c Allerton Press, Inc., 2014. ISSN 1068-3623, Journal of Contemporary Mathematical Analysis, 2014, Vol. 49, No. 3, pp. 126–138.  c V. N. Margaryan, 2014, published in Izvestiya NAN Armenii. Matematika, 2014, No. 3, pp. 50-67. Original Russian Text 

DIFFERENTIAL EQUATIONS

On Polynomials Hypoelliptic with Respect to a Group of Variables V. N. Margaryan1* 1

Russian-Armenian (Slavonic) University, Yerevan, Armenia Received November 23, 2012

Abstract—For a class of polynomials a necessary and sufficient condition is found for those polynomials to be hypoelliptic with respect to a group of variables. MSC2010 numbers : 12E10, 26C05 DOI: 10.3103/S1068362314030030 Keywords: Hypoelliptic operator; hypoelliptic equation; regular operator; almost hypoelliptic equation; hypoellipticity with respect to a group of variables.

1. BASIC NOTATION AND DEFINITIONS Let N be the set of natural numbers, N0 = N ∪ {0}, N0n be the set of n-dimensional multiindices, that is, the set of points α = (α1 , ..., αn ), αj ∈ N0 , j = 1, ..., n, Rn be the n-dimensional real Euclidean n = {ξ ∈ Rn , ξ ≥ 0, j = 1, ..., n} and space of points ξ = (ξ1 , ξ2 , ..., ξn ), C = R × iR (i2 = −1), R+ j n n R0 = {ξ ∈ R , ξ1 ...ξn = 0}. n and α ∈ N n we denote (ξ, η) = For ξ, η ∈ Rn , k, r ∈ N0 , 1 ≤ k < n, 1 ≤ r ≤ n, t ∈ R+ , λ ∈ R+ 0 ξ1 η1 + ... + ξn ηn , ξ · η = (ξ1 · η1 , ..., ξn · ηn ), ξ  = (ξ1 , ..., ξk ), ξ  = (ξk+1 , ..., ξn ), ξ = (ξ  , ξ  ), ξ(r) = (ξ1 , ..., ξr−1 , ξr+1 , ..., ξn ), |ξ| = (ξ12 + ... + ξn2 )1/2 , tλ = (tλ1 , ..., tλn ), ξ α = ξ1α1 ...ξnαn , |α| = α1 + ... + αn , α! = α1 !...αn ! and Dα = D1α1 ...Dnαn , where Dj = ∂/∂ξj , j = 1, ..., n.  Let P (ξ) = P (ξ1 , ..., ξn ) = aα ξ α be a polynomial, where the sum is over the finite collection of α

numbers (P ) = {α : α ∈ N0n , aα = 0}, aα ∈ C. In what follows we assume that D1 P · ... · Dn P = 0. The Newton’s polyhedron or characteristic polyhedron (c.p.) of the collection (P ), (of the polynomial n containing the set (P ) ∪ {0}. The P ) is defined to be the minimal, convex polyhedron (P ) ⊂ R+ n polyhedron  ⊂ R+ is said to be regular (completely regular), if the coordinates of all exterior (with respect to ) normals to the (n − 1)-dimensional non-coordinate faces of  are nonnegative (positive). For a c.p. (P ) of the collection (P ) ⊂ N0n we denote 0 (P )= the set of vertices of the polyhedron (P ), Λ((P ))= the set of unit exterior (with respect to (P )) normals to the (n − 1)-dimensional non-coordinate faces of (P ), and n ∩ R0n , λ > 0}, Λ+ ((P )) = {λ ∈ Λ((P )); λ ∈ R+

∂(P ) = {ν ∈ (P ); ∃λ ∈ Λ((P )), (ν, λ) = d(λ), d(λ) = max (ν, λ)}. ν∈(P )

n is called principal, if there exists an exterior normal λ to Γ with at The face Γ of a polyhedron  ⊂ R+ least one positive coordinate.

Definition 1.1. (see [1], Definition 11.1.2) A polynomial P (ξ) = P (ξ1 , ..., ξn ) is said to be hypoelliptic if for any nonzero α ∈ N0n we have Dα P (ξ)/P (ξ) → 0 as |ξ| → ∞. *

E-mail: [email protected]

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Definition 1.2. (see [2]) A polynomial P (ξ) = P (ξ1 , ..., ξn ) is said to be partially hypoelliptic with respect to ξ  ≡ (ξ1 , ..., ξk ) (k ∈ N, k < n), if any of the following equivalent conditions is satisfied: 1) For any nonzero α ∈ N0n we have Dα P (ξ)/P (ξ) → 0 as |ξ| → ∞ and ξ  = (ξk+1 , ..., ξn ) remains bounded.   (ξ  )α Pα (ξ  ), then 2) If the polynomial P is represented in the form P (ξ) = α ∈N0n−k

a) P0 (ξ  ) is a hypoelliptic polynomial of ξ  , b) Pα (ξ  )/P0 (ξ  ) → 0 as |ξ  | → ∞ (ξ  ∈ Rk ) for any nonzero α ∈ N0n−k . Definition 1.3. (see [3]). A polynomial P (ξ) = P (ξ1 , ..., ξn ) is said to be hypoelliptic with respect n to ξ  = (ξ1 , ..., ξk ), k < n, if for any nonzero α ∈ N0n and any sequence {ξ s }∞ s=1 ⊂ R we have D α P (ξ s )/P (ξ s ) → 0 provided that |(ξ s ) | → ∞ as s → ∞. Definition 1.4. (see [4] or [5]). A polynomial P (ξ) = P (ξ1 , ..., ξn ) is said to be regular (non degenerate), if there exists a constant c > 0 to satisfy  |ξ α | ≤ c(|P (ξ)| + 1) ξ ∈ Rn . α∈0 (P )

Remark 1.1. Below we will use the following known results (see [4] - [6]). 1) If the characteristic polyhedron (P ) of a regular polynomial P is completely regular, then P is a hypoelliptic polynomial. 2) If P is a hypoelliptic polynomial, then its characteristic polyhedron (P ) is completely regular. n is 3) Any (n − 1)-dimensional non-coordinate face of the characteristic polyhedron  ⊂ R+ principal.

