Geometry on G L -Systems of Homogenous Polynomials

Transcription

1 Int. J. Contemp. Math. Sciences, Vol. 9, 014, no. 6, HIKARI Ltd, Geometry on G L -Systems of Homogenous Polynomials Linfan Mao Chinese Academy of Mathematics and System Science Beijing , P.R. China Copyright c 014 Linfan Mao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract For integers n, m 1, a G L -system (ESm n+1 ) is such a system consisting of m homogenous polynomials P 1 (x 1, x,, x n+1 ), P (x 1, x,, x n+1 ),, P m (x 1, x,, x n+1 ) in n + 1 variables with coefficients in C, which inherits a topological graph G[ESm n+1] in Pn C. We characterize the geometrical properties of system (ESm n+1 ) in P n C, find its combinatorial invariants under linear transformations, and determine its normalization, i.e., a combinatorial manifold M such that φ : M S is a homeomorphism by combinatorics, where S = m S i with S i a hypesurface determined by P i (x) = 0 in P n C for integers 1 i m. Indeed, such a combinatorial manifold M is nothing else but a manifold of dimensional n. Particularly, if n =, its genus is determined in this paper also. Mathematics Subject Classification: 05C15, 34A34, 37C75, 70F10, 9B05 Keywords: homogeneous polynomial, G L -system of homogenous polynomial, complex projective space, hypersurface, combinatorial manifold, topological graph, non-solvable differential equation 1. Introduction A polynomial P(x 1,x,,x n+1 ) in n + 1 variables with coefficients in C is homogeneous of degree d 1 if for any constant λ 0, there is P(λx 1,λx,,λx n+1 ) = λ d P(x 1,x,,x n+1 ). Such a polynomial P(x 1,x,,x n+1 ) determines a hypersurface defined by P(x 1,x,,x n+1 ) = 0 in the projective space P n C, a set of complex one-dimensional subspace, i.e., hyperplanes passing through origin of the complex vector space C n+1,

2 88 Linfan Mao which can be classified into two classes P n C = {(x 1,x,,x n,1) C n+1, x i C,1 i n} {(x1,x,,x n,0) C n+1, x i C,1 i n}. Geometrically, such a one-dimensional subspace spanned by (x 1,x,,x n,1) P n C can be identified with that of (x 1,x,,x n ) C n, and that of spanned by (x 1,x,,x n,0) P n C can be identified with that of the hyperplane at the infinite, i.e., P n C = C n P n 1 C. Whence, we can characterize the behavior of one-dimensional subspace C by its affine behavior P(x 1,x,,x n,1) in C n and that of P(x 1,x,,x n,0) in P n 1 C. Particularly, if C is replaced by R, then the behavior of C is divided into to parts: the reality in R n combined with those of in the infinite. Let x = (x 1,x,,x n+1 ), k = (k 1,k,,k n+1 ), a k = a k1,k,,k n+1, x k = x k 1 1 xk xk n+1 n+1 and P(x) = k 1 + +k n=d a k x k be a homogenous polynomial of degree d, an algebraic covariant of weight p is a homogenous polynomial C(a k,x) such that C(a k,x ) = p C(a k,x) under an invertible linear transformation T: x 1 = α 11 x 1 + α 1x + α 1.n+1x n+1 x = α 1 x 1 + α x + α.n+1x n x n+1 = α n1 x 1 + α nx + α n+1.n+1x n+1, i.e., T = (α ij ) (n+1) (n+1) (x) t, where x t denotes the transpose of vector x, a is the k coefficient of x k in P(x 1,x,,x n+1 ) and = det(α ij) (n+1) (n+1). Clearly, P(x) is itself a covariant of weight 0 by definition. The main interesting of this paper is in the combinatorially geometrical properties of systems consisting of homogenous polynomials, particularly, the case of n =. In such a case, the subspace determined by P(x,y,z) = 0 is called an algebraic curve C in P C. A point (a,b,c) of a curve C in P C defined by a homogenous polynomial P(x, y, z) is called singular if P P P (a,b,c) = (a,b,c) = (a,b,c) = 0. x y z Denoted by Sing(P) all singular points in the curve C determined by P(x, y, z) and it is called non-singular if Sing(P) =. By the Noether s result, we know that for any curve C determined by a homogenous polynomial P(x,y,z) of degree d in P C, there is a compact connected Riemann surface S such that h : S h 1 (Sing(C)) C Sing(C)

3 Geometry on G L -systems of homogenous polynomials 89 is a homeomorphism with genus g(s) = 1 (d 1)(d ) p Sing(C) δ(p), where δ(p) is a positive integer associated with the singular point p in C. Particularly, if Sing(C) =, i.e., C is non-singular then there is a compact connected Riemann surface S homeomorphism to C with genus 1 (d 1)(d ). For an integer m 1, let M 1,M,,M m be respectively dimensional n 1,n,,n m manifolds in a Euclidean space. A finite combinatorial manifold is defined following. Definition 1.1([10]) For manifolds M 1,M,,M m with respectively dimensional n 1,n,, n m in a topological space, where n max{n 1,n,,n m },m 1 and 0 < n 1 < n < < n m, a finite combinatorial manifold M of M 1,M,,M m is the union m M i underlying a topological graph G L [ M] following: V (G L [ M]) = {M 1,M,,M m }, E(G L [ M]) = {(M i,m j ) M i Mj, 1 i j m} with a labeling L : M i L(M i ) = M i and L : (M i,m j ) L(M i,m j ) = M i Mj for integers 1 i j m. For example, let M 0 = {(x,y,z) x +y +z = 1}, M 1 = {(x,y,z) (x ) +y + z = 1}, M = {(x,y,z) (x+) +y +z = 1}, M 3 = {(x,y,z) x +(y ) +z = 1}, M 4 = {(x,y,z) x + (y + ) + z = 1}, M 5 = {(x,y,z) x + y + (z ) = 1} and M 6 = {(x,y,z) x + y + (z + ) = 1} be 7 spheres in R 3. Notice that M 0 M 1 = {(1,0,0)}, M 0 M = {( 1,0,0)}, M 0 M 3 = {(0,1,0)}, M 0 M 4 = {(0, 1,0)}, M 0 M 5 = {(0,0,1)}, M 0 M 6 = {(0,0, 1)} and M i M j = for 1 i j 6. Whence, its topological graph is a wheel W 6 with labels such as those shown in Fig.1. M 4 (0,0,1) (0,-1,0) (1,0,0) M 5 M 0 M (-1,0,0) (0,1,0) (0,0,-1) M 3 M 1 M 6 Fig.1 In references [11]-[14], we discussed combinatorial geometries on non-solvable systems of algebraic equations, ordinary or partial differential equations with variables

