V. P. MIKHAILOV

ON THE BEHAVIOR AT INFINITY OF CERTAIN CLASSES OF POLYNOMIALS

(Presented by Academician I. G. Petrovskii, February 11, 1965)

Let

\[ A(x,\xi)=\sum_{\alpha} a_{\alpha}(x)\xi^{\alpha} \tag{1} \]

be a polynomial with respect to the vector \(\xi=(\xi_1,\ldots,\xi_n)\), with continuous complex coefficients \(a_{\alpha}(x)\) depending on the parameter \(x=(x_1,\ldots,x_n)\in \overline Q\), where \(Q\) is a certain bounded domain of \(m\)-dimensional space, \(\alpha=(\alpha_1,\ldots,\alpha_n)\) is an integer vector, and \(\xi^\alpha=\xi_1^{\alpha_1}\cdots \xi_n^{\alpha_n}\). We shall assume that \(a_0(x)=A(x,0)\ne 0\) for all \(x\in\overline Q\); it will be clear from what follows that this assumption does not diminish generality. In addition, in all subsequent considerations the vector \(\xi\) will be assumed real.

Certain problems of the general theory of partial differential equations are closely connected with the study of the behavior of a general polynomial of the form (1) for real \(\xi\to\infty\). One such problem will be considered below (in Sec. V).

Definition 1. A polynomial \(\widetilde A(x,\xi)\) will be called the principal term of the polynomial \(A(x,\xi)\) if: 1) for all \(x\in\overline Q\),

\[ A(x,\xi)=\widetilde A(x,\xi)(1+o(1)) \]

as \(\xi\to\infty\); 2) the polynomial \(\widetilde A(x,\xi)\) is the sum of a minimal number of monomials of the polynomial \(A(x,\xi)\).

It can be proved that every polynomial \(A(x,\xi)\) has a unique principal term. The study of the behavior at infinity of the principal term \(\widetilde A\) is, generally speaking, simpler than the analogous study of the polynomial \(A\) itself; it is therefore natural to single out those classes of polynomials (1) for which there exists a sufficiently simple algorithm for extracting the principal terms.

I. Let us first consider the polynomial \(A(x,\xi)\) (1) with real coefficients \(a_{\alpha}(x)\). Let \(\mathfrak C(x)\) be the set of integer points of the space \(\{\alpha\}\) lying in the region \(\alpha_1\ge 0,\ldots,\alpha_n\ge 0\) and such that \(a_{\alpha}(x)\ne 0\), and let \(\mathfrak N(x)\equiv\mathfrak N^n(x)\) be the minimal convex polyhedron containing the set \(\mathfrak C(x)\). According to our assumption, the origin \(O\in\mathfrak N(x)\). Suppose, moreover, that \(\mathfrak N(x)\equiv\mathfrak N\) for all \(x\in Q\). (For this condition to hold it is necessary and sufficient that \(a_{e^k(x)}(x)\ne 0\) for all \(x\in Q\) and for all vertices \(e^k(x)\) of the polyhedron \(\mathfrak N(x)\).) The open faces of dimension \(k\), \(0\le k\le n-1\), will be denoted by \(\mathfrak N_i^k\), \(i=1,\ldots,N_k\), where \(N_k\) is the number of such faces. The vertices of the polyhedron \(\mathfrak N\), i.e. the faces \(\mathfrak N_i^{(0)}\), \(i=1,\ldots,N_0\), of zero dimension, will for subsequent convenience be denoted by \(e^i=\mathfrak N_i^{(0)}\), \(i=1,\ldots,N_0\). The surface of the polyhedron \(\mathfrak N\) will be denoted by \(\mathfrak N^{n-1}\):

\[ \mathfrak N^{n-1}=\bigcup_{k=0}^{n-1}\bigcup_{i=1}^{N_k}\overline{\mathfrak N_i^k}, \]

where \(\overline{\mathfrak N_i^k}\) is the closure of the face \(\mathfrak N_i^k\).

Let \(m^i=(m_1^i,\ldots,m_n^i)\) be the vector of the normal to the face \(\mathfrak N_i^{\,n-1}\), outward with respect to the polyhedron \(\mathfrak N\).

Definition 2. A face \(\mathfrak N_i^{\,n-1}\) of the polyhedron \(\mathfrak N\) will be called principal if among the coordinates \(m_s^i\), \(s=1,\ldots,n\), of its outward normal at least one is positive. A point \(a\in \mathfrak N^{\,n-1}\) will be called principal if \(a\) is a limit point for at least one principal \((n-1)\)-dimensional face. A face \(\mathfrak N_i^k\), for \(k=0,\ldots,n-2\), will be called principal if it consists of principal points of \(\mathfrak N^{\,n-1}\).

Consider, for arbitrary \(k\) and \(i\), \(k=0,\ldots,n-1;\ i=1,\ldots,N_k\), the polynomial

\[ A_{\mathfrak N_i^k}(\xi)=\sum_{a\in \mathfrak N_i^k} a_a(x)\xi^a . \]

Definition 3. A face \(\mathfrak N_i^k\), \(i=1,\ldots,N_k,\ k=0,\ldots,n-1\), will be called nondegenerate if on the real sphere \(|\xi|=1\) the polynomial \(A_{\mathfrak N_i^k}(x,\xi)\) can vanish only at points of intersection of this sphere with the coordinate planes.

Definition 4. The polynomial \(A(x,\xi)\) (1) is called nondegenerate if all its principal faces \(\mathfrak N_i^k\) are nondegenerate.

The polynomials \(A_{\mathfrak N_i^k}(x,\xi)\) are generalized homogeneous and become homogeneous after the substitution \(\xi=\zeta^{m^l}\) \((\xi_1=\zeta_1^{m_1^l},\ldots,\xi_n=\zeta_n^{m_n^l})\), where \(m^l\) is the vector of the outward normal of one of those faces \(\mathfrak N_l^{\,n-1}\) for which the face \(\mathfrak N_i^k\) is limiting. (We note that if any of the numbers \(m_i^l=0\), then the corresponding variables \(\xi_i\) should be put equal to arbitrary constant numbers.) Therefore the verification of nondegeneracy of the polynomial \(A_{\mathfrak N_i^k}(x,\xi)\) reduces to checking the “generalized ellipticity” of the polynomials \(A_{\mathfrak N_i^k}(x,\xi)\) for all principal faces \(\mathfrak N_i^k\) of the polyhedron.

Definition 5. The polynomial \(A(x,\xi)\) is called complete if the vertex \(O=(0,\ldots,0)\) of its polyhedron \(\mathfrak N\) is not principal.

If the polynomial \(A(x,\xi)\) is complete, then one can show that its polyhedron \(\mathfrak N\) is \(n\)-dimensional, its only nonprincipal vertex is the vertex \(O\), on each coordinate axis of the space \(\{a\}\) the polyhedron \(\mathfrak N\) has one principal vertex, and all faces of the polyhedron \(\mathfrak N\) which are not principal lie in the coordinate planes.

Theorem 1. If the polynomial \(A(x,\xi)\) is complete and nondegenerate, then \(\lim_{\xi\to\infty}|A(x,\xi)|=\infty\). Moreover \(\lim_{\xi\to\infty} A(x,\xi)=(\operatorname{sign} a_{e^k}(x))\infty\), where \(e^k\) is any principal vertex of the polyhedron \(\mathfrak N\).

Theorem 2. If the polynomial \(A(x,\xi)\) is complete and nondegenerate, and \(\mathfrak N(x)\equiv \mathfrak N\) for all \(x\in Q\), then its principal part

\[ \tilde A(x,\xi)=\sum_{\alpha\in \mathfrak N_{\mathrm{pr}}^{\,n-1}} a_\alpha(x)\xi^\alpha , \tag{2} \]

where \(\mathfrak N_{\mathrm{pr}}^{\,n-1}\) is the aggregate of all principal points of the surface \(\mathfrak N^{\,n-1}\) of the polyhedron \(\mathfrak N\).

Together with the concept of the principal part of the polynomial \(A(x,\xi)\), it is convenient also to introduce the somewhat cruder concept of a majorizing polynomial.

Definition 6. The polynomial \(\hat A(x,\xi)\) is called a majorizing polynomial for the polynomial \(A(x,\xi)\), if there exist

such constants \(\sigma \geqslant 0\) and \(\gamma > 0\) that the inequalities

\[ \gamma^{-1}\widetilde A(x,\xi) \leqslant A(x,\xi)+\sigma \leqslant \gamma \widetilde A(x,\xi) \]

are satisfied for all real \(\xi\) and \(x \in Q\).

It is clear that the principal part \(\overline A(x,\xi)\) of the polynomial \(A(x,\xi)\) is at the same time one of its majorizing polynomials.

Theorem 3. If the polynomial \(A(x,\xi)\) is complete and nondegenerate, and \(\mathfrak N(x) \equiv \mathfrak N\) for all \(x \in Q\), then a majorizing polynomial for it is the polynomial

\[ \widetilde A(\xi)=\xi^{e_1}+\cdots+\xi^{e_{N_0}} . \tag{3} \]

II. Let us now consider the general case of complex coefficients \(a_\alpha(x)\). The construction of the set \(\mathfrak C(x)\) and of the polyhedron \(\mathfrak N(x)\) is carried out in the same way as in the preceding point. We shall assume that \(\mathfrak N(x) \equiv \mathfrak N\) for all \(x \in Q\). All definitions of the preceding point are also retained.

Theorem 4. If the polyhedron \(\mathfrak N(x)\) of the polynomial \(A(x,\xi)\) with complex coefficients does not depend on \(x \in Q\), and the polynomial \(A(x,\xi)\) is complete and nondegenerate, then

\[ \lim_{\xi\to\infty} A(x,\xi)=\infty . \]

The principal part in the polynomial \(A(x,\xi)\) is the polynomial (2), and as a majorizing polynomial for the polynomial \(|A(x,\xi)|^2\) one may take the polynomial \((\widetilde A(\xi))^2\), where \(\widetilde A\) is defined by formula (3).

III. Definition 7 \((^1)\). A polynomial \(A(\xi)\) with complex coefficients is called hypoelliptic if

\[ \lim_{\xi\to\infty}(A(\xi))^{-1} = \lim_{\xi\to\infty}\frac{|\operatorname{grad} A(\xi)|}{A(\xi)} =0 . \]

Theorem 5. In order that a complete nondegenerate polynomial \(A(\xi)\) be hypoelliptic, it is necessary and sufficient that the outward normals of all principal \((n-1)\)-dimensional faces of the polyhedron \(\mathfrak N\) have only positive coordinates.

From Theorem 5 there follows a criterion for hypoellipticity of a differential operator \(A(-iD)\) with complex coefficients, if the characteristic polynomial of this operator \(A(\xi)\) is complete and nondegenerate.

IV. Let

\[ \mathscr P(x,D)u \equiv \sum_{\alpha} p_\alpha(x)D^\alpha u = f(x) \tag{4} \]

be a linear differential equation with sufficiently smooth complex coefficients \(p_\alpha(x)\), where \(x=(x_1,\ldots,x_n)\in \overline Q\); \(Q\) is some bounded domain; \(D=(D_1,\ldots,D_n)\). Suppose that the characteristic polynomial for the operator \(\mathscr P(x,D)\) has the form

\[ \mathscr P(x,i\xi)\equiv \operatorname{Re}\mathscr P(x,i\xi)+i\operatorname{Im}\mathscr P(x,i\xi) = A(x,\xi)+iB(x,\xi), \tag{5} \]

\[ A(x,\xi)=\sum_{\alpha}a_\alpha(x)\xi^\alpha,\qquad B(x,\xi)=\sum_{\alpha}b_\alpha(x)\xi^\alpha, \]

where the polynomial \(A(x,\xi)\) is a complete nondegenerate polynomial with real coefficients. Together with the polyhedron \(\mathfrak N(x)\equiv \mathfrak N\) for the polynomial \(A(x,\xi)\), consider the convex polyhedron \(\mathfrak M=(\bigcup R_\alpha)\cup \mathfrak N\), where

\(R_\alpha\) is the set of those points \(\beta=(\beta_1,\ldots,\beta_n)\) for which \(0\leq \beta_i\leq \alpha_i\), \(i=1,\ldots,n\). It turns out that all vertices \(f^i=(f_1^i,\ldots,f_n^i)\) of the polyhedron \(\mathfrak M\) have even coordinates (\(f_j^i\) are even numbers). Introduce on the set of functions \(u(x)\in C_0^\infty(\bar Q)\) the scalar product

\[ (u,v)=\int_Q \sum D^{f^i/2}u\cdot D^{f^i/2}\bar v\,dx . \]

The Hilbert space obtained by completing \(C_0^\infty(Q)\) with respect to this scalar product will be denoted by \(\mathfrak H(Q)\).

Definition 8. A function \(u(x)\in \mathfrak H(Q)\) will be called a solution of the first boundary-value problem in the domain \(Q\) for equation (4), with \(f(x)\in \mathscr L_2(Q)\), if

\[ (u,\mathcal P^*(x,D)v)=(f,v) \]

for every \(v\in C_0^\infty(Q)\). (Here \(\mathcal P^*(x,D)\) is the operator formally adjoint to the operator \(\mathcal P(x,D)\).)

Theorem 6. The problem of finding a solution of the first boundary-value problem for equation (4) is Fredholm if the polynomial \(B(x,\xi)\) (5) is subordinate to the polynomial \(A(x,\xi)\). Under the same assumption, for the operator \(\mathcal P(x,D)+\lambda\), for sufficiently large \(\lambda\), the first boundary-value problem is uniquely solvable.

Here by subordination of the polynomial \(B(x,\xi)\) to the polynomial \(A(x,\xi)\) we mean the following. Suppose first that the coefficients \(p_\alpha(x)\) are complex. Then every point \(\alpha\) for which \(p_\alpha(x)\neq 0\) in \(\bar Q\) must be either an interior point of \(\mathfrak M\), or such a boundary point of \(\mathfrak M\) that lies on no principal face.

If, however, the coefficients \(\{p_\alpha(x)\}\) are real, then the subordination condition is weaker: every \(\alpha\) for which \(p_\alpha(x)\neq 0\) in \(\bar Q\): a) either lies inside \(\mathfrak M\); b) or, lying on the boundary of \(\mathfrak M\), lies on none of its principal faces; c) or lies on the boundary of \(\mathfrak M\), or even outside \(\mathfrak M\), but \(\alpha-i_k\) (if \(\alpha-i_k\geq 0\)) for any \(k=1,\ldots,n\) lies either inside \(\mathfrak M\), or on the boundary of \(\mathfrak M\), but not on a principal face (here \(i_k\) is the direction vector of the \(k\)-th coordinate axis, and the inequality \(\beta\geq 0\) means that \(\beta_i\geq 0\), \(i=1,\ldots,n\)).

V. The first boundary-value problem can also be considered for a “parabolic” equation constructed with the aid of the “elliptic” operator in (4), in the same way as in Section IV of paper \((^2)\). In this case Theorem 4 from \((^2)\) remains valid as well.

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
22 I 1965

REFERENCES CITED

\(^{1}\) L. Hörmander, Comm. Pure and Appl. Math., 11, 197 (1958).
\(^{2}\) V. P. Mikhailov, DAN, 151, No. 2, 282 (1963).