4) If the characteristic polyhedron (P ) of a collection (P ) ⊂ N0n is completely regular, then any (n − 1)-dimensional non-coordinate face of (P ) is principal. 2. AUXILIARY RESULTS As an immediate consequence of the Definitions 1 - 3 we have the following proposition. Proposition 2.1. Let k, n ∈ N and k < n. If a polynomial P (ξ) = P (ξ1 , ..., ξn ) is hypoelliptic (resp., partially hypoelliptic) with respect to ξ  = (ξ1 , ..., ξk ), then the following assertions hold: 1) For any j : 1 ≤ j ≤ k, the polynomial Qj (ξj ; ξ  ) = P (0, ..., 0, ξj , 0, ..., 0, ξ  ), where ξ  = (ξk+1 , ..., ξn ), is hypoelliptic (resp., partially hypoelliptic) with respect to ξj . 2) For any j : k + 1 ≤ j ≤ n, the polynomial Qj (ξ  ; ξj ) = P (ξ  , 0, ..., 0, ξj , 0, ..., 0) is hypoelliptic (resp., partially hypoelliptic) with respect to ξ  = (ξ1 , ..., ξk ). 3) If a polynomial P is hypoelliptic with respect to ξ  , then it is partially hypoelliptic with respect to ξ  . ˜ ξ  (ξ  ) = P (ξ  ; ξ  ) is hypoelliptic with respect to ξ  , and 4) For any ξ  ∈ Rn−k , the polynomial Q k is completely regular, implying that ˜ ξ  ) ⊂ R+ hence, by Remark 1.1 2), the polyhedron (Q ˜ ξ  )), ˜ ξ  )) = Λ+ ((Q Λ((Q

ξ  ∈ Rn−k .

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Lemma 2.1. (see [8]) Let k, n ∈ N and k < n. If the polynomial P (ξ) = P (ξ1 , ..., ξn ) =



aα ξ α

α∈(P )

is partially hypoelliptic with respect to ξ  = (ξ1 , ..., ξk ), then the following assertions hold: 1) For any α ∈ (P ), α = (α1 , ..., αk ) ∈ N0k , 0 = α = (αk+1 , ..., αn ) ∈ N0n−k , ˜ 0 )\∂(Q ˜ 0 ), α ∈ (Q

˜ 0 (ξ  ) = P (ξ  ; 0 ). Q

˜ ξ  ) = (Q ˜ 0 ). 2) For any ξ  = (ξk+1 , ..., ξn ) ∈ Rn−k , (Q 3) Let λ = (λ1 , ..., λn ) ∈ Λ((P )). If for some j : 1 ≤ j ≤ k we have λj > 0, then there exists an index l : k + 1 ≤ l ≤ n such that λl > 0.  aα ξ α Lemma 2.2. (see [8]) Let k, n ∈ N and k < n. If the polynomial P (ξ) = P (ξ1 , ..., ξn ) = is hypoelliptic with respect to α

ξ α

α∈(P )

= (ξ1 , ..., ξk ), then the following assertions hold: ˜ 0 )\∂(Q ˜ 0 ), where Q ˜ 0 (ξ  ) = P (ξ  ; 0 ). ∈ (Q

= 0, 1. For any α ∈ (P ), ˜ 0 ) for any ξ  ∈ Rn−k . ˜ ξ  ) = (Q 2. (Q 3. For any c > 0 there exists a constant T > 0 to satisfy

|P (ξ)| ≥ c, ∀ξ ∈ Rn , |ξ  | ≥ T.

(2.1)

4. Let λ ∈ Λ((P )). If for some j : 1 ≤ j ≤ k we have λj > 0, then λ ∈ Λ+ ((P )). ˜ 0 )) ⊂ Rk is 5. If λ ∈ Λ((P )) is the normal to the face for which some non-coordinate face (Q + a subface, then λ ∈ Λ ((P )). Corollary 2.1. Under the conditions of Lemma 2.2 the following assertions hold: 1. inf |P (ξ)| → ∞ as |ξ  | → ∞. ξ  ∈Rn−k

2. If k = n − 1, then for any λ = (λ1 , ..., λn ) ∈ Λ((P )) we have λn > 0. Proof. The first assertion immediately follows from Lemma 2.2,3), while the second assertion follows from Lemma 2.2,4) and Remark 1.1,3). Corollary 2.2. A polynomial P (ξ) = P (ξ1 , ..., ξn ) is hypoelliptic with respect to ξ  = (ξ1 , ..., ξk ) (k < n, k, n ∈ N ) if and only if for any a ∈ C the polynomial P + a is hypoelliptic with respect to ξ  . Proof. The result immediately follows from Corollary 2.1,1) and Definition 1.3. Let k, n ∈ N, k < n, and j = 1, ..., k. We denote Π (P ) = {α ∈ N0n−k , ∃α ∈ N0k α ≡ (α ; α ) ∈ (P )}, Πj (P ) = {α ∈ N0n−k , ∃α = (0, ..., 0, αj , 0, ..., 0) ∈ N0k α = (α ; α ) ∈ (P )}, and let (Π (P )) and (Πj (P )) be the characteristic polyhedrons in Rn−k of the collections Π (P ) and Πj (P ) (j = 1, ..., k), respectively. Lemma 2.3. Let k, n ∈ N and k < n. If the polynomial P (ξ) = P (ξ1 , ..., ξn ) =



aα ξ α is hypoel-

α∈(P )

liptic with respect to ξ  = (ξ1 , ..., ξk ), then (Π (P )) = (Πj (P )) for j = 1, ..., k.

Proof. Assume the opposite, that there exists an index j0 : 1 ≤ j0 ≤ k such that (Πj0 (P )) = (Π (P )). Since Πj0 (P ) ⊂ Π (P ), we have (Πj0 (P )) ⊂ (Π (P )), implying that there exists λ ∈ Λ((Π (P ))) to satisfy dΠ (P ) (λ ) =

sup

ν  ∈(Π (P ))

(ν  , λ ) >

sup

ν  ∈(Π j0 (P ))

(ν  , λ ) = dΠj

0

 (P ) (λ )

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≥ 0.