4 90 Linfan Mao in R n. Now let P 1 (x),p (x),,p m (x) (ES n+1 m ) be m homogeneous polynomials in n + 1 variables with coefficients in C and each equation P i (x) = 0 determine a hypersurface M i, 1 i m in R n+1, particularly, a curve C i if n =. Then how can we characterize the system (ESm n+1 )? and how can we determine its geometrical behaviors? In this paper, we characterize the system (ESm n+1 ), find its combinatorial invariants and answer these questions, particularly for n = by a combinatorial approach and determine such a geometrical space is nothing else but a combinatorial Riemann surface M defined by Definition 1.1. Furthermore, we show such a combinatorial Riemann surface M is homeomorphic to a Riemann surface with genus determined in this paper. All terminologies and notations not defined in this paper are standard. We follow [1] for geometry, [], [4], [7] for those of algebra and algebraic curves, [5]-[6] for algebraic invariants, [8]-[11] for combinatorics, particularly, topological graphs and [3], [15] for partial differential equations.. G L -System of Homogeneous Polynomials in n + 1 Variables For two curves P(x,y), Q(x,y), particularly two lines in R, if they are not intersect, we can always say that P,Q are parallel in terminology. Such a property enables one to classify the bundle of lines in R by that of parallel or not parallel. Notice that by the famous Bézout s theorem, if P(x, y, z), Q(x, y, z) are two complex homogenous polynomials of degrees m, n and have no common components, then their curves determined by P(x,y,z), Q(x,y,z) have precisely mn intersection points counting multiplicities in P C ([7]). Thus any two curves, particularly, lines in P C intersect each other and there are no parallel subspace in projective space ([1]). However, denoted by I(P, Q) the set of intersection points of homogenous polynomials P(x) with Q(x) in n + 1 variables. We can also introduce the parallel hypersurfaces with degree 1 or not in P n C by distinguishing their intersections at infinite or not following. Definition.1 Let P(x), Q(x) be two complex homogenous polynomials of degree d in n + 1 variables. They are said to be parallel, denoted by P Q if d 1 and there are constants a,b,,c (not all zero) such that for x I(P,Q), ax 1 + bx + + cx n+1 = 0, i.e., all intersections of P(x) with Q(x) appear at a hyperplane on P n C, or d = 1 with all intersections at the infinite x n+1 = 0. Otherwise, P(x) are not parallel to Q(x), denoted by P Q. For example, let P( x z, y z,1)zm and Q = P( x z, y z,1)zm +kz m for integers k,m 1. It is clear that their only intersection points appear at the infinite line z = 0. Thus they are parallel in P C by Definition.1. The parallel property naturally enables one to introduce the underlying graph G[ESm n+1 ] of (ESm n+1 ) following. Definition. Let P 1 (x),p (x),,p m (x) be homogenous polynomials in (ES n+1 m ).

5 Geometry on G L -systems of homogenous polynomials 91 Define a topological graph G L [ESm n+1 ] in C n+1 by V (G L [ESm n+1]) = {P 1(x),P (x),,p m (x)}; E(G L [ESm n+1 ]) = {(P i (x),p j (x)) P i P j, 1 i,j m} with a labeling L : P i (x) P i (x), (P i (x),p j (x)) I(P i,p j ), where 1 i j m. Such a system (ESm n+1 ) is called a G L -system and the topological graph of G L [ESm n+1 ] without labels is called the underlying graph of G L - system (ESm n+1 ), denoted by G[ESm n+1 ]. Notice that the parallel property is symmetric and transitive. We can classify polynomials in (ESm n+1 ) by this property into classes C 1,C,,C l such that P i P j if P i,p j C k for an integer 1 k l, where 1 i j m. Such a classification is parallel maximal if each C i is maximal for integers 1 i l, i.e., for P {P k (x), 1 k m}\c i, there is a polynomial Q(x) C i such that P Q. Then we know the following result by definition. Theorem.3 Let n be an integer. For a system (ESm n+1 ) of homogenous polynomials with a parallel maximal classification C 1,C,,C l, G[ES n+1 m ] K(C 1,C,,C l ) and with equality holds if and only if P i P j and P s P i implies that P s P j, where K(C 1,C,,C l ) denotes a complete l-partite graphs Conversely, for any subgraph G K(C 1,C,,C l ), there are systems (ESm n+1 ) of homogenous polynomials with a parallel maximal classification C 1,C,,C l such that G G[ESm n+1 ]. Proof Clearly, for any P(x),Q(x) C k, 1 k l, P Q by definition. So (P(x),Q(x)) E(G[ESm n+1 ]), which implies that Notice that the equality G[ES n+1 m ] K(C 1,C,,C l ). G[ES n+1 m ] = K(C 1,C,,C l ) holds implies that for P(x) C i, Q(x) C j with 1 i j l, there must be P Q, i.e., P P k if P k Q. Conversely, if P s P i and P i P j implies that P s P j, then it is clear that for P(x) C i, Q(x) C j with 1 i j l, there must be (P(x),Q(x)) E(G[ESm n+1 ]). Thus G[ES n+1 m ] = K(C 1,C,,C l ). Conversely, if G K(C 1,C,,C l ), we construct a system (ESm n+1 ) following:

6 9 Linfan Mao For v V (G), choose homogenous polynomials P v (x) and P u (x) with P u P v if (v,u) E(G), otherwise, P u P v if (v,u) E(G). Then it is clear that G G[ESm n+1 ] by definition. ¾ Notice that for three hyperplanes P 1 (x),p (x),p 3 (x), if P 1 P and P P 3, then there must be that P 1 P 3. By Theorem.3, we get the following conclusion. Corollary.4 Let all polynomials in (ESm n+1 ) be degree 1, i.e., hyperplanes with a parallel maximal classification C 1,C,,C l, then G[ES n+1 m ] = K(C 1,C,,C l ). P 4 (x,y,z) P 6 (x,y,z) P 1 (x,y,z) P (x,y,z)p 3 (x,y,z) p 14 p 15 p 16 p 4 p 5 p 6 p 34 p 35 p 36 P 5 (x,y,z) P 1 (x,y,z) P (x,y,z) P 3 (x,y,z) p 4 p p 6 5 p 14 p 15 p 16 p 34 p 35 p 36 P 4 (x,y,z) P 5 (x,y,z)p 6 (x,y,z) Fig. For example, let the system (ES 3 m) be consisted of polynomials P 1 (x,y,z) = x z, P (x,y,z) = x z, P 3 (x,y,z) = x 3z and P 4 (x,y,z) = y z, P 5 (x,y,z) = y z, P 6 (x,y,z) = y 3z. Then they are all lines in P R with a figure and topological graph shown in Fig.. Furthermore, we consider the existence of G L -systems of homogenous polynomial for a topological graph G L with labels following: Problem.5 Let G L be a topological graph with labels I(e) P n C for e E(G). Are there systems (ESm n+1) of homogenous polynomials with GL [ESm n+1] GL? The results following answer this question in case of n =, i.e., algebraic curves. Theorem.6 Let n 1,n,,n l be l positive integers and let G L K(n 1,n,,n l ) be a topological graph with labels I(e) for e E(G), where I(e) P C is a finite point set. Then (1) There always exists a system (ES 3 m) of homogenous polynomials of degree d I(e) with a maximal parallel classification C 1,C,,C l such that G[ES 3 m] G with C i = n i, 1 i l and I (P u,p v ) I(e) for e = (u,v) E(G).

7 Geometry on G L -systems of homogenous polynomials 93 () For e = (u,v) E(G L ) labeled with I(e), there are homogenous polynomials P u (x,y,z), P v (x,y,z) of degrees D, d with D d such that I(P u,p v ) = I(e) if (D,d) {(1,1),(,1), (,),(3,3)} or (D,d) = (3,1), I(e) = {(x 1,y 1,z 1 ),(x,y,z ), (x 3,y 3,z 3 )} with det(m 3 ) = 0 or (D,d) = (3,), I(e) = {(x 1,y 1,z 1 ), (x,y,z ), (x 3,y 3,z 3 ),(x 4,y 4,z 4 ),(x 5,y 5,z 5 ),(x 6,y 6,z 6 )} with det(m 6 ) = 0, where M 3 = x 1 y 1 z 1 x y z x 3 y 3 z 3, M 6 = x 1 y 1 z 1 x 1 y 1 x 1 z 1 y 1 z 1 x y z x y x z y z x 3 y 3 z 3 x 3 y 3 x 3 z 3 y 3 z 3 x 1 4 y 4 z 4 x 4 y 4 x 4 z 4 y 4 z 4 x 5 y 5 z 5 x 5 y 5 x 5 z 5 y 5 z 5 x 6 y 6 z 6 x 6 y 6 x 6 z 6 y 6 z 6. Proof Let e E(G). Notice that a homogenous polynomial of degree d is in the form P(x,y,z) = a ijk x i y j z k, which has i.e., (d + 1)(d + ) i+j+k=d terms. Choose a new point p P C \ I(e). If (d + 1)(d + ) I(e) {p} = I(e) + 1, d I(e), there exists a homogenous polynomial P(x,y,z) passing through points in I(e) {p} with coefficients a ijk, i + j + k = d not all equal to zero by solving linear equations a ijk x i 0 yj 0 zk 0 = 0, (x 0,y 0,z 0 ) I(e). i+j+k=d Thus if e = (u,v) E(G), we can always choose p u p v P C \ I(e) and respectively find two distinct homogenous polynomials P u (x,y,z), P v (x,y,z) of degrees D, d passing through I(e) {p u }, I(e) {p v } such that I(P u,p v ) I(e). Assuming u C i, 1 i l. For w C i, w u, let P w (x,y,z) = P u (x,y,z) + λ w z d, where λ w is a complex number. Then it is clear that all polynomials in C i, 1 i l are parallel. Now let system (ES 3 m) be consisted all of homogenous polynomials P v (x,y,z), v V (G). Then such a system (ES 3 m) is clearly with a maximal parallel classification C i, 1 i l such that G[ES 3 m] G such that C i = n i, 1 i l and I (P u,p v ) I(e) for e = (u,v) E(G) by construction. Thus we get the conclusion (1). For (), by noticing that (D + 1)(D + ) > D Dd, (d + 1)(d + ) > Dd

8 94 Linfan Mao if (D,d) {(1,1),(1,),(,), (3, 3)} and (D + 1)(D + ) > D Dd, (d + 1)(d + ) = Dd if (D,d) = (3,) or (3,1). Thus the system of liner equations a i,j,k x i l yj l zk l = 0, 1 l Dd i+j+k=d and i+j+k=d b i,j,k x i l yj l zk l = 0, 1 l Dd have non-zero solutions a i,j,k, b i,j,k for i+j+k = D or d if (D,d) {(1,1),(1,),(,), (3,3)}, or (D,d) = (3,1), I(e) = {(x 1,y 1,z 1 ),(x,y,z ),(x 3,y 3,z 3 )} with det(m 3 ) = 0, or (D,d) = (3,), I(e) = {(x 1,y 1,z 1 ),(x,y,z ),(x 3,y 3,z 3 ),(x 4,y 4,z 4 ),(x 5,y 5,z 5 ), (x 6,y 6,z 6 )} with det(m 6 ) = 0 by linear algebra. Thus we can find homogenous polynomials P u (x,y,z) and P v (x,y,z) of degrees D, d such that I(P u,p v ) = I(e). ¾. The conclusion () in Theorem.6 implies the nine points theorem holds in the geometry of algebraic curves. Furthermore, we get the following conclusion. Corollary.7 Let L be a finite point set on projective algebraic curve P(x,y,z) of degree d 3 with L = d. Then there always exists a projective algebraic curve Q(x,y,z) P(x,y,z) of degree d passing through points in L. Corollary.7 partially answers Question.5 and enables one to get the following result. Theorem.8 Let G L be a topological graph with labels I(e) P C on e E(G). If I(e) = d with d 3, then there are systems (ESm 3 ) of homogenous polynomials P v (x,y,z) of degree d for v V (G L ) such that G L [ESm] 3 G L. The inverse of Bézout s theorem is not true in general. For example, let p 1,p,,p d be d points on a projective algebraic curve P(x,y,z) of degree d, which are not collinear. Clearly, there are no lines Q(x, y, z) passing through all points p 1,p,,p d. But the Bézout s theorem immediately implies the result following, which is useful for constructing systems (ES 3 m) underlying topological graphs G L [ES 3 m]. Theorem.9 Let P 1 (x,y,z),p (x,y,z),,p k (x,y,z), Q 1 (x,y,z),q (x,y,z),, Q l (x,y,z) be homogenous polynomials for integers k,l 1. Then there are homogenous polynomials P(x,y,z), Q(x,y,z) such that I(P,Q) = 1 i k,1 j l I(P i,p j ).

