Topological K-equivalence of analytic function-germs

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1 Cent. Eur. J. Math. 8(2) DOI: /s Central European Journal of Mathematics Topological K-equivalence of analytic function-germs Research Article Sérgio Alvarez 1, Lev Birbrair 2 João Carlos Ferreira Costa 3 Alexandre Fernandes 2 1 Departamento de Matematica, Universidade Estadual Santa Cruz, Ilheus, Brazil 2 Departamento de Matematica, Universidade Federal do Ceara, Fortaleza, Brazil 3 Departamento de Matematica, Universidade Estadual Paulista, Sao Jose do Rio Preto, Brazil Received 15 July 2009; accepted 20 January 2010 Abstract: MSC: We study the topological K-equivalence of function-germs (R n, 0) (R, 0). We present some special classes of piece-wise linear functions and prove that they are normal forms for equivalence classes with respect to topological K-equivalence for definable functions-germs. For the case n = 2 we present polynomial models for analytic function-germs. 32S15, 32S05 Keywords: Topological K-equivalence Topological equivalence Function-germs Versita Sp. z o.o. 1. Introduction Let f, g: (R n, 0) (R, 0) be two germs of continuous functions. The germs are called topologically K-equivalent there exist germs of homeomorphisms if h: (R n, 0) (R n, 0) and H : (R n R, 0) (R n R, 0) such that H(R n {0}) = R n {0} and the following diagram is commutative: (Id,f) (R n, 0) (R n R, 0) π (R n, 0) h H h (R n (Id,g), 0) (R n R, 0) π (R n, 0) where Id: R n R n is the identity map and π : R n R R n is the canonical projection π(x, t) = x, (x, t) R n R. lev.birbrair@pq.cnpq.br jcosta@ibilce.unesp.br alexandre.fernandes@ufc.br 338

2 S. Alvarez, L. Birbrair, J.C.F. Costa, A. Fernandes The K-equivalence of singularities of differential maps was introduced by J. Mather and it was considered as an important step of investigation of A-equivalence of these singularities. Two germs of continuous functions f, g: (R n, 0) (R, 0) are called topologically A-equivalent if there exist germs of homeomorphisms h: (R, 0) (R, 0) and k : (R n, 0) (R n, 0) such that f k = h g. In [6], T. Fukuda proved a so-called finiteness theorem about topological A-equivalence of polynomial function-germs. Namely, he proved that for any integer number K 0 the number of equivalence classes, with respect to topological A-equivalence, of polynomial function-germs of degree less than or equal to K is finite. M.Coste [5] generalized this result for functions definable in o-minimal structures. He proved that, for a given o-minimal structure A, the number of equivalence classes, with respect to topological A-equivalence, of the germs of the functions definable in A is countable. That is why the finiteness results of Fukuda (see also results of R.Benedetti and M.Shiota [4]) and Coste are true for topological K-equivalence. Moreover, finiteness theorem for topological K-equivalence remains true for germs of polynomial maps from R m to R p (see [2] for maps from R m to R 2 and [11] for maps from R m to R p ). Results of L.Birbrair, J.Costa, A.Fernandes and M.Ruas [1] show that a finiteness theorem is true for polynomial function-germs, with respect to bi-lipschitz K-equivalence. This result was also announced for polynomial maps from R m to R p in [11]. The finiteness results, above cited, are of the existence type and do not give any hint to the solution of the respective recognition problem. The paper is devoted to a classification, with respect to topological K-equivalence, of germs of functions definable in o- minimal structure. In Section 2, we present some special classes of piece-wise linear functions, so-called tent functions. We prove that these tent functions are normal forms for equivalence classes. Namely, we prove that, for each definable germ f, there exists a tent-function g such that f and g are topologically K-equivalent. The main technical tool of this section is a definable version of the main lemma of T.Nishimura [7]. Section 3 is devoted to a special case: analytic function-germs of two variables. For this case, the equivalence classes of K-decomposition defined in Section 2 can be described as equivalence classes of finite collections of elements 1, 0 or 1 by cyclic permutations. These classes are called K-symbols. We say that K-symbol is analytic if the number of elements is even and there are no elements equal to 0. By the results of Section 2, K-symbol is a complete invariant of definable function-germs, with respect to topological K-equivalence. In this section we consider so-called recognition problem. It is easy to see that a K-symbol of a germ of an analytic function is analytic. The question is the following. Let us fix an analytic K-symbol. Does there exist a germ of a polynomial function, such that the K-symbol of this germ of this function coincides to that given K-symbol? We give a positive answer to this question. As an application of our method we present normal forms for analytic functions with an isolated critical point. These normal forms were discovered by A. Prishlyak [9] for the problem of topological classification of germs of smooth functions. We would like to thank Maria Ruas, Ilia Itenberg, Eugenii Shustin for very interesting discussions and anonymous referee for usefull suggestions. 2. Tent functions Let M R n be the boundary of a n dimensional convex polytope B of R n, such that the origin belongs to int(b). Let Z M be a closed subset such that: Z is a union of some faces of M; Codimension of Z in M is not equal to zero. Let {U i } be the set of connected components of M Z. The collection {U i } is called a K-decomposition of M and the set Z is called the zero-locus of {U i }. Let M be a norm in R n such that the polytope M is realized as the unit sphere S n 1. Let Z = {x R n {0} : x x M Z} and Ũ i = {x R n {0} : x x M U i }. 339