(2.2)

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It follows from the definitions of the sets Π (P ), Πj0 (P ) and formula (2.2) that there exists β ∈ (P ) such  that β  ∈ Π (P )\Πj0 (P ), βj = 0 and (β  , λ ) = dΠ (P ) (λ ). We write the polynomial P in the j=1,j=j0

form of homogeneous polynomials λ = (0 ; λ ), 0 ∈ Rk : P (ξ) =

M 

Pj (ξ) =

j=0

M 



j=0

α∈(P ),(α,λ)=dj

aα ξ α ,

where d0 > ... > dM . From the definitions of the vector λ and the set Π (P ) we have d0 = max (α, λ) = max (α , λ ) = α∈(P )

α∈(P )

max (ν  , λ ) = dΠ (P ) (λ ),

ν  ∈Π (P )

(2.3)

which, together with (2.2), implies d0 = dΠ (P ) (λ ) > 0. Let G(P ) = {α ∈ (P0 ), αj = βj , j = 1, ..., k, j = j0 }, mj0 (0) = max αj0 and α∈G(P )

G0 (P ) = {α ∈ G(P ), αj0 = mj0 (0)} = {α =

(γ  ; α )

∈ G(P )}, where

γ

= (β1 , ..., βj0 −1 , mj0 (0), βj0 +1 , ..., βk ). It follows from the definition of the set G0 (P ) that G0 (P ) = ∅, and hence there exists a point a ∈ R0n−k such that   aα · aα ≡ c1 = 0. (2.4) α∈G0 (P ) 

Now we examine the behavior of the polynomials Pj and Dξγ Pj (ξ) for j = 0, ..., M on the sequence 

n s ε  s λ {ξ s }∞ s=1 ⊂ R , where ξj = 0, j = 1, ..., k, j = j0 , ξj0 = s , (ξ ) = a · s , s = 1, 2, ..., and ε > 0 is an arbitrary number.  It follows from the definitions of the sequence {ξ s }∞ s=1 and the sets Πj0 (P ) (recall that if for α ∈ (P ),  / Πj0 (P ), then ((ξ  )s )α = 0, s = 1, 2, ...) that with some constant c2 > 0 we have for all s = 1, 2, ..., α ∈                aα (ξ s )α  =  aα (ξjs0 )αj0 ((ξ  )s )α  |P (ξ s )| =    α∈(P )  α∈(P ),α ∈Π (P )  j0

       dΠ (P ) (λ )+ε·mj0      aα sεαj0 aα s(α ,λ )  ≤ c2 · s j0 , =      α∈(P ),α ∈Πj0 (P ) where mj0 =

max

α∈(P ),α ∈Π j (P )

(2.5)

αj0 .

0

Next, using the formula (2.1) and the definitions of the polynomial P0 (α ∈ (P0 ) <=> α ∈ (P ), (α, λ) = (α , λ ) = d0 ), the multiindex γ  , the number mj0 (0), the set G0 (P ) and the sequence {ξ s }∞ s=1 for all s = 1, 2, ... we can write         !        α       γ s  s α −γ  s α   α (α ,λ )  d0   ((ξ ) ) aα  ((ξ ) )  =  aα γ ! a s |Dξ  P0 (ξ )| =   = c1 · s .  (α − γ )!       α∈(P ),α ≥γ

α∈G0 (P )

(2.6) Similarly, with some constant c3 > 0 for all j = 1, ..., M and s = 1, 2, ... we obtain 

|Dξγ P0 (ξ s )| ≤ c3 · sdj +ε(mj0 (j)−mj0 (0)) , 

|Dξγ Pj (ξ s )| = 0,

mj0 (j) ≥ mj0 (0),

(2.8)

mj0 (j) < mj0 (0),

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where mj0 (j) =

max

α∈(Pj ),α ≥γ 

αj0 .

In view of (2.2) and (2.4), from estimates (2.5)-(2.8) we infer that for sufficiently small ε > 0 

|

Dξγ P (ξ s ) P (ξ s )

|≥

c1 · sd0 (1 + 0(1)) c2 · s

dΠ

j0

 (P ) (λ )+εmj0

→∞

s → ∞.

as

The last relation contradicts the assumption of the lemma, because γ  = 0 (mj0 (0) > 0), (ξ  )s → ∞ as s → ∞ (ξjs0 = sε , 1 ≤ j0 ≤ k), and the result follows. Lemma 2.3 is proved. Corollary 2.3. Under the conditions of Lemma 2.3 for any 1 ≤ j1 < ... < jl ≤ k, l ≤ k we have (Πj1 ,...,jl (P )) = (Π (P )), where (Πj1 ,...,jl (P )) is the characteristic polyhedron of the collection Πj1 ,...,jl (P ) = {α ∈ N0n−k , ∃α ∈ N0k , αj = 0, j = j1 , ..., jl , α = (α ; α ) ∈ (P )}.

Proof. The result immediately follows from Lemma 2.3, since Πj1 (P ) ⊂ Πj1 ,...,jl (P ) ⊂ Π (P ), implying that (Π (P )) = (Πj1 (P )) ⊂ (Πj1 ,...,jl (P )) ⊂ (Π (P )). Corollary 2.4. Under the conditions of Lemma 2.3 we have ordj P = max αj = ordk+1 Qj ,

j = k + 1, ..., n,

α∈(P )

where Qj ((ξ  , ξj )) = P ((ξ  , 0, ..., 0, ξj , 0...0)),

j = k + 1, ..., n.

Proof. The result immediately follows from Lemma 2.3, since Π (Qj ) = Πj (P ) for j = k + 1, ..., n. 3. SOME PROPERTIES OF THE POLYNOMIALS Lemma 3.1. Let n ∈ N . If the polynomial  aα1 ,β · ξ1α1 η β P ((ξ1 ; η)) = P ((ξ1 , η1 , ..., ηn )) = α=(α1 ,β)∈(P ),β∈N0n n is regular, where is hypoelliptic with respect to ξ1 , then the polyhedron (Π (P )) ⊂ R+

Π (P ) = {β ∈ N0n : there exists α1 ∈ N0

s.t. (α1 , β) ∈ (P )}.

n and an index j : 1 ≤ j ≤ n such Proof. Assume the opposite, that there exist μ ∈ Λ((Π (P ))) ⊂ R+ that μj < 0. We assume for simplicity that (see Remark 1.1,3)):

μ = (μ1 , ..., μl ) ≥ 0

(1 ≤ l < n),

μ = (μl+1 , ..., μn ) < 0.

By dΠ (P ) (μ) and dΠ(Π (P )) (μ) we denote dΠ (P ) (μ) =

max

ν∈(Π (P ))

(ν, μ) =

dΠ(Π (P )) (μ) =

max

β∈0 (Π (P ))

max

(β, μ) = max (β, μ), β∈Π (P )

max (ν, μ),

β∈0 (Π (P )) ν∈Π(β)

n , ν ≤ β} for given β ∈ N n . Since μ ∈ Λ((Π (P ))), there exists γ ∈ where Π(β) ≡ {ν, ν ∈ R+ 0 0    (Π (P )) such that γ = 0 and

(γ, μ) = dΠ (P ) (μ).