9 Geometry on G L -systems of homogenous polynomials 95 Proof Clearly, I(P i,p j ) for integers 1 i k,1 j l by Bézout s theorem. Define P(x,y,z) = k P i (x,y,z) and Q(x,y,z) = Applying Bézout s theorem, we then know that I(P,Q) = I(P i,p j ). 1 i k,1 j l l Q i (x,y,z). Then we have a criterion for a topological graph G L being that a underlying graph of system (ES 3 m ) following. Theorem.10 Let G L be a topological graph labeled with I(e) for e E(G L ). Then there is a system (ESm) 3 of homogenous polynomials such that G L [ESm] 3 G L if and only if there are homogenous polynomials P vi (x,y,z),1 i ρ(v) for v V (G L ) such that ρ(u) I(e) = I( ρ(v) P ui, P vi ) for e = (u,v) E(G L ), where ρ(v) denotes the valency of vertex v in G L. Proof If (ES 3 m ) is a system of homogenous polynomials P v,v V (G L ) such that G l [ES 3 m] G L, for v V (G L ) let P v1 = P v and P vi = 1 for integers i ρ(v). Then, by definition. we know that ρ(u) ρ(v) I(e) = I(P u,p v ) = I( P ui, P vi ) for e = (u,v) E(G L ). Conversely, if there are homogenous polynomials P vi (x,y,z),1 i ρ(v) for v V (G L ) such that ρ(u) ρ(v) I(e) = I( P ui, P vi ) for e = (u,v) E(G L ), let ρ(v) L : v P vi (x,y,z) = P v (x,y,z) for v V (G L ). Then the system (ES 3 m) consisting of homogenous polynomials P v (x,y,z),v V (G L ) is such a system with the conclusion holds. In fact, the identity mapping 1 G : v V (G L ) v V (G L ) is such an isomorphism with ρ(u) ρ(v) I(e) = I( P ui, P vi ) = I(P u,p v ) ¾

10 96 Linfan Mao for e = (u,v) E(G L ). Thus, G L [ES 3 m] G L. Choosing each P vi (x,y,z),1 i ρ(v) in Theorem.10 being line, we get a special conclusion following. Corollary.11 Let p ij be the intersection point of lines L 1 i with L j for integers 1 i D, 1 j d in P C and I = {p ij, 1 i D, 1 j d}. Then there are homogenous polynomials P(x,y,z) of degree D and Q(x,y,z) of degree d such that I(P,Q) = I. ¾ 3. Automorphisms of G L -System of Homogenous Polynomials in n + 1 Variables Classifying isomorphic systems (ESm n+1 ) enables one to introduce isomorphic covariants on homogenous polynomials following. Definition 3.1 Let ( 1 ESm n+1 ) and ( ESm n+1 ) be systems respectively consisting of covariants Ci 1(a k,x), C i (a k,x) on homogenous polynomials P i(x) for integers 1 i m. If there is an invertible linear transformation T such that for Ci 1(a k,x) ( 1 ESm n+1), there is C i (a k,x) ( ESm n+1) with C i (a k,x ) = p C 1 i (a k,x) holds for integers 1 i m, where p is a constant, then the system ( ESm n+1) is said to be linear isomorphic to system ( 1 ESm n+1 ), denoted by ( 1 ESm n+1 ) T ( ESm n+1 ), where is the determinant of T. Notice that a homogenous polynomial P(x) is itself a covariant of weight 0. Whence, a system (ESm n+1 ) consisting of homogenous polynomials P i (x),1 i m is itself a covariant system by definition. For example, let the systems ( 1 ES3 3) be 6x + 19y + 18z + 0xy + 14xz + 3yz ( 1 ES3 3 ) 10x + 33y + 14z + 37xy + 33yz + 53yz x 1 + 3y + 4z + 4xy + 5xz + 7yz and let ( ES 3 3 ) be ( ES 3 3 ) x + y + z xy + yz xz. Then ( 1 ES3 3) T ( ES3 3 ) because there is an invertible linear transformation x = x + y + z T : y = x + 3y + z z = x + 3y + 4z such that ( 1 ES3 3) is transformed to ( ES3 3 ) under the transformation T. The following result shows that G L [ESm n+1 ] is an invariant on isomorphic systems (ESm n+1 ).

11 Geometry on G L -systems of homogenous polynomials 97 Theorem 3. Let ( 1 ESm n+1 ), ( ESm n+1 ) be systems consisting of covariants Ci 1(a k,x), Ci (a k,x) on homogenous polynomials P i(x) for integers 1 i m of weight p, respectively. Then ( 1 ESm n+1 ) T ( ESm n+1 ) if and only if G L [ 1 ESm n+1 ] T G L [ ESm n+1 ] and for any integer i, 1 i m, C (a i T k,x ) = p Ci 1(a k,x) holds for a constant p, where is the determinant of T. Proof If ( 1 ESm n+1) T ( ESm n+1 ), we show T is an isomorphism between topological graphs G L [ 1 ESm n+1] and GL [ ESm n+1]. In fact, T : V (GL [ 1 ESm n+1]) V (G L [ ESm n+1 ]) is 1 1 and onto by definition. Notice that a hyperplane is transferred to a hyperplane by a linear transformation on P n C. Thus, Cu T Cv T in ( ESm n+1 ) if and only if C u C v in ( 1 ESm n+1 ), which implies that (Cu T,Cv T ) E(G[ ESm n+1 ]) if and only if (C u,c v ) E(G[ 1 ESm n+1 ]), i.e., G[ 1 ESm n+1 ] G[ ESm n+1 ]. Clearly, I ( Cu T,Cv T ) = T (I(Cu,C v )) for (C u,c v ) E(G L [ 1 ESm n+1 ]). Consequently, the linear transformation T : V (G[ 1 ES n+1 m ]) V (G[ ES n+1 m ]), E(G[ 1 ES n+1 m ]) E(G[ ES n+1 m ]) is commutative with that of labeling L, i.e., T L = L T, i.e., G L [ 1 ESm n+1] T G L [ ESm n+1]. By assumption, Ci 1(a k,x) is a covariant on homogenous polynomials P i(x) for integers 1 i m. Let T : Ci 1(a k,x) C i (a k,x). Then Ci (a k,x ) = C (a i T k,x ) = p Ci 1(a k,x) for integers 1 i m, where p is a constant. Notice that ( 1 ESm n+1 ) = {C1 v (a k,x) v V (GL [ 1 ESm n+1 ])}, ( ESm n+1 ) = {C u (a k,x) u V (GL [ ESm n+1 ])}. If G L [ 1 ESm n+1 ] T G L [ ESm n+1 ] with C (a i T k,x ) = p Ci 1(a k,x) holds for a constant p, let T = [α ij ] (n+1) (n+1) x t. Then T must be a linear isomorphism between systems ( 1 ESm n+1) and ( ESm n+1). In fact, for C1 v (a k,x) (1 ESm n+1 ), let T : Cv 1(a k,x) C (a v T k,x) ( ESm n+1). Then C (a v T k,x ) = p Cv 1(a k,x). Consequently, ( 1 ESm n+1 ) T ( ESm n+1 ) by definition. ¾ Theorem 3. enables one immediately knowing the following result. Corollary 3.3 Let (ES n+1 m ) be a system consisting of covariants C1 i (a k,x) on homogenous polynomials P i (x) for integers 1 i m of weight p and T be an invertible linear transformation. Then for any integer k G L [ES n+1 m ] G L [T k (ES n+1 m )] Particularly, let p = 0, i.e., (ESm n+1 ) consisting of homogenous polynomials P 1 (x),p (x),,p m (x) in Theorem 3.. Then it also implies the following conclusion.