3 Topological K-equivalence of analytic function-germs The collection {Ũi} is called the K-decomposition of R n associated to {U i } and the set Z is called the zero-locus of {Ũi}. Let M 1, M 2 be two boundaries of polytopes B 1 and B 2, Z 1, Z 2 be two zero-loci and {U 1 i }, {U 2 j } be two K-decompositions. The K-decompositions are called combinatorially equivalent if there exist triangulations of pairs (M 1, Z 1 ), (M 2, Z 2 ) and a simplicial isomorphism between these triangulations. A function T i : R n R is called an elementary tent function associated with {U i } if it has the following form: T i (x) = { dist(x, Ũ i ) se x Ũi, 0 otherwise. The functions i a i T i, where a i is equal to 1, 0 or 1, are called tent functions associated with {U i }. Let {Ũi} and {Ṽj} be K-decompositions of R n. Let α, β be two tent functions associated with Ũi and Ṽj respectively. The functions α and β are called combinatorially equivalent if there exists a germ of simplicial isomorphism h: (R n, 0) (R n, 0) such that, for each Ũ i, there exist Ṽ j such that h(ũ i ) = Ṽ j and sign[α(x)] = sign[β(h(x))]. Clearly, if two tent functions are combinatorially equivalent then K-decompositions of the corresponding polytopes are combinatorially equivalent. Definition 2.1. Let A be an o-minimal structure on R. Let f, g: (R n, 0) (R, 0) be two germs of definable in A continuous functions. The germs are called topologically K-equivalent in A if there exist germs of definable in A homeomorphisms h: (R n, 0) (R n, 0) and H : (R n R, 0) (R n R, 0) such that H(R n {0}) = R n {0} and the following diagram is commutative: (Id,f) (R n, 0) (R n R, 0) π (R n, 0) h H h (R n (Id,g), 0) (R n R, 0) π (R n, 0) where Id: R n R n is the identity map and π : R n R R n is the canonical projection π(x, t) = x, (x, t) R n R. Theorem 2.1. Let A be an o-minimal structure on R. 1. Let f : (R n, 0) (R, 0) be a germ of a definable in A continuous function. Then there exist a K-decomposition of R n and a tent function α : (R n, 0) (R, 0) such that f and α are topologically K-equivalent in A. 2. Two tent functions are topologically K-equivalent in A if and only if they are combinatorially equivalent. We use the following version of Nishimura s Lemma [7]. Lemma 2.1. Let A be an o-minimal structure on R. Let U be a neighbourhood of 0 R n. Let f and g be two germs of definable in f(x) A continuous functions. If g(x) > 0, for each x U g 1 (0), then f and g are topologically K-equivalent in A. 340