(3.1)

Hence, taking into account that μ < 0, we obtain dΠ(Π (P )) (μ) > dΠ (P ) (μ). JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS

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Let A(P, γ  ) := {α = (α1 ; β) ∈ (P ), β  ≥ γ  } = A1 (P, γ  ) ∪ A˜1 (P, γ  ) := {α = (α1 ; β) ∈ A(P, γ  ), (β  , μ ) = dΠ(Π (P )) (μ)} ∪ {α = (α1 ; β) ∈ A(P, γ  ), (β  , μ ) < dΠ(Π (P )) (μ)}, A0 (P, γ  ) = {α = (α1 ; β) ∈ A1 (P, γ  ), β  = γ  },

m1 =

max

(α1 ;β)∈A1 (P,γ  )

α1 ,

A˜0 (P, γ  ) = {α = (m1 ; β) ∈ A0 (P, γ  )}, where for ν ∈ N0n , ν  = (ν1 , ..., νl ) and ν  = (νl+1 , ..., νn ). It follows from the definitions of the multiindex γ and the number m1 that A˜0 (P, γ  ) = ∅. Hence there exists a point a ∈ R0l to satisfy   aα1 ;β · γ  !aβ = c1 = 0. (3.3) ˜0 (P,γ  ) α=(m1 ;β)∈A

It is easy to check that for any α = (α1 , β) ∈ A˜1 (P, γ  ), (β  , μ ) + (β  − γ  , μ ) ≤ d˜Π(Π (P )) (μ),

(3.4)

where dΠ(Π (P )) (μ) > d˜Π(Π (P )) (μ) =

(β  , μ ),

(3.5)

min μj .

(3.6)

max

˜1 (P,γ  ) (α1 ;β)∈A

for any α = (α1 , β) ∈ A1 (P, γ  )\A0 (P, γ  ) (β  , μ ) + (β  − γ  , μ ) ≤ dΠ(Π (P )) (μ) −

l+1≤j≤n



Now we examine the behavior of the ratio Dηγ P ((ξ1 ; η))/P ((ξ1 ; η)) on the sequence {(ξ1s ; η s )}∞ s=1 ⊂ 



Rn+1 , where ξ1s = sε , (η  )s = a · sμ , (η  )s = sμ , s = 1, 2, ..., ε > 0 is an arbitrary number and m01 := ord1 P ≡

max (α1 ;β)∈(P )

≥ m1 .

α1

Taking into account that A(P, γ  ) = A˜0 (P, γ  ) ∪ (A0 (P, γ  )\A˜0 (P, γ  )) ∪ (A1 (P, γ  )\A0 (P, γ  )) ∪A˜1 (P, γ  ) := A˜0 (P, γ  ) ∪ B1 (P, γ  ) ∪ B2 (P, γ  ) ∪ A˜1 (P, γ  ), we can use formulas (3.3)-(3.4), (3.6), the definitions of the sets A˜0 (P, γ  ), B1 (P, γ  ), B2 (P, γ  ), ˜ γ  ) and the numbers m1 , m0 , dΠ (P ) (μ), dΠ(Π (P )) (μ), dΠ(Π (P )) (μ), to write with some constant A(P, 1 c2 > 0 

|Dηγ P ((ξ1s ; η s ))| = | ≥|

 ˜0 (P,γ  ) (α1 ;β)∈A

−| −|

aα1 ;β ·

aα1 ;β ·

β  ! (β  −γ  )!

(α1 ;β)∈A(P,γ  )    γ  ! · aβ · sεα1 +(β ,μ ) |



(α1 ;β)∈B2 (P,γ  )





−|







· aβ · sεα1 +(β ,μ )+(β 

 −γ  ,μ ) 

(α1 ;β)∈B1 (P,γ  )

aα1 ;β ·

β  !

(β  −γ  )!

· a · sεα1 +(β ,μ )+(β

aα1 ;β ·

β  ! (β  −γ  )!

· aβ · sεα1 +(β ,μ )+(β

| 



aα1 ;β · γ  ! · aβ · sεα1 +(β ,μ ) |

β





 −γ  ,μ )

|







 −γ  ,μ )

|

˜1 (P,γ  ) (α1 ;β)∈A ≥ c1 · sεm1 +dΠ(Π (P )) (μ) − c2 · sε(m1 −1)+dΠ(Π (P )) (μ) εm01 +dΠ(Π (P )) (μ)− min μj 0 ˜ l+1≤j≤n −c2 · s − c2 · sεm1 +dΠ(Π (P )) (μ) .

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Therefore, using (3.5) for sufficiently small ε > 0, we have 

|Dηγ P ((ξ1s ; η s ))| ≥ c1 · sεm1 +dΠ(Π (P )) (μ) (1 + 0(1)).

(3.7)

For the polynomial P , with some constant c3 , we have for all s = 1, 2, ...,               εm01 +dΠ (P ) (μ) s s s α1 s β  β  εα1 +(β,μ)    aα1 ;β (ξ1 ) (η )  =  aα1 ;β a s . |P ((ξ1 ; η ))| =   ≤ c3 s  β∈Π (P ),(α1 ;β)∈(P )  (α1 ;β)∈(P ) (3.8) It follows from (3.6), (3.7) and (3.2) that for sufficiently small ε > 0  c1 dΠ(Π (P )) (μ)−dΠ (P ) (μ)+ε(m1 −m01 ) ·s → ∞, as s → ∞. |Dηγ P ((ξ1s ; η s ))|/|P ((ξ1s ; η s ))| ≥ c3 This contradicts the condition of the lemma because γ  = 0 and ξ1s = sε → ∞ as s → ∞, showing that n is regular. Lemma 3.1 is proved. under the condition of lemma the polyhedron (Π (P )) ⊂ R+ Corollary 3.1. Let k < n. If the polynomial P (ξ) = P (ξ1 , ..., ξn ) is hypoelliptic with respect to ξ  = (ξ1 , ..., ξk ), then the polyhedron (Π (P )) is regular. Proof. The result follows from lemma 3.1, if we take into account that by Proposition 2.1 1) the polynomial Q1 (ξ1 ; ξ  ) = P (ξ1 , 0, ..., 0, ξ  ) (ξ  = (ξk+1 , ..., ξn )) is hypoelliptic with respect to ξ1 , and by Lemma 2.3 and Corollary 2.4 we have (Π (P )) = (Π (Q1 )).  Lemma 3.2. Let k, n ∈ N and k < n. If the polynomial P (ξ) = P (ξ1 , ..., ξn ) = α∈(P ) aα ξ α is hyn is regular, where ˜  (P )) ⊂ R+ poelliptic with respect to ξ  = (ξ1 , ..., ξk ), then the polyhedron (P ∪ Π    n   ˜ (P ) ≡ {(0 ; α ) ∈ N , α ∈ Π (P )}. Π 0 ˜  (P ))) and an index j0 : 1 ≤ j0 ≤ n such Proof. Assume the opposite, that there exist λ ∈ Λ((P ∪ Π that λj0 < 0. We write the polynomial P in the form of a sum of λ-homogeneous polynomials P (ξ) =