12 98 Linfan Mao Corollary 3.4 Let ( 1 ESm n+1 ),( ESm n+1 ) be systems of homogenous polynomials P i (x),1 i m. Then ( 1 ESm n+1 ) T ( ESm n+1 ) if and only if G L [ 1 ESm n+1 T ] G L [ ESm n+1 ] with T an invertible linear transformation on P n C. Furthermore, if ( 1 ESm n+1) = ( ESm n+1) = (ESn+1 m ) in Definition 3.1, a linear isomorphism is called a automorphism of (ESm n+1 ). Clearly, all automorphisms of (ESm n+1 ) form a group Aut[ESn+1 m ] under the composition operation, which can be determined following. Theorem 3.5 Let (ESm n+1 ) be a covariant system of C i (a k,x), 1 i m of weight p on homogenous polynomials P 1 (x),p (x),,p m (x) for an integer n 1. Then Aut[ES n+1 m ] = AutG L ; v V (H) Aut(C v ) : H L G L [ES n+1 m ] PGL(n), where H L G L, Aut(C v ) denote respectively an induced topological subgraph H of graph G with labeling L on G L and automorphism group of covariant C v (a k,x) at vertex v. Proof Let (ESm n+1) be a covariant system of C i(a k,x), 1 i m of weight p on homogenous polynomials P 1 (x),p (x),,p m (x) for an integer n 1 with an invertible linear isomorphism T : (ESm n+1) (ESn+1 m ) such that CT v = C v for v V (H), where H G[ESm n+1]. Clearly, if H =, then T AutGL [ESm n+1 ]. Now if H, by Theorem 3. there must be T : G L [ESm n+1 ] H L G L [ESm n+1 ] H L with T Aut(C v ). Thus v V (H) Aut[ES n+1 m ] AutG L ; v V (H) PGL(n). Aut(C v ) : H L G L [ESm n+1 ] Conversely, such linear transformations ω are indeed automorphisms of system ). In fact, let (ES n+1 m ω : C v C v, v V (H L ), V L (G[ES n+1 m ] H) V L (G[ES n+1 m ] H) be an invertible linear transformation. Then, T Aut[ESm n+1 ]. Thus PGL(n). Aut[ESm n+1 ] AutG L ; Aut(C v ) : H L G L [ESm ] n+1 ¾ v V (H) Particularly, let p = 0. We get the automorphism group of system (ESm n+1) of homogenous polynomials following. Corollary 3.6 Let (ESm n+1) be a system of homogenous polynomials P 1(x),P (x),,p m (x) for an integer n 1. Then Aut[ES n+1 m ] = AutG L ; v V (H) Aut(P v (x)) : H L G L [ES n+1 m ] PGL(n).

13 Geometry on G L -systems of homogenous polynomials 99 For example, let (ES 3 4 ) = {P i(x,y,z) = a i x + b i y + c i z, 1 i 4} be 4 distinct homogenous polynomials of degree 1, such as those shown in Fig.3. x + y + z x + y + z x + y + z x + yc + z x + 4y + z x + y + z x + y + zx + y + 3z K L 4 {(P 1,P 4 )} K L 1.3 Fig.3 According to the ratio of coefficients, G[ES4 ] and Aut[ES 4 ] are listed in Table 1 following, Case Ratio G[ES4 ] Aut[ES 4 ] 1 a 1 : b 1 a : b a 3 : b 3 a 4 : b 4 K 4 S {P1,P,P 3,P 4 } a 1 : a : a 3 : a 4 = b 1 : b : b 3 : b 4 K 4 S {P1,P,P 3,P 4 } 3 a 1 : a : a 3 = b 1 : b : b 3, a 4 : b 4 K 1.3 S {P1,P,P 3,P 4 } 4 a 1 : a = b 1 : b, a 3 : b 3, a 4 : b 4 C 4,K 4 {(P 1,P )} S {P1,P,P 3,P 4 } Table 1 where S {P1,P,P 3,P 4 } denotes the symmetric group on P 1,P,P 3,P 4 and the last column is obtained by the four lines lemma in algebraic geometry. Clearly, AutK L 4 = AutK L 4 = AutC4 L = S {P 1,P,P 3,P 4 }, (P 1 P )(P 3 P 4 ),(P 1 P 3 ) = S {P1,P,P 3,P 4 }, S P1,P,P 3, (P 1 P 4 ) = S {P1,P,P 3,P 4 }. Thus, Aut[ES4 3] S 4, a finite group. It should be noted that Aut[ESm n+1 ] maybe not finite. But if it is indeed finite, a natural inverse question is the following: Problem 3.7 Let H PGL(n) be a finite group. Is there a finite system (ES n+1 of homogenous polynomials with Aut[ESm n+1] H? The answer for this question is positive. Thus m ) Theorem 3.8 For any finite subgroup H PGL(n), there always exists a finite system (ESm n+1 ) of homogenous polynomials with Aut[ESm n+1 ] = H. Proof Let P(x) be a homogenous polynomial in n + 1 variables and system (ESm n+1) = {P h (x), h H}. Clearly, P h (x) P g (x) for h,g H if h g. Notice that H is finite. Thus m <, i.e., a finite system (ESm n+1 ). Clearly, H Aut[ESm n+1 ] by definition. If Aut[ESm n+1 ] H, let θ Aut[ESm n+1 ] \ H, then there must be P θ (x) (ESm n+1 ). By the construction of (ESm n+1 ), there is an element h H such that P h (x) = P θ (x), i.e., θ H, a contradiction. The topological graph G L [ESm n+1 ] of system (ESm n+1 ) constructed in the proof of Theorem 3.8 is dependent on P(x) and group H. For example, let P(x) = x 1 and ¾

14 300 Linfan Mao the invertible matrix for h H is Thus, [ ] α h ij. Then (n+1) (n+1) P h (x) = α h 11x 1 + α h α h 1.n+1x n+1. P h (x) P(x) {(0,x,,x n+1 ), x i C, i n + 1} and P h P g if and only if α h 11 α g 11 = αh 1 α g 1 for h, g H. Consequently, V (G[ESm n+1 ]) = { x h 1, h H } = = αh 1n α g 1n αh 1.n+1 α g 1.n+1 { E(G[ESm n+1 ]) = (x h 1, xg 1 ), h, g H without α h 11 α g = αh 1 11 α g = = αh 1n 1 α g αh 1.n+1 1n α g 1.n+1 with a labeling L : x h 1 xh 1 and (xh 1,xg 1 ) xh 1 x g 1, h,g H. } 4. Topology on G L -Systems of Homogeneous Polynomials in n + 1 Variables Let π : C n+1 \{0} P n C determined by π(x 1,x,,x n+1 ) = [x 1,x,,x n+1 ] be the projection from C n+1 \ {0} to P n C, where [x 1,x,,x n+1 ] is the homogenous coordinates for points in P n C. Clearly, the equation P(x) = 0 determines a smooth function x n+1 = f(x 1,x,,x n ) by the implicit theorem if P(x) is homogenous. It is easily to verify that the subspace Γ[f] defined by Γ[f] = { (x 1,x,,x n,x n+1 ) C n+1 : x n+1 = f(x 1,x,,x n ),x i C,1 i n } is a complex n-manifold, i.e., a hypersurface in C n+1 and π : Γ[f] C n+1 π(γ[f]) P n C is a bijection. Thus π(γ[f]) is a hypersurface in P n C. Generally, let (ESm n+1) be a GL -system of homogenous polynomials P(x 1 ),P(x ),,P(x m ) in n + 1 variables with respectively hypersurfaces S i = π(γ[f i ]), where f i is the implicit function x n+1 = f i (x 1,x,,x n ) determined by P i (x) = 0 and let M = m M i with M i = Γ[f i ] for integers 1 i m. Clearly, for p M, there is an open neighborhood U(p) homeomorphic to C n, M is Hausdorff and second countable if m <. Thus M is also an n-manifold and π : M S is 1 1 by definition. We get a result following. Theorem 4.1 Let (ES n+1 m ) be a G L -system consisting of homogenous polynomials P(x 1 ),P(x ),,P(x m ) in n + 1 variables with respectively hypersurfaces

15 Geometry on G L -systems of homogenous polynomials 301 S 1,S,,S m. Then there is a combinatorial manifold M in C n+1 such that π : M S is 1 1 with G L [ M] G L [ S], where, S = m S i. Furthermore, such a M is also a complex n-manifold. Notice that the geometrical figure of a combinatorial manifold M consisting of manifolds M 1,M,,M m naturally induces a topological graph G M in C n+1 following: V (G M ) = {maximal subsets {M i1,,m ik } {M 1,,M m }, M i1 Mik }; E(G M ) = {M 1,M,,M m }. Clearly, G L [ M] without labels in Definition 1.1 is the line graph of G M, i.e., G L [ M] = L(G M ) by definition. Particularly, for n = we can further calculate the genus of surface M in R 3 by applying G M such that π : M S is 1 1 in Theorem 4.1. Firstly, we prove a general result on the genus of surfaces underlying a topological graph in R 3 following. Theorem 4. Let S be a combinatorial surface consisting of m orientable surfaces S 1,S,,S m underlying a topological graph G L [ S]. Then g( S) = β(g S ) m + ( 1) i+1 i S kl l=1 [ ) ) ] g (S k1 Ski c (S k1 Ski + 1, where g(s k1 Sk Skl ), c(s k1 Sk Skl ) are respectively the genus and number of path-connected components in surface S k1 Sk Skl, β(g S ) is the Betti number of topological graph G S. Proof The proof is by induction on m. If m =, by the classification theorem of surfaces, we assume S i is a connected sum T #T # #T of g (S }{{} i ) toruses for g(s i ) 1 i m. By this geometrical model, the contribution of genus of surfaces S to S 1 clearly is ) ) g (S ) g (S 1 S + c (S 1 S 1 because these c(s 1 S )) path-connected components contribute c(s 1 S )) 1 new tori to S 1 and β(g S ) = 0 in this case. Whence, ) ) ) g (S 1 S = g (S 1 ) + g (S ) g (S 1 S + c (S 1 S 1.

16 30 Linfan Mao Thus the conclusion holds with m =. Now assume the conclusion holds with m s. Notice that S = s+1 ( s ) Ss+1 S i = S i. Applying the inclusion-exclusion principle and the contribution of new tori by pathconnected components in their ( intersection, the contribution of genus of surface S s+1 s to the combinatorial surface g(s s+1 ) c Whence, S k1 Ski S i ) is [ )) ( 1) i g (S s+1 (S k1 Sk Ski )) ] (S s+1 (S k1 Sk Ski loops of S s+1 with S 1,S,,S s. g( S) = g( S \ S s+1 ) + loops of S s+1 with S \ S s+1 [ )) = g( S \ S s+1 ) + g(s s+1 ) ( 1) i g (S s+1 (S k1 Ski S k1 Ski )) ] c (S s+1 (S k1 Ski loops of S s+1 with S 1, S,, S s = β(g S \ Ss+1 ) m [ ) ) ] + ( 1) i+1 g (S k1 Ski c (S k1 Ski + 1 S k1 Ski [ )) +g(s s+1 ) + ( 1) i+1 g (S s+1 (S k1 Ski 1 k 1,,k i s )) c (S s+1 (S k1 Ski = β(g S ) s+1 + ( 1) i+1 S k1 Ski ] loops of S s+1 with S 1, S,, S s [ ) ) ] g (S k1 Ski c (S k1 Ski + 1. Thus, the conclusion holds also with m = s + 1. ¾ Corollary 4.3 Let S be a combinatorial surface consisting of orientable surfaces S 1,S,,S m with S k1 Ski path-connected or empty for integers 1 k 1,k,,k i m. Then g( S) = β(g S ) + m ( 1) i+1 S k1 Ski ) g (S k1 Ski.

