THE INDEX OF ANALYTIC VECTOR FIELDS AND NEWTON POLYHEDRA CARLES BIVI AAUSINA


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1 THE INDEX OF ANALYTIC VECTOR FIELDS AND NEWTON POLYHEDRA CARLES BIVI AAUSINA Abstract. In this work we study a condition of nondeeneracy on analytic maps (R n ; 0)! (R n ; 0) usin the lanuae of Newton polyhedra. When f : (R n ; 0)! (R n ; 0) satises this condition and f 1 (0) = f0, we obtain a result on the invariance of the local index of f under the summation of suitable monomials to each component function of f. 1. Introduction Let f : (R n ; 0)! (R n ; 0) be a continuous map erm such that 0 is isolated in f 1 (0), then the index or mappin deree of f, denoted by ind 0 (f), is a well known topoloical invariant of f (see [4], [5] or [11] for the denition and properties of this invariant). We shall ive a qualitative result on the index of f. The motivation of our work is a theorem of Cima GasullTorrerosa that we shall state after some preliminary denitions. The results of these notes are included and proved in [1]. Let : (R n ; 0)! (R n ; 0) be a weihted homoeneous map erm of weihts w 1 ; : : : ; w n and derees d 1 ; : : : ; d n. Suppose that 1 (0) = f0. Let G : (R n ; 0)! (R n ; 0) be another analytic map erm such that the deree, with respect to the weihts w 1 ; : : : ; w n, of each monomial appearin in the Taylor expansion of G i is strictly reater than d i. Then, the map f = + G is said to be semiweihted homoeneous and that is the weihted homoeneous part of f or the principal part of f with respect to the weihs w 1 ; : : : ; w n. Theorem 1.1. [3] Let f = + G : (R n ; 0)! (R n ; 0) be a semiweihtedhomoeneous map erm, where is the weihted homoeneous part of f. Then ( + G) 1 (0) = f0 and ind 0 () = ind 0 ( + G). In view of the above result, if : (R n ; 0)! (R n ; 0) is analytic map erm such that 1 (0) = f0 and G : (R n ; 0)! (R n ; 0) is an arbitrary analytic map erm, we consider the problem of ndin `ood conditions' on and G in order to obtain that ( + G) 1 (0) = f0 and ind 0 () = ind 0 (+G). In this task we have applied the techniques of [2] on the estimation of Lojasiewicz exponents. 2. Conditions of nondeeneracy and the main result In order to show the main result, we need to introduce some preliminary denitions. We say that a subset + R n + is a Newton polyhedron when there exists some A R n + such 2000 Mathematics Subject Classication. Primary 32S05; Secondary 57R45. Key words and phrases. The index of a vector eld, real analytic functions, Newton polyhedra. Work supported by DGICYT Grant BFM2000{
2 2 CARLES BIVI AAUSINA that + is equal to the convex hull of the set fk + v : k 2 A; v 2 R n +. If we x a Newton polyhedron + R n + and a vector v 2 R n +, v 6= 0, we denote by `(v; +) the minimum between the scalar products hv; ki where k 2 +. We also denote by (v; +) the subset of + where this minimum is attained. The faces of + are the subsets of the form = (v; +), for some v 2R n +, v 6= 0; in this case, we say that v supports the face. The union of the compact faces of + is denoted by. We dene the dimension of a face of + as the dimension of the minimal ane subspace containin. A vector v 2R n + is said to be primitive when the components of v are positive relative prime inteer numbers. We observe that any face of + of dimension n 1 is supported by a unique primitive vector. The family of primitive vectors supportin some face of + of dimension n 1 will be denoted by F( + ). Moreover, we denote by F c ( + ) the family of primitive vectors supportin some compact face of + of dimension n 1, that is, F c ( + ) = F( + ) \ (R + r f0) n. Let A n denote the local rin of analytic function erms (R n ; 0)!R. If h 2 A n, h 6= 0, and h = P k a kx k is the Taylor expansion of h, then the support of h, denoted by supp(h), is the set of those k 2Z n + such that k 6= 0. Then, the Newton polyhedron of h is +(h) = + (supp(h)). If h = 0, we set supp(h) = ; and + (h) = ;. The union of the compact faces of +(h) is denoted by (h). Let us x a vector v 2R