M 

Pj (ξ) =

j=0

M 



j=0

(α,λ)=dj

aα ξ α ,

where d0 > ... > dM , d0 ≥ 0 (0 ∈ (P )). Let mj0 (j) = max αj0 j = 0, ..., M . Since λ is a normal of the (n − 1)-dimensional non-coordinate α∈(Pj )

˜  (P )) ∩ (P ) to satisfy βj > 0. Hence face, it is easy to see that there exists a multiindex β ∈ 0 (P ∪ Π 0 we have mj0 (0) > 0. Let B = {j : 0 ≤ j ≤ M, mj0 (j) ≥ mj0 (0)}, χ = max{dj − mj0 (j) · λj0 }, j∈B

B1 = {j ∈ B, dj − mj0 (j) · λj0 = χ},

m0j0 = max mj0 (j) j∈B1

(≥ mj0 > 0).

Taking into account that 0 ∈ B and λj0 < 0, we obtain χ ≥ d0 − mj0 (0)λj0 > 0.

(3.9)

Since d0 > ... > dM and λj0 < 0, and hence dj /λj0 < dl /λj0 for all j < l, j, l = 0, ..., M , there exists a unique index j1 : 0 ≤ j1 ≤ M to satisfy mj0 (j1 ) = m0j0 , because otherwise for j, l ∈ B1 , j < l we would have dj − χ dl − χ < = mj0 (l). mj0 (j) = λj0 λj0 This implies {j}M j=0 = {j1 } ∪ {j : 0 ≤ j ≤ M, mj0 (j) < mj0 (j1 )} ∪ {j : 0 ≤ j ≤ M, mj0 (j) > mj0 (j1 )} := {j1 } ∪ G1 ∪ G2 . JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS

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We consider the following two possible cases (under our assumptions). 1. λ ≤ 0. 2. λ ≤ 0, where λ = (λ1 , ..., λk ). Let λ ≤ 0, then there exists l : 1 ≤ l ≤ k such that λl > 0. We examine the behavior of the ratio mj (j1 )

n s λj j = 1, ..., n, j = j and ξ s = Dj0 0 P (ξ)/P (ξ) on the sequence {ξ s }∞ 0 s=1 ⊂ R , where ξj = aj s j0 n−1 λj0 s , s = 1, 2, ..., and the point a(j0 ) = (a1 , ..., aj0 −1 , aj0 +1 , ..., an ) ∈ R0 is chosen to satisfy  aα · aα(j0 ) (j0 ) · mj0 (j1 )! := c1 = 0. α∈(P ),αj0 =mj0 (j1 )

For sufficiently large s with some constant c2 > 0, we have |P (ξ s )| =

M 

|Pj (ξ s )| ≤ c2 · sd0 (1 + 0(1)).

(3.10)

j=0

It follows from the definitions of numbers mj0 (j1 ), χ, and formula (3.9) that for all s = 1, 2, ...  mj (j1 ) aα · aα(j0 ) (j0 ) · mj0 (j1 )! · s(α,λ)−mj0 (j1 )λj0 | |Dj0 0 Pj1 (ξ s )| = | α∈(P ),αj0 =mj0 (j1 )

= c1 · sdj1 −mj0 (j1 )λj0 = c1 · sχ .

(3.11)

Next, for j ∈ G1 from the definitions of numbers mj0 (j) and j1 we have mj0 (j1 )

Dj0

Pj1 (ξ s ) = 0 s = 1, 2, ...

(3.12)

Since λj0 < 0 and mj0 (j) > mj0 (j1 ), for j ∈ G2 with some constant c3 > 0 we can write        αj0 ! mj0 (j1 ) s s αj0 −mj0 (j1 ) s α(j0 )   (ξj0 ) Pj1 (ξ )| =  aα · · (ξ (j0 )) |Dj0  (αj0 − mj0 (j1 ))! α∈(P ),αj =mj (j1 )  0

0

= c3 · sdj −mj0 (j1 )λj0 = 0(sdj −mj0 (j1 )λj0 ) = 0(sχ ).

(3.13)

Since λj0 < 0 and mj0 (0) > 0, and by (3.8), d0 < d0 − mj0 (0) and λj0 ≤ χ, then in view of (3.9) - (3.13) for s → ∞ we have   m (j )  D j0 1 P (ξ s )    j0 =  P (ξ s )  

mj (j1 ) 0 Pj1 (ξ s )|− 0

|Dj



j∈G1 ∪G2 |P (ξ s )|

mj (j1 ) 0 Pj (ξ s )| 0

|Dj

≥

c1 sχ (1+0(1)) c2 sd0 (1+0(1))

→ ∞.

This contradicts the condition of the lemma and shows that the case 1) is impossible. Consider the case 2). We first show that in this case λ = 0, and hence k + 1 ≤ j0 ≤ n, λj0 < 0. Assume the opposite, that there exist l : 1 ≤ l ≤ k such that λl < 0. Since λ is a normal of the ˜  (P )), there exists a multiindex (n − 1)-dimensional non-coordinate face of the polyhedron (P ∪ Π 0  ˜ β ∈  (P ∪ Π (P )) ∩ (P ) such that βl = 0 and (β, λ) = d0 . By the definition of the set Π (P ) we have n−k ˜  (P )) and (β  , λ ) ≤ d0 = (β, λ). . Hence (0 , β  ) ∈ (P ∪ Π β  = (βk+1 , ..., βn ) ∈ (Π (P )) ⊂ R+   Thus, (β , λ ) ≥ 0, which contradicts the assumption λl > 0 because βl > 0, βj ≥ 0 and λj ≤ 0 for j = 1, ..., k, j = l. The obtained contradiction shows that for the case 2) we have λ = 0. Therefore we have (P0 ) = {α ∈ (P ),

(α, λ) = d0 } = {α = (α α ) ∈ (P ),

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Denoting m0 = max |α | and A0 (P ) = {α ∈ (P0 ), |α| = m0 }, we examine the behavior of the ratio

α∈(P0 ) mj0 (0) Dj0 P (ξ)/P (ξ) on the



n s ε  s λ sequence {ξ s }∞ s=1 ⊂ R , where ξj = s , j = 1, ..., k, (ξ ) = a · s , ε > 0

is an arbitrary number to be specified later, and the point a ∈ R0n−k is chosen to satisfy   aα · mj0 (0)!aα (j0 ) (j0 ) ≡ c4 = 0,