17 Geometry on G L -systems of homogenous polynomials 303 Furthermore, if S k1 Ski is simply-connected or empty for any integers k 1,k,,k i, then m g( S) = β(g S ) + g (S i ). Proof Notice that for integers 1 k 1,k,,k i m, if S k1 Ski pathconnected, then c(s k1 Ski ) = 1 and if S k1 Ski is simply-connected, then g (S k1 Ski ) = 0. According to Theorem 4., this conclusion follows. ¾ Theorem 4. enables one to determine the genus of surface S by applying Noether s formula in algebraic geometry, particularly, the non-singular case. Notice that the Bèzout s theorem claims that the number of intersection points counting multiplicities of two projective curves C 1, C in P C is I(P,Q) = deg(p)deg(q) if they are determined by P(x,y,z) and Q(x,y,z) without common component. Thus, if S i is the normalizations of C i for 1 i, then surfaces S 1 and S are tangent each other at deg(p)deg(q) points. Applying Theorem 4. and Corollary 4.3, we get the genus of the normalization S of complex curves C 1,C,,C m following. Theorem 4.4 Let C 1,C,,C m be complex curves determined by homogenous polynomials P 1 (x,y,z),p (x,y,z),,p m (x,y,z) without common component, and let R Pi,P j = deg(p i )deg(p j ) k=1 (c ij k z bij k y)eij k, ωi,j = deg(p i )deg(p j ) k=1 e ij k 0 1 be the resultant of P i (x,y,z),p j (x,y,z) for 1 i j m. Then there is an orientable surface S in R 3 of genus g( S) = β(g C ) + + m (deg(p i) 1)(deg(P i ) ) [ c 1 i j m(ω i,j 1) + i 3 ( 1) i C k1 Cki p i Sing(C i ) δ(p i ) (C k1 Cki ) 1 m with a homeomorphism ϕ : S C = C i. Furthermore, if C 1,C,,C m are non-singular, then m (deg(p i ) 1)(deg(P i ) ) g( S) = β(g C ) i j m(ω i,j 1) + ( 1) i i 3 C k1 Cki [ ) ] c (C k1 Cki 1, ] where δ(p i ) = 1 ( ( I p i P i, P ) ) i ν φ (p i ) + π 1 (p i ) y

18 304 Linfan Mao is a positive integer with a ramification index ν φ (p i ) for p i Sing(C i ),1 i m. Proof This result is an immediately consequence of Theorem 4. and Noether s result. For its genus, let S i be the normalization of C i, i.e., ϕ i : S i C i in R 3 for integers 1 i m and S = m S i. Define ϕ : S C by ϕ(p) = ϕi (p) if p S i, 1 i m. This definition is well-defined because if p S k1 Ski, then ϕ(p) C k1 Cki. Notice that each ϕ i is a homeomorphism. Thus ϕ is also a homeomorphism from S to C, i.e., a normalization of surface to m C i with G S G C. Clearly, C k1 Cki, i.e., S k1 Ski only consists of isolated points. Whence, g (S k1 Ski ) = 0 for any subset {k 1,,k i } {1,,,m} with i and 1 i j m [ ) ] c (S i Sj 1 = = 1 i j m 1 i j [ ) ] c (C i Cj 1 (ω i,j 1), ) ) c (S k1 Ski 1 = c (C k1 Cki 1 if C k1 Cki. Substituting the Noether s formula into Theorem 4., we get that g( S) = β(g S ) m [ ) ) ] + ( 1) i+1 g (S k1 Ski c (S k1 Ski 1 = β(g C ) + = β(g C ) + + C k1 Cki 1 i =j m (ω i,j 1) + i 3 m g (S i ) + ( 1) i i C k1 Cki m (deg(p i) 1)(deg(P i ) ) [ ( 1) i c C k1 Cki [ ) ] c (C k1 Cki 1 p i Sing(C i) δ(p i ) (C k1 Cki ) 1 ]. ¾ Theorem 4.4 enables us to get consequences following. Corollary 4.5 Let C 1,C,,C m be complex non-singular curves determined by homogenous polynomials P 1 (x,y,z),p (x,y,z),,p m (x,y,z) without common component, any intersection point p I(P i,p j ) with multiplicity e p and P i (x,y,z) = 0 P j (x,y,z) = 0, P k (x,y,z) = 0 i,j,k {1,,,m}

19 Geometry on G L -systems of homogenous polynomials 305 has zero-solution only. Then the genus of normalization S of curves C 1,C,,C m is g( S) = m deg(p i )(deg(p i ) 3) + ω i,j. 1 i j m Particularly, if e ij p = 1 for p I(P i,p j ), 1 i j m, then g( S) = m deg(p i )(deg(p i ) 3) + Proof Notice that G C K m with β(k m ) = m(m + 1) 1 i j m deg(p i )deg(p j ). m + 1 and C k1 Cki =, i 3 by assumption. Applying Theorem 4.4, we know that g( S) = β(g C ) + + m (deg(p i ) 1)(deg(P i ) ) ( 1) i (ω i,j 1) + 1 i =j m i 3 C k1 Cki m (deg(p i ) 1)(deg(P i ) ) = β(k m ) + + = m deg(p i )(deg(p i ) 3) + 1 i =j m [ ) ] c (C k1 Cki 1 1 i =j m ω i,j. (ω i,j 1) Particularly, if e ij p = 1 for p I(P i,p j ), then ω i,j = deg(p i )deg(p j ) by Bézout s theorem for integers 1 i j m, we get that g( S) = m deg(p i )(deg(p i ) 3) + deg(p i )deg(p j ). 1 i j m ¾ Corollary 4.6 Let C 1,C,,C m be complex non-singular curves determined by homogenous polynomials P 1 (x,y,z),p (x,y,z),,p m (x,y,z) without common component and C i Cj = m m C i with C i = κ > 0 for integers 1 i j m. Then the genus of normalization S of curves C 1,C,,C m is g( S) = g( S) = (κ 1)(m 1) + m (deg(p i ) 1)(deg(P i ) ). Proof Notice that G S S 1.m+1 with β(s 1.m+1 ) = 0, ω i,j = κ for integers

