A Basis for Polynomial Solutions to Systems of Linear Constant Coefficient PDE's

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1 advances in mathematics 7, 5763 (996) article no A Basis for Polynomial Solutions to Systems of Linear Constant Coefficient PDE's Paul S. Pedersen Los Alamos National Laboratory, Mail Stop B265, Los Alamos, New Mexico Received January, 992 Let K represent either the real or the complex numbers. Let P k, k=, 2,..., r be constant coefficient (with coefficients from K) polynomials in n variables and let N M =[u(x)#k[x] M P k (x,..., x n ) u(x)=0; k=, 2,..., r] be the set of all polynomial solutions (of degree M) to this system of partial differential equations. We solve the problem of finding an easily computed basis for the vector space N M. To do this we use a certain associative, and commutative algebra (defined over K), namely K[;]=K[;, ; 2,..., ; n ] where [P k (;)=0 k=,..., r] and [; m ;m 2 2 }}};m n =0 m n +}}}+m n =M+]. Let the vector space K[;] M equal the span over K of [; m ;m 2 2 }}};m n n m +}}}+m n M]. We show how the expression M j=0 (x ; +}}}+x n ; n ) j j! can be used to find an easily computed basis for N M. 996 Academic Press, Inc. Introduction Let P k, k=, 2,..., r be polynomials in n variables (with coefficients from K) and let K[x] M represent the set of all polynomials of degree less than or equal M in x=(x, x 2,..., x n ) (throughout this paper K will represent either the complex or the real numbers). We will find an easily computed basis for the vector space N M = {u # K[x] M } P k\ x,..., x n+ u=0; k=, 2,..., r =. () As the introduction in [] points out, the problem of finding a basis for the vector space () is both natural and basic. [] also points out that little has been done in addressing the general problem. In that same paper a method is discussed for finding a basis for N M when [w # C n P k (w), k=,..., r] is smooth and irreducible. In [2] and [3] a method was Copyright 996 by Academic Press, Inc. All rights of reproduction in any form reserved.

2 58 paul s. pedersen developed for finding a basis in a large number of cases where the P k are of homogeneous order. In discussing a closely related question, Stiller and, in earlier work, Michelli established various facts regarding the dimension of N M when the P k are also of homogeneous order (see [4]). There have also been numerous publications discussing polynomial solutions for special cases of (). For example, much has been written about polynomial solutions to Laplace's equation in several variables (see any book discussing spherical harmonics; see e.g., []). The method presented in this paper extends the method appearing in [2] and [3] and is distinct from the methods presented by other authors. Furthermore, our method is simple and will give a basis for all cases of (). It is well know that the real and imaginary parts of polynomials in z=x +x 2 i form a basis for the set of all polynomial solutions to Laplace's equation ( 2 x 2 +2 x 2 2 ) u(x, x 2 )=0. We will be guided by the similarity between the algebraic relationship 2 +i 2 =0 (which relates the elements which form a basis for the complex number system) and the form of the 2 dimensional Laplacean operator. Specifically, to find all polynomial solutions in (), we use a finitely generated, commutative, and associative algebra K[;]=K[;, ; 2,..., ; n ] whose generators satisfy the system of algebraic equations and [P k (;)=0 k=, 2,..., r] [; m ;m 2 2 }}};m n n =0 m +}}}+m n =M+] (2) We show how the expression M j= (n i= x i ; i ) j j! can be used to find a basis for N M. Notation. We let N be the set of non-negative integers and we let C represent the complex numbers. A symbol w will generally represent the ordered n-tuple (w, w 2,..., w n ) and [w] will represent the set [w, w 2,..., w n ]. As is usual, K[w] will represent the set of all polynomials in w, w 2,..., w n. For M in N, K[w] M will equal the vector space spanned by all polynomials of degree M and K[w] M will equal the vector space spanned by all polynomials degree less than or equal M. We use the standard notation of letting w m represent w m w m 2 2 }}}wm n n where m= (m, m 2,..., m n )isinn n. For m in N n we define m =m +m 2 +}}}+m n and let N n =[m # M Nn m M], X m =x m xm 2 2 }}}xm n m n!m 2!}}}m n! and D m =D m }}}Dm n n =m +}}}+m n x m }}}xm n. Also, for m in n Zn &N n we set X m =0. So, for example, for any i, j in N n we have D i X j =X j&i.