(3.14)

α∈A0 (P )

where for b = (bk+1 , ..., bn ) ∈ R0n−k and j0 : k + 1 ≤ j0 ≤ n we put b(j0 ) = (bk+1 , ..., bj0 −1 , bj0 +1 , ..., bn ). Since λ = 0, implying (α , λ ) = dj for any α ∈ (Pj ), j = 0, ..., M , for all s with some constant c5 > 0, we have    M M          s s ε α λ α  |Pj (ξ )| ≤ aα · (s ) · (a · s )  |P (ξ )| ≤   j=0 j=0 α∈(P ),(α ,λ )=dj ≤ c5

M 





sdj +ε·m ≤ c5 · sd0 +ε·m (1 + 0(1)),

(3.15)

j=0

where m = max |α | ≥ m0 . Using the definition of the number mj0 (0) and formula (3.14), for all α∈(P )

s = 1, 2, ... we can write

    mj0 (0) s P0 (ξ )| =  |Dj0 α∈(P0 ),αj =mj 0

0

α (j

aα mj0 (0)!a

0)

(j0 ) · s

ε|α |

s

(α ,λ )−mj0 (0)·λj0

(0)

     

        α(j ) 0 aα · mj0 (0)!a (j0 ) · sd0 −mj0 (0)·λj0 +εm0 ≥   α∈A0 (P )          α(j0 )  − aα mj0 (0)!a (j0 ) sd0 −mj0 (0)λj0 +ε(m0 −1) = c4 sd0 −mj0 (0)·λj0 +εm0 (1 + 0(1)). (3.16) α∈(P0 )\A0 (P )  Similarly, with some constant c6 > 0, for j = 1, ..., M we obtain     mj0 (0) s  Pj (ξ ) ≤ c6 · sdj −mj0 (0)·λj0 +εm . Dj0

(3.17)

Taking into account that dj < d0 , j = 1, ..., k, λj0 < 0 and mj0 (0) > 0, from (3.15)-(3.17) for sufficiently small ε > 0 and s → ∞ we obtain   m (0)  D j0 P (ξ s )    j0 =  P (ξ s )   ≥

c4 ·s

M mj (0) mj (0)  0 P0 (ξ s )|− |Dj 0 Pj (ξ s )| 0 0 j=1

|Dj

|P (ξ s )|

d0 −mj (0)·λj +εm0 d −mj (0)·λj +εm 0 0 0 0 −c6 ·s 1 (1+0(1))  d +εm c5 ·s 0

=

≥

c4 −mj0 (0)·λj0 +ε(m0 −m ) c5 s

→ ∞.

This contradicts the condition of the lemma because (ξ s ) → ∞ as s → ∞, showing that the polyhedron ˜  (P )) is regular. Lemma 3.2 is proved. (P ∪ Π ˜  (P ))) we have λ = Lemma 3.3. Under the conditions of Lemma 3.2 for any λ ∈ Λ((P ∪ Π (λk+1 , ..., λn ) > 0. ˜  (P ))) we have λ ≥ 0. Assume the opposite, that there Proof. By Lemma 3.2, for any λ ∈ Λ((P ∪ Π  ˜ (P ))) and an index j0 : k + 1 ≤ j0 ≤ n (without loss of generality, we can take exist λ ∈ Λ((P ∪ Π j0 = k + 1) such that λk+1 = 0. JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS

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Using the arguments similar to the one used in the proof of Lemma 2.2 4), and taking into account that λk+1 = 0, we obtain λ = (λ1 , ..., λk ) = 0. Since λ is a normal of the (n − 1)-dimensional non˜  (P ))), there exists a multiindex j1 : k + 2 ≤ j1 ≤ n coordinate face of the polyhedron λ ∈ Λ((P ∪ Π such that λj1 > 0. For simplicity we assume that λj = 0, j = 1, ..., k + l, 1 ≤ l < n − k and λj > 0, j = k + l + 1, ..., n. We write the polynomial P in the form of a sum of λ-homogeneous polynomials: P (ξ) =

M 

Pj (ξ) =

j=0

M 



j=0

α∈(P ),(α,λ)=dj

aα · ξ α ,

where d0 > ... > dM . By the assumption λj = 0, j = 1, ..., k + l we have ⎧ ⎫ n ⎨ ⎬  αj λj = d0 = {α ∈ (P ), (α , λ ) = d0 }. (P0 ) = {α ∈ (P ), (α, λ) = d0 } = α ∈ (P ) : ⎩ ⎭ j=k+l+1

˜  (P )), Since λ is a normal of the (n − 1)-dimensional non-coordinate face of the polyhedron (P ∪ Π there exists a multiindex β ∈ (P0 ) such that βk+1 = 0. Denote mk+1 (0) = max αk+1 , α∈(P0 )

m = max |α |, α∈A(P0 )

A(P0 ) = {α ∈ (P0 ), αk+1 = mk+1 (0)}, A1 (P0 ) = {α ∈ A(P0 ), |α | = m }.

By definition of the set A1 (P0 ) we have A1 (P0 ) = ∅, hence there exists a point b ∈ R0n−k−1 such that  aα · mk+1 (0)!bα(k+1) = c1 = 0, (3.18) α∈A1 (P0 ) m

(0)

k+1 P (ξ)/P (ξ) on the where α(k + 1) = (αk+2 , ..., αn ). We examine the behavior of the ratio Dk+1 ∞ s ε s s sequence {ξs=1 }, where ξj = aj · s , j = 1, ..., k, aj ∈ R, j = 1, ..., k, ξk+1 = 0, ξj = bj · sλj , j = k + 2, ..., n, and ε > 0 is an arbitrary number. By the form of the vector λ and the definition of the number m with some constant c2 > 0 for all s = 1, 2, ... we have

|P (ξ s )| ≤ |P0 (ξ s )| +

M 



|Pj (ξ s )| ≤ c2 · sd0 +ε·m + c2

j=1

where

m ˜

= max α∈(P )

|α |.

M 





˜ sdj +ε·m = c2 · sd0 +ε·m (1 + 0(1)).

(3.19)

j=1

From the definition of the number mk+1 (0) and the form of λ (λk+1 = 0), in view

of (3.18) for sufficiently large s with some constant we can write        mk+1 (0) s α(k+1) εα (α ,λ )−αk+1 ·λk+1   P0 (ξ )| =  aα · mk+1 (0)!b ·s ·s |Dk+1  α∈A(P0 )              d +εm   α(k+1) d0 +ε|α |  α(k+1)  0   aα · mk+1 (0)b ·s a · m (0)b ≥ = α k+1   ·s  α∈A1 (P0 )  α∈A(P0 ) −











|aα · mk+1 (0)bα(k+1) | · sd0 +ε|α | ≥ c1 · sd0 +εm − c3 · sd0 +ε(m −1) = c1 · sd0 +εm (1 + 0(1))

α∈A(P0 )\A1 (P0 )

(3.20) Similarly, with some constant c4 > 0, for j = 1, ..., M and all s = 1, 2, ... we obtain     mk+1 (0) s Pj (ξ )| =  aα |Dk+1 α∈(Pj ),αk+1 ≥mk+1 (0) JOURNAL OF CONTEMPORARY MATHEMATICAL ANALYSIS Vol. 49

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  αk+1 ! ε|α | (α ,λ )−(αk+1 −mk+1 (0))·λk+1  ˜ ·s · ·s ≤ c3 · sdj +εm  (αk+1 − mk+1 (0))!

(3.21)

Taking into account that dj < d0 , j = 1, ..., M , from (3.19)-(3.21) for sufficiently small ε > 0, we obtain M  mk+1 (0)  |Dmk+1 (0) P0 (ξ s )| −  m (0) |Dk+1 Pj (ξ s )|  k+1  D k+1 P (ξ s )  c1 sd0 +εm (1 + 0(1)) c1 j=1   k+1 ≥ ≥ ≥ (1 + 0(1)).    s s d +εm   P (ξ ) |P (ξ )| c2 c2 s 0

The obtained contradiction completes the proof of Lemma 3.3. k such Lemma 3.4. Under the conditions of Lemma 3.2 for all ν  ∈ Π (P ) for which there is 0 = ν  ∈ R+    ˜ (P )) the polyhedron that (ν ; ν ) ∈ (P ∪ Π k ˜  (P ))} ⊂ Rk , (β  ; ν  ) ∈ (P ∪ Π (P, ν  ) ≡ {β  ∈ R+ +

is completely regular. ˜  Proof. Let {λj }m j=1 = Λ((P ∪ Π (P ))), m ∈ N. Then n ˜  (P )) = {ν ∈ R+ , (ν, λj ) ≤ d(λj ), j = 1, ..., m}, (P ∪ Π

where d(λj ) =

max

˜  (P )) ν∈(P ∪Π

(ν, λj ), j = 1, ..., m.

By Lemmas 2.2,5) and 3.2 there exists r ∈ N, r ≤ m, such that the sequence (possibly after enumerj j j j j  j  j ation) {λj }m j=1 satisfies λ > 0, j = 1, ..., r and (λ ) = (λ1 , ..., λk ) = 0, (λ ) = (λk+1 , ..., λn ) > 0 for j = r + 1, ..., m. Taking into account that k , (ν  , (λ )j ) ≤ d(λj ) − (ν  , (λ )j ) j = 1, ..., m}, (P, ν  ) = {β  ∈ R+

˜  (P )), ν  = 0 of lemma implies (ν  , (λ )j ) < d(λj ), j = and that the assumption (ν  , ν  ) ∈ (P ∪ Π   j j   j 1, ..., r, (ν , (λ ) ) ≤ d(λ ), (ν , (λ ) ) = 0, j = r + 1, ..., m, we can write k , (ν  , (λ )j ) ≤ d(λj ) − (ν  , (λ )j ) j = 1, ..., r}. (P, ν  ) = {β  ∈ R+

Since (λ )j > 0 for j = 1, ..., r, from the last formula we obtain Λ((P, ν  )) ⊂ {(λ )j /|(λ )j |}rj=1 , k is a completely regular polyhedron. Lemma 3.4 is proved. showing that (P, ν  ) ∈ R+ Corollary 3.2. If under the condition of Lemma 3.2 for α = (α ; α ) ∈ (P ), αj = 0, 1 ≤ j ≤ k, then α − ej ∈ (P, α )\∂(P, α ), where ej = (0, ..., 0, 1, 0, ..., 0) is the unit vector of j-th direction in Rk . Proof. The result is an immediate consequence of the fact that (P, α ) is a completely regular polyhedron (see Lemma 3.4). 4. THE MAIN RESULT Let k < n be given numbers, (P ) ⊂ N0n be a given collection and let the sets Π (P ), Πj (P ), j = 1, ..., k, be defined by (P ) and k as in §2. We denote ˜  (P )), G(P ) = 0 (P ) ∩ 0 (P ∪ Π ˜  (P ) is defined by Π (P ) as in т §3. It is clear that G(P ) = 0 (P ) if and only if (P ) is where the set Π a regular polyhedron. Lemma 4.1. Let k, n ∈ N, k < n, and let P (ξ) = P (ξ1 , ..., ξn ) be a polynomial such that with some constant c1 > 0  |ξ α | ≤ c1 (|P (ξ)| + 1) ∀ξ ∈ Rn . (4.1) α∈G(P )

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If (Πj (P )) = (Π (P )) for any j = 1, ..., k, then there exists a constant c2 > 0 such that  |ξ α | ≤ c2 (|P (ξ)| + 1) ∀ξ ∈ Rn , |ξ  | ≥ 1.

(4.2)

˜  (P )) α∈0 (P ∪Π

˜  (P ) = {(0 ; α ) ∈ N n , Proof. Taking into account that for any A ⊂ N0n we have 0 (A) ⊂ A and Π 0 ˜  (P )) = G(P ) ∪ {α ∈ 0 (Π (P ))}. Therefore β  = ∃α ∈ N0k , (α ; α ) ∈ (P )}, we can write 0 (P ∪ Π ˜  (P ))\G(P ). 0 for any β ∈ 0 (P ∪ Π ˜  (P ))\G(P ) and Since by assumption (Πj (P )) = (Π (P )) for j = 1, ..., k, for any β ∈ 0 (P ∪ Π any index j : 1 ≤ j ≤ k there is a multiindex αj (β  ) = (0, ..., 0, αj (β  ), 0, ..., 0, β  ), αj (β  ) = 0 such that (αj (β  ), β  ) ∈ 0 (Qj ) (see Corollary 2.4), where the polynomial Qj is determined by P and ˜  (P )), and hence αj (β  ) ∈ G(P ) j : 1 ≤ j ≤ k as in Proposition 2.1 1). This implies αj (β  ) ∈ 0 (P ∪ Π  (αj (β ) = 0), j = 1, ..., k. In view of (4.1), for any ξ ∈ Rn with |ξ  | ≥ 1 and with some constants c3 , c4 > 0, we can write    |ξ β | = |ξ β | + |ξ β | = ˜  (P )) β∈0 (P ∪Π

=



|ξ β | +

β∈G(P )

=

 β∈G(P )





β  ∈0 (Π (P ))

|ξ β | + c3

˜  (P ))\G(P ) β∈0 (P ∪Π

β∈G(P )

|(ξ  )β | ≤



|ξ β | + c3

β∈G(P )



k 

β  ∈0 (Π (P ))

j=1

j (β)

|ξ α



k 

β  ∈0 (Π (P ))

j=1



| ≤ c4



α (β  )

|(ξ  )|β |ξj j

|=

|ξ α | ≤ c4 · c1 (|P (ξ)| + 1).