4 S. Alvarez, L. Birbrair, J.C.F. Costa, A. Fernandes Remark 2.1. In the original version, see [7], the lemma is proved for some continuous maps. But it is easy to see that the arguments are definable in A and the homeomorphism constructed by Nishimura are definable in A. Proof of Theorem Let f : (R n, 0) (R, 0) be a germ of a definable in A continuous function. Let us consider a triangulation of the pair (R n, f 1 (0)). Let (M, Y ) be a link of this triangulation at 0. Let cy be a cone over Y considered as a union of all simplexes of this triangulation, such that {0} is a vertex and all other vertices belong to Y. By [10], there exists a germ of a definable in A homeomorphism h: (R n, cy ) (R n, f 1 (0)). Let Z be a union of all edges of Y with codimension more than or equal to zero. Let {U i } be a K-decomposition corresponding to M and Z. Let f(x) = f(h(x)). Let α = a i T i be a tent function with a i = sign[ f Ũi ]. By Nishimura s i Lemma, f and α are topologically K-equivalent in A. 2. Let α and β be two tent functions associated to different K-decompositions of R n. Suppose that α and β are topologically K-equivalent. Then, by the definition of topological K-equivalence, a definable in A homeomorphism H : R n R R n R maps the graph of α to the graph of β. Since H is a definable homeomorphism, it can be triangulated. A triangulation of H makes a combinatorial equivalence between α and β. Conversely, let α and β be two combinatorially equivalent tent functions which are associated with the following K-decompositions of R n {Ũi} and {Ṽ i } respectively. Let h: (R n, 0) (R n, 0) be a simplicial isomorphism such that, for each Ũ i, there exist Ṽ j such that h(ũ i ) = Ṽ j and sign[α(x)] = sign[β(h(x))]. By Lemma 2.1, α and β h are topologically K-equivalent. Finally, since h is a homeomorphism, α and β are topologically K-equivalent too. 3. Analytic function-germs of 2 variables and K-symbols A K-decomposition of R 2 can be described as a finite collection of rays with initial point (0, 0) R 2. The sets Ũ i are sectors between these rays. An equivalence class of tent functions by a combinatorial equivalence described in Section 2 is a function η: {U i } { 1, 0, 1}. Figure 1. In other words, this invariant can be described as an equivalence class of finite collections of elements 1, 0 or 1 by cyclic permutations. An equivalence class η described above is called a K-symbol. A K-symbol is called analytic if 1. the number of sectors is even; 2. for all i, η(i) 0. Clearly, if a function-germ f : (R 2, 0) (R, 0) is analytic and not constant then the corresponding K-symbol is analytic, because an analytic non-zero function cannot be identically equal to zero on an open subset. We say that a K-symbol η admits an algebraic realization if there exists a polynomial f(x, y) of two variables, with f(0, 0) = 0, such that the K-symbol of the germ of this polynomial at 0 R 2 is equal to η. 341

5 Topological K-equivalence of analytic function-germs Theorem 3.1. Every analytic K-symbol admits an algebraic realization. We need the following lemma. Lemma 3.1. Let γ 1, γ 2 : [0, ε) R 2 be two semialgebraic arcs, such that γ 1 (0) = γ 2 (0) = 0 and the tangent vectors at 0 to γ 1 and γ 2 are the same. Then there exists a polynomial p(x, y) such that: 1. the germ of set {(x, y) R 2 : p(x, y) = 0} at 0 R 2 is a curve with two half-branches γ 1 and γ 2 ; 2. the curve {(x, y) R 2 : p(x, y) = 0} have a singular point at 0 R 2 ; 3. the unit tangent vectors to γ 1 and γ 2 at 0 R 2 are the same and equal to the unit tangent vector to γ 1 at 0 R 2 ; 4. for small t 0 we have p(γ 1 (t)) > 0 and p(γ 2 (t)) > 0; 5. the germ of the set {(x, y) R 2 : p(x, y) 0} is bounded by the curves γ 1 γ 2. Figure 2. Proof. Let Γ 1 and Γ 2 be the Zariski closures of the arcs γ 1 and γ 2 respectively. By the real algebraic version of the embedded uniformization of singularities (see [3]) there exist a two-dimensional real algebraic variety S and a real birational blow-up map π : S R 2 such that the strict transforms of the curves Γ 1 and Γ 2 are nonsingular and have a normal crossing intersection at S. Let β 1,..., β k be the connected real algebraic curves representing the exceptional divisor of the map π. Let V R 2 be the subset bounded by the arcs γ 1 and γ 2 and the small circle S 1 (0, ε). The set π 1 (V ) is bounded by the arcs γ 1 = π 1 (γ 1 ) and γ 2 = π 1 (γ 2 ), γ 1 (0) γ 2 (0), some arcs β ij belonging to the curves β i and the inverse image of the curve x 2 + y 2 = ε 2. The map π can be chosen in such a way that the arcs β ij are not tangent to the arcs γ 1 and γ 2. Let us choose a real algebraic curve M such that M π 1 (V ), M is tangent to one of the arcs β ij and M does not intersect to the others arcs β ij. For example, M can be chosen as a small circle in the corresponding local coordinates, tangent to one of the arcs β ij and not intersecting to the other arcs from the family { β ij }. By the construction, π(m) is an algebraic subset of R 2. Let p(x, y) be a polynomial defining π(m). Since 0 π(m), the set {(x, y) R 2 : p(x, y) 0} will be locally bounded by arcs γ 1 and γ 2 (notice that M does not intersect other curves from family { β ij }), satisfying the lemma. We say that an analytic K-symbol η is alternative if, for each i, η(i) η(i + 1). 342