20 306 Linfan Mao 1 i j m and i,j 1) + 1 i j m(ω [ ) ] ( 1) i c (C k1 Cki 1 i 3 C k1 Cki = ( ) i ( 1) i m (κ 1) = (κ 1)(m 1). i Applying Theorem 4.4, we get that g( S) = (κ 1)(m 1) + m (deg(p i ) 1)(deg(P i ) ). ¾ For homogenous polynomials with small degrees, Theorem 4.4 also enables one to get interesting conclusions. Corollary 4.7 Let L 1,L,,L m be distinct lines in P C with respective normalizations of spheres S 1,S,,S m. Then there is a normalization of surface S of L 1,L,,L m with genus β(g L ). Particularly, if G L ) is a tree, then S is homeomorphic to a sphere. Corollary 4.8 Let C 1,C,,C m be non-singular conics determined by homogenous polynomials P 1 (x,y,z),p (x,y,z),,p m (x,y,z) of degree without common component in P C and let S 1,S,,S m be respective normalizations of surfaces conics C 1,C,,C m. Then there is a normalization surface S in of genus g( S ) = β(g C ) + ( 1) i i C k1 Cki [ ) ] c (C k1 Cki 1 with c(c k1 Cki ) 4 for integers C k1 Cki. Particularly, if no 3 conics of C 1,C,,C m pass through a common point, then g( S ) = β(g C ) + 1 i j m ( m ω i,j ). 5. Application to Elliptic Differential Equations As we known, a Dirichlet problem on Laplace equation is to find functions u(x) in a region D R n with { u = 0, x D u D = ϕ(x) holds, where = x + 1 x + + x.

21 Geometry on G L -systems of homogenous polynomials 307 If D R n is nothing else but a ball B n and ϕ(x) is continuous for x = a, the Poission s integral formula ([3]) u ( ξ ) = x =a 1 a π n Γ ( ) a ξ 1 n x ξ ϕ(x)ds n provides the solution of Dirichlet problem. Particularly, if n =, we get u(ρ,θ) = 1 π a ρ π a + ρ aρcos(θ α) ϕ(α)dα 0 in spherical coordinates (ρ,θ). Let ϕ 1 (x),ϕ (x),,ϕ m (x) be m respective continuous functions, particularly, m homogenous polynomials for x = a. A natural question on the Laplace equation is that whether a system of } u = 0, x D 1 i m (PDES u D = ϕ i (x) m) D is solvable or not? We consider a special case of D = B n and denote by Φ(ϕ 1,n),Φ(ϕ,n),, Φ(ϕ m,n) the hypersurfaces respectively determined by x n+1 = ϕ i (x) in R n+1 with a topological graph G Φ. Then such a question can be also viewed as the Dirichlet problem on the Laplace equation prescribed with a boundary value G Φ with Φ = m Φ(ϕ i,n). Similarly, let S(ϕ 1,n),S(ϕ,n),,S(ϕ m,n) be hypersurfaces determined by x n+1 = u(x) in R n+1 with a topological graph G S. By a geometrical view, such a system (PDES D m ) is solvable if m S(ϕ i,n). Otherwise, non-solvable in classical meaning. However, S and Φ are both n-manifold in R n by Theorem 4.1 and there is a bijection τ : G Φ G S. Thus, it is also meaningful for knowing the solution behaviors of Laplace equation by generalizing solution in classical meaning to G L - solutions of systems (PDES D m) following. Definition 5.1 A G L -solution of system (PDESm) D is such an n-manifold S consisting of hypersurfaces S(ϕ 1,n),S(ϕ,n),,S(ϕ m,n) underlying a topological graph G S in R n+1. Then, all of the previous discussions implies an interesting conclusion following.

22 308 Linfan Mao Theorem 5. Let Φ be a G Φ -system consisting of continuous functions ϕ 1 (x), ϕ (x),,ϕ m (x) for x = a. Then the system (PDES D m) of the Dirichlet problem on Laplace equation prescribed with boundary value Φ is G L -solvable. Particularly, if n = with m ϕ i (θ), we get a surface consisting of m surfaces S(ϕ 1,n), S(ϕ,n),,S(ϕ m,n) in R 3 with genus determined by Theorem 4.. References [1] M.Berger, Geometry(I), Springer Verlag Berlin Heidelberg, [] H.M.Farkas and Irwin Kra, Riemann Surfaces, Springer Verlag New York Inc., [3] Fritz John. Partial Differential Equations(4th Edition). New York, USA: Springer- Verlag, 198. [4] P.A.Griffiths, Algebraic Curves (in Chinese), Peking University Press, 000. [5] G.B.Gurevich, Foundations of the Theory of Algebraic Invariants, P.Noordhoff Ltd. Groningen, the Netherlands, [6] David Hilbert, Theory of Algebraic Invariants, Cambridge University Press,1993. [7] Frances Kirwan, Complex Algebraic Curves, Cambridge University Press & Beijing World Publishing Corporation, 008. [8] Linfan Mao, Automorphism Groups of Maps, Surfaces and Smarandache Geometries (Second edition), The Education Publisher Inc., USA, 011. [9] Linfan Mao, Smarandache Multi-Space Theory (Second edition), The Education Publisher Inc., USA, 011. [10] Linfan Mao, Combinatorial Geometry with Applications to Field Theory (Second edition), The Education Publisher Inc., USA, 011. [11] Linfan Mao, Graph structure of manifolds with listing, International J.Contemp. Math. Science, Vol.5, 011, No., [1] Linfan Mao, Non-solvable spaces of linear equation systems. International J. Math. Combin., Vol. (01), 9-3. [13] Linfan Mao, Global stability of non-solvable ordinary differential equations with applications, International J.Math. Combin., Vol.1 (013), [14] Linfan Mao, Non-solvable equation systems with graphs embedded in R n, International J.Math. Combin., Vol. (013), 8-3. [15] M.Renardy, R.C.Rogers. An Introduction to Partial Differential Equations(th Edition). Berlin Heidelberg, USA: Springer-Verlag, 004. Received: March 14, 014