α∈G(P )

Lemma 4.1 is proved. Remark 4.1. (see [4], [5]) 1) If Q(ξ) = Q(ξ1 , ..., ξn ) is a regular polynomial, then ξ α /Q(ξ) → 0 as |ξ| → ∞ for any α ∈ (Q)\∂(Q), and there exists a constant c > 0 such that  |ξ α | ≤ c(|Q(ξ)| + 1), ξ ∈ Rn . α∈(Q)

2) If the characteristic polyhedron of a regular polynomial is regular, then Q(ξ) → ∞ as |ξ| → ∞. Theorem 4.1. Let k < n, and let P (ξ) = P (ξ1 , ..., ξn ) be a polynomial such that 1. (Π (P )) = (Πj (P )) for any 1 ≤ j ≤ k, 2. the inequality (4.1) is satisfied with some constant c1 > 0, ˜  (P )) is regular, 3. (P ∪ Π ˜  (P )) = ∅ and λ = (λk+1 , ..., λn ) > 0 for any λ ∈ Λ((P ∪ Π ˜  (P ))), 4. Λ+ (P ∪ Π 5. for any α = (α ; α ) ∈ (P ) with α = 0 the polyhedron (P, α ) is completely regular. Then the polynomial P is hypoelliptic with respect to ξ  = (ξ1 , ..., ξk ). n Proof. It is enough to show (see [7]) that for any sequence {ξ}∞ s=1 ⊂ R n 

|Dj P (ξ s )|/|P (ξ s )| → 0 as

s→∞

(4.3)

j=1

as long as |(ξ  )s | → ∞ for s → ∞. By Lemma 4.1 and Remark 4.1,1) there exists a constant c2 > 0 such that  |ξ α | ≤ c2 (|P (ξ)| + 1), ξ ∈ Rn , |ξ  | ≥ 1. (4.4) hP (ξ) = ˜  (P ))∩N n α∈(P ∪Π 0

It follows from Remark 4.1,2), the condition 3) of the theorem and formula (4.4) that there exist constants c3 > 0 and t > 0 such that hP (ξ) ≤ c4 |P (ξ)|,

ξ ∈ Rn ,

|ξ  | ≥ t.

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It follows from the definition of function hP and formula (4.5) that to prove (4.3) it is enough to show that for any α ∈ (P ) and j : 1 ≤ j ≤ n with αj ≥ 1 j

(ξ s )α−e /hP (ξ s ) → 0.

(4.6)

as s → ∞, where ej = (0, ..., 0, 1, 0, ..., 0) is the unit vector of the j-th direction in Rn . If k + 1 ≤ j ≤ n, then by the condition 4) of the theorem we have ˜  (P ))\∂(P ∪ Π ˜  (P )), α − ej ∈ (P ∪ Π and for such values of α the relation (4.6) follows from Remark 4.1,1). Now let α = (α ; α ) ∈ (P ) and αj ≥ 1, 1 ≤ j ≤ k. Then by the condition 5) of the theorem and k is completely regular, and hence by Lemma 3.4 we conclude that the polyhedron (P, α ) ⊂ R+ Corollary 3.2 we have α − (e )j ≡ (α1 , ..., αj−1 , αj − 1, αj+1 , ..., αk ) ∈ (P, α )\∂(P, α ). Denote   |ξ  |β , hP (ξ  , α ) = β  ∈0 (P,α )

Using Remark 4.1,1) and taking into account that |(ξ  )s | → ∞ as s → ∞, we obtain 

 j

|((ξ  )s )α −|e | |/hP ((ξ  )s , α ) → ∞,

as s → ∞. α

Finally, it is easy to see that with some constant c3 > 0, |(ξ  ) | · hP (ξ  , α ) ≤ c3 hP (ξ), ξ ∈ Rn , and hence, taking into account that hP (ξ s ) → ∞ as s → ∞, we obtain     (ξ s )α−ej  α|((ξ  )s )α | · o(hP ((ξ  )s , α ))   → 0, as s → ∞. ≤    hP (ξ s )  |((ξ  )s )α | · hP ((ξ  )s , α ) + hP (ξ s ) Theorem 4.1 is proved. Theorem 4.2. Let k < n and let P (ξ) = P (ξ1 , ..., ξn ) be a polynomial satisfying (4.1). Then P is hypoelliptic with respect to ξ  = (ξ1 , ..., ξk ) if and only if the conditions 1), 3)-5) of Theorem 4.1 are fulfilled. The result is an immediate consequence of Lemmas 3.1-3.4 and Theorem 4.1. REFERENCES ¨ 1. L. Hormander, The Analysis of Linear Partial Differential Operators, vol. 2 (Springer-Verlag, 1983). ¨ 2. L. Gording, B. Malgrange, “Operateurs differentiels partillement hypoelliptiques”, Rend. Acad. Sci. Paris, 247 (23), 2083-2085, 1958. 3. R. J. Elliot, “Almost hypoelliptic differential operators”, Proceed of the London Math. Society, 53-19 (3), 537552, 1969. 4. V. P. Mikhailov, “On the behavior at infinity of a class of polynomials”, Trudi MIAN SSSR, 91, 59-81, 1967. 5. S. Gindikin, L. R. Volevich, The Method of Newton’s Polyhedron in Theory of Partial Differential Equations (Kluwer Academic Publishers, London, 1992). 6. S. M. Nikolskii, “The first boundary value problem for a general linear equation”, DAN SSSR, 144 (4), 767769, 1962. 7. H. G. Ghazaryan, V. N. Margaryan, “The support of hypoellipticity for linear differential operators”, Izv. AN. Arm. SSR, 21 (5), 453-470, 1986. 8. V. N. Margaryan, S. R. Hayrapetyan, “On some properties of polynomials hypoelliptic with respect to a group of variables”, Vestnik RAU, 1, 16-26, 2013.

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