6 S. Alvarez, L. Birbrair, J.C.F. Costa, A. Fernandes Claim 3.1. Let η be an analytic alternative K-symbol. Then η admits the following algebraic realization: p(x, y) = (x + y)(2x + y) (sx + y). Proof. The set {(x, y) R 2 : p(x, y) = 0} is the collection of the lines l i = {(x, y) R 2 : ix + y = 0}. Let Ũi, Ũi+1 be two sectors such that Ũi Ũi+1 is a half-line of the line l i. Observe that the function ix + y have different signs in int(ũi) and in int(ũi+1). So, the function p also have different signs in int(ũi) and in int(ũi+1). Thus, the K-symbol of p is alternative. An analytic K-symbol η is called double alternative if, for each i, η(i) η(i + 2). The corresponding sequence looks like the following 1, 1, 1, 1, 1, 1 Claim 3.2. An analytic double alternative K-symbol admits the following algebraic realization: p(x, y) = (x + y)(2x + y) 2 (3x + y)(4x + y) 2 (2sx + y) 2. Figure 3. Proof. Let θ S 1. Then, for all r > 0, we have sign[p(rθ)] = sign[p(θ)]. Suppose that θ moves along S 1. Then p(θ) does not change the sign when θ crosses l i for i even, and p(θ) change the sign when θ crosses l i, for i odd, where : ix + y = 0 (see figure 4). Thus, the K-symbol of p is double alternative. l i Proof of Theorm 3.1. By the definition of analytic K-symbol, for an analytic K-symbol the number of sectors is even, that is, 2r, for some integer r > 0. We use induction on r. If r = 1 the statement is trivial. In fact, a sequence ( 1, 1) is realized for example by the function p(x, y) = x, a sequence (1, 1) is realized by p(x, y) = x 2 and a sequence ( 1, 1) is realized by p(x, y) = x 2. Let us suppose that all K-symbols, for all r r 0, are algebraically realized. Consider now a sequence with 2r elements. If the sequence is double alternative, then it is realized (cf Claim 3.2). Thus we can suppose that the sequence is not double alternative and, thus there exist three consecutive elements η(i), η(i + 1), η(i + 2) such that η(i) = η(i + 2). Let us consider another K-symbol obtained from the first one by taking these elements away and putting η(i) instead of them: Figure

7 Topological K-equivalence of analytic function-germs By the induction hypotheses, the new configuration is algebraically realized. Let p 1 (x, y) be a germ of a real polynomial realizing this new K-symbol. Let γ 1 and γ 2 be the semialgebraic arcs bounding the area corresponding to η(i). By Lemma 3.1, there exists a polynomial p(x, y) satisfying the conditions of the lemma and positive outside the area bounded by the curve {(x, y) R 2 : p(x, y) = 0}. Thus, the realization p of our K-symbol can be obtained as a product p = p 1.p or p = p 1.p 2. Corollary 3.1. Let f(x, y) be an analytic function at 0 R 2. Then there exists a polynomial p(x, y) such that the germ of f at 0 R 2 is topologically K-equivalent to the germ of p at 0 R 2. The following result, which was obtained in [9], gives normal forms for all germs of functions (R 2, 0) (R, 0) with isolated critical point. In fact, here we include an elementary proof for the case of K-equivalence of the result of A. Prishlyak for topological equivalence. Corollary 3.2. Let f(x, y) be a germ of a definable in A function at 0 R 2, such that f 1 (0) Sing(f) = {0} and 0 is not an extreme point of f. Then f is topologically K-equivalent to a product of linear functions p(x, y) = (x + y)(2x + y) (sx + y). Proof. Let γ be a half-branch of the curve {(x, y) R 2 : f(x, y) = 0}. Since f 1 (0) Sing(f) = {0}, (γ {0}) Sing(f) = and the function f changes a sign if one crosses γ. Thus, the K-symbol of f is alternative. By the result of Claim 3.1, f is topologically K-equivalent to a product of linear functions p(x, y) = (x + y)(2x + y) (sx + y). Remark 3.1. It is clear that the normal forms p(x, y) = (x + y)(2x + y) (sx + y) given above are topologically K-equivalent to the normal forms re(z s ) (z C) given by A. Prishlyak in [9]. A map-germ is called Topologically K-determined when there exists an integer k N such that any map-germ g with j k (f) = j k (g) is topologically K-equivalent to f. These map-germs were amply studied by T. Nishimura in [8]. In [12], it is proved that a function-germ f is topologically K-determined if, and only if, f 1 (0) Sing(f) = {0}. Corollary 3.3. Two topologically K-determined germs are topologically K-equivalent if and only if their zero-sets have the same number of half-branches. Acknowledgements Research of the second author is supported under CNPq grant N /93-2. Research of the third author is supported under FAPESP grant N 2007/ Research of the fourth author is supported under CNPq grant N / and FUNCAP/CNPq/PPP. This paper was revised during my posdoctoral stage at Universidade de São Paulo. 344

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