3 bases 59 A Pairing of Vector Spaces General references for the material in this section are [5, pp. 628] and [6, p. 39] and [, p. 558]. The function (, ): C[x]_C[D] C defined by (f(x), P(D))=P (D) f(x) 0 (we apply the complex conjugate of the operator P(D) tof(x) and then evaluate the result at 0) is bilinear. Since D j X j = we see that (, ) is non-degenerate. Also since (X i, D j )=$ ij, [D j ] and [X j ] for j in N n are dual bases. Let C[D]*=Hom(C[D], C). Since (, ) is non-degenerate there is a linear isomorphism 8: C[D]*C[x] given by the following conditions: For 9 # C[D]* we have 8(9)=f (f,q)=9(q) for all Q # C[D]. Let P (D),..., P r (D) be polynomials in C[D] of any degrees. We extend this set of polynomials to P (D),..., P r (D),..., P v (D), by joining the elements of [D m m =M+]. Let J=(P (D),..., P v (D)) /C[D] (the ideal generated by P (D),..., P v (D)) and let J M =J & C[D] M. We also let N=[f(x)#C[x] (f(x), P i (D))=0 for i=, 2,..., v] and N M = N & C[x] M (it is easy to verify that this is the same N M that we defined in ()). J M and N M are, of course, vector spaces. The restriction (, ) M of (, ) to C[x] M _C[D] M is still bilinear and non-degenerate. Therefore 8=C[D]* M C[x] M is given by 8(9)=f (f,q) M = 9(Q) for all Q # C[D] M. Lemma. N M $J = M. Proof. Choose a basis Q (D),..., Q t (D) for C[D] M so that Q s+ (D),..., Q t (D) is a basis for J M. (3) Therefore N M =[f(x)#c[x] M (f(x), Q i (D)) M =0 for i=s+,..., t]. Let 9,..., 9 t # C[D]* M be dual to Q (D),..., Q t (D) which is to say that 9 i (Q j (D))=$ ij for all i, j such that i, jt. Set f i =8(9 i ) for i=,..., t. (4) Since the 9 i are linearly independent in C[D]* M the f i form a basis for C[x] M. Now (f i (x), Q j (D)) M =9 i (Q j (D))=$ ij implies that f (x),..., f s (x) forms a basis for C[x] M.

4 60 paul s. pedersen Lemma 2. Let z=z(d, M)= : m M Then 8(9)=f(x) if and only if (9)z=f(x). X m D m. (5) Proof. Let f(x)= m M a m X m #C[x] M. Then ( 9)z= m M X m 9(D m )= m M X m (f(x), D m ) M = m M X m ( p M a p X p, D m ) M =f(x). Corollary 3. For Q i (D) in (3), f i (x) in (4) and z in (5) we have t z= : i= f i (x)q i (D). (6) Proof. By a change of basis in (5) we can find h i (x) so that z= t h i= i(x)q i (D). Hence f j (x)=(9 j )z= t h i= i(x) 9 j (Q i (D))= h j (x). The point of Corollary 3 is that for any pair of dual bases [f i (x)] and [Q i (D)] we have (6). Theorem 4. Let ; i represent the class of x i in the residue class ring C[x,..., x n ](P (x,..., x n ),..., P v (x,..., x n )). Let C[D] M have basis [Q i (D) i=,..., t] where [Q i (;)=0 i=s+,..., t]. Then there exist f i (x) so that z(;, M)= s i= f i (x)q i (;) and [ f i (x) i=,..., s] forms a basis for N M. Proof. By definition of the ; i we have that P j (;)=0 for j=, 2,..., v, that Q j (;)=0 for j=s+,..., t and that Q (;),..., Q s (;) forms a basis for C[;] M. We apply vector space homomorphism C[x]C[D] C[x]C[;] (where D j ; j )toz(d,m) to get z(;, M)= t f i= i(x) Q i (;)= s f i= i(x)q i (;). The proof of Lemma shows that [f i (x) i=,..., s] forms a basis for N M. Remarks. () The preceeding arguments work if we are finding real solutions to a system of real coefficient PDE's. (2) There always exist subsets I of N n so that M [;m m#i] is a basis for C[;] M. (3) It is easy to verify that z(;, M)= n j=0 (M x i= i ; i ) j j!. (4) If J M =(P (D),..., P v (D)) contains then N M =[0] and there are no polynomial solutions.

5 bases 6 (5) If the P k (D), k=,..., r are of homogeneous order then N= N M=0 M and there is a basis of N consisting of homogeneous elements. We can find a basis for N M by finding a basis for C[;] M and rewriting ( n x i= i ; i ) M in terms of this basis. So if C[;] M has basis Q (;),..., Q s (;) then there exist f (x),..., f s (x) (homogeneous, degree M) so that ( n x i= i ; i ) M = s f i= i(x)q i (;) and we may conclude that f (x),..., f s (x) forms a basis for N M. Examples. () Let P(D)=( 2 x 2 +2 x 2 ). Let R[x 2,x 2 ] (P(D), D M+, D M D 2,..., D M+ 2 )$R[;, ; 2 ]. Therefore ; 2 +;2 2 =0 and it is not hard to verify that R[;] M has basis [; k, ;k& ; 2 0kM]. Now we can find f k, (x), f k,2 (x), k=0,,..., M so that z(;, M)= M f k=0 k,(x); k +f k,2(x); k& ; 2. Theorem 4 implies that f k, (x), f k,2 (x), k=0,,..., M form a basis for the set of all polynomial solutions to Laplace's equation having degree M. These solutions are in fact the same as the solutions one gets by finding the real and imaginary parts of the functions (x +x 2 i) k, k=0,..., M. If we use Remark 5 in this example, we have that R[;] M has basis [; M, ;M& ; 2 ]. We then get that (x ; + x 2 ; 2 ) M =f M, (x); M +f M,2(x); M& ; 2 and we may conclude that f M, (x) and f M,2 (x) form a basis for N M. (2) Let P(D)=( 2 x 2 &x 2) and let R[x 2,x 2](P(D), D 4, D 3 D 2,..., D 4 2 ) $R[;, ; 2 ] where ; 2 &; 2=0 and [; m =0 m =4]. Now R[;] 3 has basis, ;, ; 2, ; ; 2 and using this basis we have that z(;, 3)=+x ; +(x 2 +x 2 2); 2+(x x 2 +x 3 6); ; 2. We conclude that the four coefficients, x, (x 2 +x 2 2), (x x 2 +x 3 6) form a basis for all real polynomial solutions to the heat equation having degree 3. (3) Let P (D)= 2 x 2 +2 x x 2 3 and P 2 (D)=(x ) (x 2 )+(x )(x 3 )+(x 2 )(x 3 ). We will find a basis for the complex vector space N 3 by using Remark 5. To find these solutions we form the algebra C[;] generated by ; i, i=, 2, 3 where ; 2 +;2 2 +;2=0, 3 ; ; 2 +; ; 3 +; 2 ; 3 =0 and [; m =0 m =4]. Now C[;] 3 has basis ; 3, ; 2 ; 2, ; 2 ; 3, ; ; 2 2. We leave it to the reader to verify that these elements are indeed linearly independent in C[;] 3. The following table shows how the other elements of C[;] 3 can be written in terms of these elements. ; ; 2 3 =&;3 &; ; 2 2, ; ; 2 ; 3 =&; 2 ; 2&; 2 ; 3, ; 3 2 =&;3 &2;2 ; 2&; 2 ; 3&; ; 2 2, ;2 2 ; 3=; 2 ; 2+; 2 ; 3&; ; 2 2, ; 2 ; 2 3 =;3 +;2 ; 2+; 2 ; 3+; ; 2 2, ;3 3 =&;2 ; 2&2; 2 ; 3+; ; 2 2.

6 62 paul s. pedersen In ( 3 i= x i ; i ) 3 we replace the ; m by the appropriate linear combination of our basis elements as given in the table. Collecting terms we get 3 3 \ : x i ; i= i+ =(x 3 6&x x 2 3 2&x3 2 6+x 2x 2 3 2);3 +(x 2 x 2 2&x x 2 x 3 &x 3 23+x 2 2x 3 2+x 2 x 2 32&x 3 36); 2 ; 2 +(x 2 x 32&x x 2 x 3 &x x2x 2 32+x 2 x 2 3 2&x3 3 3);2; 3 +(x x 2 2&x 2 x 2 3 2&x3 2 6&x2x 2 32+x 2 x x36); 3 ; 2. 2 Theorem 4 tells us that these four coefficients of the basis elements form a basis for N 3. The Hilbert Characteristic Function If P (x,..., x n ),..., P r (x,..., x n ) are of homogeneous order, then J=(P (x,..., x n ),..., P r (x,..., x n )) is a homogeneous ideal and the number of linearly independent forms of degree M in the residue class ring C[x,..., x n ]J is designated /(J;M). The function / is called the Hilbert characteristic function of the ideal. A consequence of Lemma is that /(J; M)=dim N M. Other papers have noted that this equality holds for sufficiently large M (see [4]). As a corollary to this observation we have that dim N M is a polynomial in M for sufficiently large M (see, for example, [7, pp ]). Polynomial and C Solutions to Systems References [8, 9, and ] discuss the relationship between polynomial solutions and C solutions to systems of constant coefficient PDE's. [0] and [9] discuss how one may find a basis for the C solutions to the system [u # C P k (x,..., x n ) u=0; k=, 2,..., r] by using functions of the form Q(x)e n i=0 x ia i where [: # C n P k (:)=0, k=, 2,..., r] and where the Q(x) are certain unspecified polynomials. However this later work does not allow one to find a polynomial basis for the system. Acknowledgment The author thanks an anonymous referee who made suggestions that greatly simplified an earlier form of this paper.

7 bases 63 References. S. P. Smith, Polynomial solutions to constant coefficient differential equations, Trans. Amer. Math. Soc. 329, No. 2 (992), P. Pedersen, A function theory for finding all polynomial solutions to a linear constant coefficient PDE's of homogeneous order, J. Complex Variables 24 (993), P. Pedersen, Analytic solutions to a class of PDE's with constant coefficients, J. Differential Integral Equations 3, No. 4 (990), P. J. Stiller, Vector bundles on complex projective spaces and systems of partial differential equations, I, Trans. Amer. Math. Soc. 298, No. 2 (986), W. H. Greub, ``Linear Algebra,'' Springer-Verlag, New York, E. M. Stein and G. Weiss, ``Introduction to Fourier Analysis on Euclidean Spaces,'' Princeton Univ. Press, Princeton, NJ, O. Zariski and P. Samuel, ``Commutative Algebra, Vol. II,'' van Nostrand, New York, B. Malgrange, Existence et approximation des solutions des e quations de convolution, Ann. Inst. Fourier (Grenoble) 6 (955), F. Treves, ``Linear Partial Differential Operators,'' Gordon and Breach, New York, L. Ehrenpreis, A fundamental principle for systems of linear differential equations with constant coefficients, in ``Proceedings International Symposium on Linear Spaces, Jerusalem, 96,'' p. 6.. O. Kellogg, ``Foundations of Potential Theory,'' Verlag von Julius Springer, Berlin, 929. Printed in Belgium Uitgever: Academic Press, Inc. Verantwoordelijke uitgever voor Belgie : Hubert Van Maele Altenastraat 20, B-830 Sint-Kruis

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