images of locally finite or lo-
cally nilpotent derivations, of certain associative algebras (e.g., see
THE RADICAL OF THE KERNEL OF A DIFFERENTIAL OPERATOR 3
[Z1, Z2, EWZ, EZ] and [Z4]–[Z7], etc.). For some MSs arisen from
the kernels of some ordinary differential operators of univariate poly-
nomial algebras over a field, see [EN, EH].
In this paper we study the radicals of the kernels of some ordinary or
partial differential operators of A and show that for certain differential
operators Φ, the kernel KerΦ is also a MS of A. We also apply some
results proved in this paper to study R-derivations of A that are locally
algebraic or locally integrable over R (see Definition 4.1). In particu-
lar, we show that if R is an integral domain of characteristic zero and
A is reduced and torsion-free as an R-module, then A has no nonzero
R-derivation that is locally algebraic over R (see Theorem 4.6). Fur-
thermore, we also show a formula for the determinant of a differential
vandemonde matrix over commutative algebras (see Proposition 5.1).
This formula not only provides some information for the radicals of the
kernels of ordinary differential operators of commutative algebras, but
also is interesting on its own right.
Arrangement and Content: In Section 2, we assume that A is
commutative andderive some necessarily conditions for theelements in
the radical of the kernel of an arbitrary differential operators of A (see
Theorem 2.1 and Corollary 2.4). In particular, for every differential
operator Φ ∈ W(A) such that Φ1A is not zero nor a zero-divisor of A,
the kernel KerΦ forms a MS of A.
In Section 3, we drop the commutativity assumption on A but as-
sume that (R,+) is torsion-free and A is reduced and torsion-free as an
R-module. We first derive in Theorem 3.1 some necessary conditions
satisfied by the elements in the radical of the kernel of a differential
operator P(D) of A, where P(·) is a polynomial in n commutative free
variables and D = (D ,D ,...,D ) are n commutating R-derivations
1 2 n
of A such that for each 1 ≤ i ≤ n, A can be decomposed as a direct
sum of the generalized eigen-subspaces of D .
i
We then show in Proposition 3.6 that if R also is an integral do-
main of characteristic zero, then the conclusions in Theorem 3.1 also
hold for the differential operators of A which are multivariate polyno-
mials in commuting locally finite R-derivations of A. Finally, we show
in Proposition 3.7 that similar conclusions as those in Proposition 3.6
(with the same assumptions on R and A) also hold for all ordinary dif-
ferential operators of A. In particular, for all the differential operators
Φ in Theorem 3.1 and Propositions 3.6 and 3.7 with Φ1A 6= 0, KerΦ
forms a MS of A.
In Section 4, we apply some results proved in Sections 2 and 3 to
study some properties of R-derivation of A that are locally algebraic or
4 WENHUAZHAO
locally integral over R (see Definition 4.1). We first show in Theorem
4.3 that if A is commutative and (A,+) is torsion-free, then every
locally integral D of A has its image in the nil-radical nil(A) of A.
We also show in Theorem 4.6 that if R also is an integral domain of
characteristic zero and A is reduced and torsion-free as an R-module,
then A has no nonzero R-derivation that is locally algebraic over R.
In Section 5, we assume that A is commutative and first show in
Proposition 5.1 a formula for the determinant of a differential vande-
monde matrix over A. We then apply this formula in Proposition 5.4
to derive more necessary conditions satisfied by the elements in the
radicals of the kernels of all ordinary differential operators of A. we
pointoutinRemark5.3thattheformula derived inProposition5.1can
also be used to derive formulas for the determinants of several other
families of matrices.
2. The Commutative Algebra Case
In this section, unless stated otherwise, R denotes a unital commu-
tative ring, A a commutative unital R-algebra and ξ = (ξ ,ξ ,...,ξ )
1 2 n
n noncommutative free variables. We denote by A[ξ] the polynomial
algebra in ξ over A, and ∂ (1 ≤ i ≤ n) the A-derivation ∂/∂ξ of A[ξ].
i i
Once and for all, we fix in this section a nonzero P(ξ) ∈ A[ξ] and n
R-derivations D (1 ≤ i ≤ n) of A. Write D = (D ,D ,...,D ) and
i 1 2 n
P(ξ) = a + d P (ξ) for some a ∈ A, d ≥ 1 and homogeneous
0 k=1 k 0
polynomials P (ξ) (1 ≤ k ≤ d) of degree k in ξ.
k
For each u ∈PA, we set ∇ u:= (D u,D u,...,D u), and call it the
D 1 2 n
gradient of u with respect to D. When D is clear in the context, we
will simply write ∇ u as ∇u.
D
We define P(D) and P(∇u) by first writing P(ξ) as a polynomial
in ξ with all the coefficients on the most left (of the monomials), and
then replacing ξ by D and D u, respectively, for each 1 ≤ i ≤ n.
i i i
The main result of this section is the following
Theorem 2.1. With the setting as above, let u ∈ A such that um ∈
KerP(D) for all 1 ≤ m ≤ d. Then
(2.1) a ud = (−1)dd!P (∇u).
0 d
Furthermore, if ud+1 also lies in KerP(D), then
(2.2) a ud+1 = 0.
0
To show the theorem above, we first need the following two lemmas.
The first lemma is well-known and can also be easily verified by using
THE RADICAL OF THE KERNEL OF A DIFFERENTIAL OPERATOR 5
the mathematical induction, which is similar as the proof for the usual
binomial formula.
Lemma 2.2. Let u ∈ A and ℓ : A → A that maps a ∈ A to ua.
u
Denote by ad : W(A) → W(A) that maps each Λ ∈ W(A) to [u,Λ]:=
u
ℓ Λ−Λℓ . Then for all Φ ∈ W(A) and k ≥ 1, we have
u u
k
k
(2.3) (ad )k(Φ) = (−1)i uk−iΦ◦ℓi.
u i u
i=0 (cid:18) (cid:19)
X
Lemma 2.3. Let u ∈ A. Then the following statements hold:
1) there exists Q(ξ) ∈ A[ξ] with either Q(ξ) = 0 or degQ(ξ) ≤
d−2 such that
n
(2.4) ad P(D) = (D u)(∂ P)(D)+Q(D).
−u i i
i=1
X
2) (ad )dP(D) = d!P(∇u).
−u
Proof: 1) First, if degP(ξ) = 0, then the statement holds triv-
ially, for A is commutative and hence ad P(ξ) = 0. So we assume
−u
degP(ξ) ≥ 1. By the linearity and also the commutativity of A we
may assume P(ξ) = ξ ξ ···ξ with 1 ≤ i ≤ n for all 1 ≤ j ≤ k.
i1 i2 ik j
We use the induction on k ≥ 1. If k = 1, then ad D = ℓ .
−u i1 Di1u
Hence the statement holds by choosing Q(ξ) = 0. Assume that the
statement holds for all 1 ≤ k ≤ m−1 and consider the case k = m.
Since ad is a derivation of W(A), we have
−u
m m
ad P(D) = D ···(ad D )···D = D ···(ℓ )···D
−u i1 −u ij im i1 Diju im
j=1 j=1
X X
m m
= (ℓ )D ···D ···D + [D ···D , ℓ ]D ···D
Diju i1 ij im i1 ij−1 Diju ij+1 im
j=1 j=2
X X
c
Here D means that the term D is omitted:
ij ij
mc m
= (D u)D ···D ···D + (ad (D ···D ))D ···D .
ij i1 ij im −Diju i1 ij−1 ij+1 im
j=1 j=2
X X
c
Applying the induction assumption to the terms ad (D ···D )
−Diju i1 ij−1
(2 ≤ j ≤ m) in the sum above we see that there exists Q(ξ) ∈ A[ξ]
6 WENHUAZHAO
with Q(ξ) = 0 or degQ(ξ) ≤ m−2 such that
m
ad P(D) = (D u)D ···D ···D +Q(D)
−u ij i1 ij im
j=1
X
n c
= (D u)(∂ P)(D)+Q(D).
i i
i=1
X
Hence by the induction statement 1) follows.
2) First, by statement 1) it is easy to see that (ad )dP(D) =
−u
(ad )dP (D). Then by the linearity and also the commutativity of A
−u d
we may assume P (ξ) = ξ ξ ···ξ with 1 ≤ i ≤ n for all 1 ≤ j ≤ d.
d i1 i2 id j
Applying statement 1) (d times) we have
(ad )dP(D) = (D u)(D u)···(D u)(∂ ∂ ···∂ P).
−u k1 k2 kd k1 k2 kd
1≤k1,kX2,...,kd≤n
Then by the equation above and the commutativity of A, it is easy to
✷
see that the equation in statement 2) follows.
Now we can prove the main result of this section.
Proof of Theorem 2.1: By Eq.(2.3) and Lemma 2.3, 2) we have
d
d
(2.5) d!P (∇u) = (−1)d (−1)i ud−iP(D)◦ℓi.
d i u
i=0 (cid:18) (cid:19)
X
By applying both sides of the equation above to 1A ∈ A and then
using the condition ui ∈ KerP(D) (1 ≤ i ≤ d), we get d!P (∇u) =
d
(−1)dudP(D)·1A. It is well-known and also easy to check that every
derivationof a commutative ring annihilates theidentity element of the
ring. Hence d!P (∇u) = (−1)duda , i.e., Eq.(2.1) follows. Similarly,
d 0
by applying Eq.(2.5) above to u ∈ A and using the condition ui ∈
KerP(D) (1 ≤ i ≤ d +1), we get d!P (∇u)u = 0. Then by Eq.(2.1)
d
✷
we get Eq.(2.2).
One immediate consequence of Theorem 2.1 is the following
Corollary 2.4. Let D, P(ξ), a be as in Theorem 2.1, and nil(A) the
0
nil-radical of A, i.e., the set of all nilpotent elements of A. Then the
following statements hold:
1) r(KerΛ) ⊆ Ann(a ), where Ann(a ) is the set of the elements
0 0
b ∈ A such that a b = 0;
0
2) if a is not zero nor a zero-divisor of A, then r(KerP(D)) =
0
nil(A) and KerP(D) is a MS of A;
THE RADICAL OF THE KERNEL OF A DIFFERENTIAL OPERATOR 7
3) if a = 0, then r(KerP(D)) ⊆ {u ∈ A|P (∇u) = 0}. In
0 d
particular, if n = 1, i.e., D is a single derivation of A, and
the leading coefficient of P(ξ) is not a zero-divisor of A, then
r(KerP(D)) ⊆ {u ∈ A|Du ∈ nil(A)}.
Example 2.5. Let R = C and A the C-algebra of all smooth complex
valued functions f(x) over R. Let D = d . Then for each nonzero
dx
univariate polynomial P(ξ) ∈ C[ξ], KerP(D) is the set of solutions
f(x) ∈ A of the ordinary differential equation P(D)f = 0.
Let λ (1 ≤ i ≤ k) be the set of all distinct roots of P(ξ) in C with
i
multiplicity m . Then it is well-known in the theory of ODE (e.g.,
i
see [L] or any other standard text book on ODE) that KerP(D) is the
C-subspace of A spanned by xjeλix for all 1 ≤ i ≤ k and 1 ≤ j ≤ m .
i
From the fact above it is easy to verify directly that r(KerP(D)) =
{0}, if P(0) 6= 0; and r(KerP(D)) = C, if P(0) = 0. Consequently,
Theorem 2.1, Corollary 2.4 and also Proposition 5.4 in Section 5 all
hold in this case.
We end this section with the following two remarks.
First, we will show in Propositions 3.7 and 5.4 that for the ordi-
nary differential operators Φ of certain R-algebras A (not necessarily
commutative), the radical r(KerΦ) also satisfies some other necessarily
conditions (other than those in Theorem 2.1).
Second, Theorem 2.1 and Corollary 2.4 do not always hold for the
differential operators of a noncommutative algebra, which can be seen
from the following
Example 2.6. Let X, Y be two noncommutative free variables and
R[X,Y] the polynomial algebra in X and Y over R. Let J be the
two-sided ideal of R[X,Y] generated by Y2 and A:= R[X,Y]/J. Let
D = ∂/∂X and P(ξ) = 1−Xξ ∈ A[ξ]. Then P(D) = I−ℓ D, where
X
I denotes the identity map of A, and ℓ the multiplication map by X
X
from the left. Let f = XY ∈ A. Then it is easy to check that for
all m ≥ 1, we have P(D)(fm) = 0 but P(D)(Xfm) = −Xfm 6= 0.
Therefore, 0 6= f ∈ r(KerP(D)) and KerP(D) is not a MS of A.
3. Some Cases for Non-Commutative Algebras
In this section, unless stated otherwise, R denotes a commutative
ring such that the abelian group (R,+) is torsion-free, and A an R-
algebra (not necessarily commutative) that is torsion-free as an R-
module.
We denote by IA or simply I the identity map of A, and nil(A) the
set of all nilpotent elements of A. We say A is reduced if nil(A) = {0}.
8 WENHUAZHAO
Furthermore, for each a ∈ A, we denote by Ann (a) the set of elements
ℓ
b ∈ A such that ab = 0.
Let D be an R-derivation of A. We say that A is decomposable w.r.t.
(with respect to) the R-derivation D if A can be written as a direct sum
of the generalized eigen-subspaces of D. More precisely, let H be the
set ofall generalized eigenvalues of D inR andA = ∞ Ker(D−λI)i
λ i=1
for each λ ∈ H. Then A = ⊕ A . It is easy to verify inductively
λ∈H λ
that for all m ≥ 1, a,b ∈ A and λ,µ ∈ R, we have P
m
m
(3.6) D −(λ+µ)I m(ab) = (D −λI)ia (D −µI)m−ib .
i
i=0 (cid:18) (cid:19)
(cid:0) (cid:1) X (cid:0) (cid:1)(cid:0) (cid:1)
Thenbytheidentity abovewehavethatA A ⊆ A forallλ,µ ∈ H.
λ µ λ+µ
In other words, the decomposition A = ∞ Ker(D − λI)i above is
λ i=1
actually an additive R-algebra grading of A.
Some examples of R-derivations with rePspect to which A is decom-
posable are semi-simple R-derivations, for which A (λ ∈ H) coincides
λ
with the eigen-space of D corresponding to the eigenvalue λ of D, and
also locally finite derivations when the base ring R is an algebraically
closed field.
Onceandforall, weletD (1 ≤ i ≤ n)bencommutingR-derivations
i
of A, i.e., D D = D D for all 1 ≤ i,j ≤ n, such that A is decom-
i j j i
posable w.r.t. each D . Then there exists a semi-subgroup Λ of the
i
abelian group (Rn,+) such that
(3.7) A = ⊕ A ,
λ∈Λ λ
where for each λ = (k ,k ,...,k ) ∈ Λ,
1 2 n
n ∞
(3.8) A = Ker(D −k I)j .
λ i i
i=1 j=1
\(cid:0)X (cid:1)
In particular,
n ∞
(3.9) A = KerDj .
0 i
i=1 j=1
\(cid:0)X (cid:1)
Note also that each A (λ ∈ Λ) is invariant under D (1 ≤ i ≤ n),
λ i
and A A ⊆ A for all λ,µ ∈ Λ.
λ µ λ+µ
Now, let ξ = (ξ ,ξ ,...,ξ ) be n commutative free variables and
1 2 n
0 6= P(ξ) ∈ R[ξ]. We set D:= (D ,D ,...,D ) and write and P(ξ) =
1 2 n
d
P (ξ) for some d ≥ 0 and homogeneous polynomials P (ξ) (1 ≤
k=0 k k
k ≤ d) of degree k in ξ. We let P(D) be the differential operator of A
P
obtained by replacing ξ by D (1 ≤ i ≤ n). Since D ’s commute with
i i i
one anther, P(D) is well-defined.
THE RADICAL OF THE KERNEL OF A DIFFERENTIAL OPERATOR 9
The first main result of this section is the following theorem which
in some sense extends Theorem 2.1 to the differential operator P(D)
of the R-algebra A which is not necessarily commutative.
Theorem 3.1. With the setting as above, assume further that A is
reduced. Then the following statements hold:
1) if P = P(0) = 0 and P (ξ) (1 ≤ k ≤ d) have no nonzero
0 k
common zeros in Rn, then r(KerP(D)) ⊆ A ;
0
2) if P = P(0) 6= 0, then r(KerP(D)) = {0}, and KerP(D) is a
0
MS of A.
In order to show the theorem above, we first need to show some
lemmas.
Lemma 3.2. Let R be an arbitrary commutative ring and A an R-
algebra that is torsion-free as an R-module. Let D and P(ξ) be fixed
as above. Then the following statements hold:
1) KerP(D) is homogeneous w.r.t. the grading of A in Eq.(3.7),
i.e.,
(3.10) KerP(D) = (A ∩KerP(D)).
λ
λ∈Λ
M
2) Let Z (P) be the set of λ ∈ Λ such that P(λ) = 0. Then
Λ
(3.11) KerP(D) ⊆ A .
λ
λ∈MZΛ(P)
Proof: 1) Since for each λ ∈ Λ, A is preserved by D (1 ≤ i ≤ n),
λ i
and hence also is preserved by P(D), from which Eq.(3.10) follows.
2) Let 0 6= u ∈ A and write u = ℓ u for some distinct λ ∈ Λ
i=1 λi i
(1 ≤ i ≤ n) and u ∈ A . Then by Eq.(3.10) we have that u ∈
λi λi P
KerP(D), if and only if u ∈ KerP(D). So we may assume ℓ = 1 and
λi
u ∈ A for some λ ∈ Λ.
λ
Write λ = (k ,k ,...,k ). For each 1 ≤ j ≤ n, we define a non-
1 2 n
negative integer r as follows.
j
First, let r be the greatest non-negative integer such that (D −
n n
k I)rnu 6= 0, and inductively, for each 1 ≤ j ≤ n−1, let r be the great
n j
non-negativeintegersuchthat(D −k I)rj n (D −k I)rs u 6= 0.
j j s=j+1 s s
Set u˜:= n (D −k I)rj u. Then 0(cid:16)Q6= u˜ ∈ A , u˜ ∈ Ke(cid:17)rP(D),
j=1 j j λ
and D u˜ = k(cid:16) u˜ for all 1 ≤ j ≤(cid:17)n. Hence 0 = P(D)u˜ = P(λ)u˜. Since A
j jQ
is torsion-free as an R-module, we have P(λ) = 0, as desired. ✷
10 WENHUAZHAO
Definition 3.3. Let A be a subset of Rn and λ ∈ A. We say λ is
an extremal element of A if for all m ≥ 1, mλ can not be written
as a linear combination of other elements of A with positive integer
coefficients whose sum is less or equal to m.
The following lemma should be well-known. But for the sake of
completeness, we here include a direct proof.
Lemma 3.4. Let R be a commutative ring such that the abelian group
(R,+) is torsion free. Then every nonempty finite subset A of Rn has
at least one extremal element.
Proof: Write A = {λ ,λ ,...,λ } with λ 6= λ for all 1 ≤ i 6= j ≤
1 2 n i j
n. We use the induction on n. If n = 1, there is nothing to show. So
we assume n ≥ 2.
Consider first the case n = 2 with λ 6= 0. If the lemma fails, then
2
m λ = k λ and m λ = k λ for some m ,k ≥ 1 with k ≤ m .
1 1 1 2 2 2 2 1 i i i i
Then m m λ = m (k λ ) = k (m λ ) = k k λ . Hence m m = k k ,
1 2 2 1 2 1 2 1 1 1 2 2 1 2 1 2
for λ 6= 0 and (R,+) is torsion-free, from which we have m = k
2 1 1
(and m = k ). By the fact that (R,+) is torsion-free again, we have
2 2
λ = λ . Contradiction.
1 2
Now assume the lemma holds for all 2 ≤ n ≤ k and consider the
case n = k+1. If λ is an extremal point of A, then there is nothing
k+1
to show. Assume otherwise. Then there exist m ≥ 1 and c ∈ N
i
(1 ≤ i ≤ k) such that
k
(3.12) mλ = c λ ,
k+1 i i
i=1
X
k
(3.13) 1 ≤ c ≤ m.
i
i=1
X
By the induction assumption the set A′ := {λ ,λ ,...,λ } has an
1 2 k
extremal element, say, λ . We claim that λ is also an extremal point
1 1
of the set A. Otherwise, there exist q ≥ 1 and c′ ∈ N (2 ≤ j ≤ k +1)
j
such that
k
(3.14) qλ = c′ λ + c′λ ,
1 k+1 k+1 j j
j=2
X
(3.15) 1 ≤ c′ + c′ ≤ q.
k+1 j
j=1
X