n +, v 6= 0, and a nonzero map h = P k a kx k 2 A n, then we denote the number `(v; +(h)) by `(v; h) and the set (v; +(h)) by (v; h). The principal part of h with respect to v is the sum of those a k x k such that k 2 (v; h). If h = 0, then we set p v (h) = 0. Let + R n be a Newton polyhedron. If J F( + ) and h = P k a kx k 2 A n, we dene the principal part of h with respect to J as the erm p J (h) 2 A n iven by the sum of those a k x k such that k 2 T v2j (v; h). The set T v2j p J (h) = 0. (v; h) could be empty, in this case, we x Denition 2.1. [2] Let = ( 1 ; : : : ; s ) : (R n ; 0)! (R s ; 0) be an analytic map erm and let + R n be a Newton polyhedron. If J F( + ), we say that satises the (C J ) condition when (1) x 2R n : p J ( 1 )(x) = = p J ( s )(x) = 0 fx 2R n : x 1 x n = 0: We say that is an adapted map to J F( + ) such that T v2j (v; +) is a compact face of +. + when satises the (C J ) condition, for each Given a subset L f1; : : : ; n, we dene R n L = fx 2 Rn : x i = 0; for all i 2 L and A L = A\R n L, for any subset A ofrn. Then, if h = P k a kx k 2 A n, we denote by h L the sum of those a k x k such that k 2 (supp(h)) L. If = ( 1 ; : : : ; s ) : (R n ; 0)! (R s ; 0) is an analytic map erm, we denote the map (( 1 ) L ; : : : ; ( s ) L ) : (R n jlj ; 0)! (R s ; 0) by L. Denition 2.2. [2] Let + R n + be a Newton polyhedron intersectin each coodinate axis. If = ( 1 ; : : : ; s ) : (R n ; 0)! (R s ; 0) is an analytic map erm, we say that is stronly adapted to + when the map L is stronly adapted to the Newton polyhedron ( + ) L, for each L f1; : : : ; n, L 6= f1; : : : ; n (considerin also L = ;).
3 THE INDEX OF ANALYTIC VECTOR FIELDS 3 Let + R n + be a Newton polyhedron intersectin each coordinate axis and let = ( 1 ; : : : ; s ) : (R n ; 0)! (R s ; 0) be an analytic map erm such that +( i ) intersects each coordinate axis, for all i = 1; : : : ; s, then it is straihtforward to prove that is adapted to + if and only if is stronly adapted to +. Proposition 2.3. Let f = + G : (R n ; 0)! (R n ; 0) be a semiweihtedhomoeneous map erm, where is the weihted homoeneous part of f. Let w 1 ; : : : ; w n be the weihts of, then f is stronly adapted to the Newton polyhedron determined by f 1 w i e i : i = 1; : : : ; n, where e 1 ; : : : ; e n denote the canonical basis in R n. We show the followin eneralization of Theorem 1.1. Theorem 2.4. Let : (R n ; 0)! (R n ; 0) be an analytic map erm such that 1 (0) = f0. Suppose that is stronly adapted to +. Let G : (R n ; 0)! (R n ; 0) be another analytic map erm such that `(v; G i ) > `(v; i ), for all i = 1; : : : ; n and all v 2 F c ( + ). ( + G) 1 (0) = f0 and ind 0 () = ind 0 ( + G). Then Next, we ive a corollary of the above theorem considerin a notion of nondeeneracy which is similar to the denition iven by Khovanskii [7]. Given an analytic map erm = ( 1 ; : : : ; s ) : (R n ; 0)! (R s ; 0) and a vector v 2 R n +, we say that satises the (K v ) condition when x 2R n : p v ( 1 )(x) = = p v ( s )(x) = 0 fx 2R n : x 1 x n = 0: The map is said to be nondeenerate when satises the (K v ) condition, for all v 2R n +, v 6= 0. We say that is stronly nondeenerate when L : (R n jlj ; 0)! (R s ; 0) is nondeenerate, for all L ( f1; : : : ; n. As before, if the Newton polyhedron of each component of intersects each coordinate axis, the map is nondeenerate if and only if is stronly nondeenerate. Moreover, an analytic map erm = ( 1 ; : : : ; s ) is nondeenerate if and only if it is adapted to the Minkowski sum + ( 1 ) ( s ). Corollary 2.5. Let : (R n ; 0)! (R n ; 0) be a stronly nondeenerate analytic map erm such that 1 (0) = f0. Let G : (R n ; 0)! (R n ; 0) be an analytic map erm such that supp(g i ) + ( i )r ( i ), for all i = 1; : : : ; n. Then ( + G) 1 (0) = f0 and ind 0 () = ind 0 ( + G). 3. A remark on planar vector fields If = ( 1 ; 2 ) : (R 2 ; 0)! (R 2 ; 0) is a nondeenerate analytic map erm such that 1 (0) = f0, then is automatically stronly nondeenerate. This is because of the fact that, when 1 (0) = f0, then supp( 1 ) [ supp( 2 ) contains a pure monomial x k i i of each variable x i, i = 1; 2. Therefore, Corollary 2.5 constitutes a eneralization of the result of [6] on the index of planar vector elds. As we shall see, Corollary 2.5 can be improved in the case n = 2. First, we ive a preliminary denition. If = ( 1 ; 2 ) : (R 2 ; 0)! (R 2 ; 0) is an analytic map erm such that F c ( 1 ) [ F c ( 2 ) 6= ;, then we dene +( i ) = k 2R 2 + : hk; vi `(v; i ); for all v 2 F c ( 1 ) [ F c ( 2 ) ; i = 1; 2:
4 4 CARLES BIVI AAUSINA It is clear that +( i ) is also a Newton polyhedron, we denote by ( i ) the union of compact faces of +( i ), i = 1; 2. Moreover, iven an i 2 f1; 2, we have that +( i ) +( i ) and that +( i ) = +( i ) when + ( i ) intersects each coordinate axis. Corollary 3.1. Let = ( 1 ; 2 ) : (R 2 ; 0)! (R 2 ; 0) be an analytic nondeenerate map erm such that 1 (0) = f0. Suppose that F c ( 1 ) [ F c ( 2 ) 6= ;. Let G = (G 1 ; G 2 ) : (R 2 ; 0)! (R 2 ; 0) be another analytic map erm such that supp(g i ) +( i )r ( i ), i = 1; 2. Then ( + G) 1 (0) = f0 and ind 0 ( + G) = ind 0 (). Given an analytic map erm : (R n ; 0)! (R n ; 0) such that 1 (0) = f0, we can consider the Lojasiewicz exponent of, that is, the minimum of those > 0 such that there exists some constant C > 0 and a neihbourhood U of 0 in R n such that jxj Cj(x)j, for all x 2 U (see [10, p. 136]). We shall denote this number by 0 (). Moreover, we shall denote by Q() the local alebra A n =I(), where I() is the ideal of A n enerated by the component functions of. If dimr Q() < 1, it is not dicult to prove that dimr Q() 0 () n. It is worth to state here the followin result of EisenbudLevine relatin the numbers ind 0 () and dimr Q(). Theorem 3.2. [4] Let : (R n ; 0)! (R n ; 0) be an analytic map erm such that dimr Q() < 1. Then (1) jind 0 ()j dimr Q() 1 1=n ; (2) if is sinular, then jind 0 ()j < 1 2 dim R Q(). As a consequence of the above result, if : (R 2 ; 0)! (R 2 ; 0) is an analytic map erm such that 1 (0) = f0, we have obtain the relation jind 0 ()j dimr Q() 1=2 [ 0 (f)]; where [a] denotes the inteer part of a iven real number a. We remark that, if : (R n ; 0)! (R n ; 0) is a nondeenerate map erm such that +( i ) intersects each coordinate axis, i = 1; : : : ; n, then there is a formula for dimr Q() in terms of the Newton polyhedra of each erm i (see [8]) and there is also is an upper estimate for the Lojasiewicz exponent 0 () (see [2]). References [1] BiviaAusina, C. The index of analytic vector elds and Newton polyhedra, to appear in Fund. Math. [2] BiviaAusina, C. Lojasiewicz exponents, the interal closure of ideals and Newton polyhedrons, J. Math. Soc. Japan. 55, 3 (2003), 655{668. [3] Cima, A., Gasull, A. and Torrerosa, J. On the relation between index and multiplicity, J. London Math. Soc. (2) 57 (1998), 757{768. [4] Eisenbud, D. and Levine, H.I. An alebraic formula for the deree of a C 1 map, Ann. of Math. 106 (1977), 19{44. [5] Guillemin, V. and Pollack, A. Dierential topoloy, PrenticeHall, Inc. Enlewood Clis, N.J. (1974). [6] Gutierrez, C. and Ruas, M.A.S. Indices of Newton nondeenerate vector elds and a conjecture of Loewner for surfaces in R 4, Real and Complex Sinularities, Lecture Notes in Pure and Appl. Math. vol. 232, Marcel Dekker (2003), 245{253. [7] Khovanskii, A.G. Newton polyhedra and toroidal varieties, Funct. Anal. Appl. 11 (1977), 289{295.
5 THE INDEX OF ANALYTIC VECTOR FIELDS 5 [8] Khovanskii, A.G. Newton polyhedra, a new formula for mixed volume, products of roots of a system of equations, Fields Inst. Commun. 24 (1999), 325{364. [9] Lejeune, M. and Teissier, B. Cl^oture interale des ideaux et equisinularite, Centre de Mathematiques, Universite Scientique et Medicale de Grenoble, [10] Lojasiewicz, S. Ensembles semianalytiques, IHES (1965). [11] Milnor, J. Topoloy from the dierential viewpoint, The University Press of Virinia, (1965). Departament de Matematica Aplicada, Universitat Politecnica de Valencia, Escola Politecnica Superior d'alcoi, Placa Ferrandiz i Carbonell 2, Alcoi (Alacant), Spain address: