Bilipschitz triviality of polynomial maps

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1 CADERNOS DE MATEMÁTICA 04, 7 26 May (2003) ARTIGO NÚMERO SMA#59 Bilipschitz triviality of polynomial maps Alexandre Fernandes Universidade Federal do Ceará, Centro de Ciências, Departamento de Matemática, Av. Humberto Monte, s/n Campus do Pici - Bloco 94 Fortaleza, Ceará. alex@mat.ufc.br Humberto Soares Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo - Campus de São Carlos, Caixa Postal 668, São Carlos SP humberto@icmc.usp.br URCA - Universidade Regional do Carirí, Curso de Matemática, Cep , Juazeiro do Norte, Ceará. We give in this paper a condition on θ : R n, 0 R p, 0, in terms of the Newton filtration, for the deformation f t = f + tθ to be bilipschitz trivial, where f : R n, 0 R p, 0 is a non-degenerate polynomial map. May, 2003 ICMC- USP. INTRODUCTION In the sixties S. Lojasiewicz obtained some metric properties of singular sets which indicated a new route to singularity theory. Twenty years later, T. Mostowski consolidated the indicatrizes of S. Lojasiewicz by presenting the following result: The Mostowski s theorem: The family of all complex algebraic sets of complexity bounded by some number k has a finite number of bilipschitz equivalence classes (see [5]). In [6], A. Parusinski presents a version of the Mostowiski s theorem for real singular sets; thereafter the study of the singular sets from the metric viewpoint (metric theory of singularities) gained substancial interest. One can see the importance of Mostowski-Parusinski theorem by considering the following Whitney example: X t = {(x, y) R 2 / xy(x + y)(x ty) = 0}, 0 < t <. Any two members X t, X t2 of this family, with t t 2, are not differentiable equivalents. However, they are bilipschitz equivalents. 7 Sob a supervisão CPq/ICMC

2 8 A. FERNANDES AND H. SOARES In [4] Henry and Parusinski gave the following example: f t (x, y) = x 3 + y 6 3t 2 xy 4 which satisfies the following: for any t s (0, 2 ) there is no bilipschitz map ϕ : C2, 0 C 2, 0 such that f t = f s ϕ. This turns the metric study of maps more interesting. Motivated by the above result, Fernandes and Ruas [3] obtained estimates to the bilipschitz determinacy of weighted homogeneous analytic function-germs. In this paper we use the concept of non-degeneracy with respect to some Newton polyhedron. This concept is used to solve localized equations using controlled vector fields and obtain bilipschitz triviality. We obtain estimates for the filtration of a map-germ θ : R n, 0 R p, 0 in such way that the deformation f t = f + tθ, of a non-degenerated map-germ f : R n, 0 R p, 0, is bilipzchitz trivial. This result complements that given in [3]. 2. THE NEWTON FILTRATION In this section we define the Newton polyhedron and the Newton filtration associated to a matrix A of rational numbers (see [], [0]). Let A = (a j i ), i =,..., n and j =,..., m be a matrix of non-negative rational numbers. We denote the rows and the columns of A respectively by: a i = (a i,..., a m i ), i =,..., n; a j = (a j,..., aj n), j =,..., m. The support, the Newton polyhedron and the Newton boundary of A are denoted respectively by: Supp(A) = {a j, j =,..., m}; Γ + (A) = Convex hull in R n of the set Supp(A) + R n +; Γ(A) = Union of all compact faces of Γ + (A). We call A a matrix of vertices if supp(a) = set of the vertices of Γ + (A). We say that A is a Newton matrix if A is a matrix of vertices and Γ + (A) is a Newton polyhedron in the sence of [2], that is, the intersection of Γ + (A) with each coordinate axis is not empty. If p R +, we denote p j i = paj i and pj = pa j. Hereafter we fix a Newton matrix A. Definition 2.. as The control function ρ(x) of the Newton polyhedron Γ + (A) is defined m ρ(x) = j= x 2pj 2p m = j= x 2pj x 2pj 2 2 x 2pj n n 2p. Sob a supervisão da CPq/ICMC

3 BILIPSCHITZ TRIVIALITY 9 We can choose p large enough such that the numbers 2p j i are integers and ρ 2p is a polynomial. As A is a Newton matrix, we can consider A = (a j i ) with aj = (0,..., 0, a j i, 0,..., 0) and a j i > 0, j =,..., n, that is, the first n n block of A is a diagonal matrix. For example for n = 2 ( ) 2 0 A =. 0 5 Definition 2.2. Let A be a Newton matrix and d a non-negative rational number. The matrix da is a Newton matrix. We define and denote: Γ + (ρ d ) := Γ + (da); Γ(ρ d ) := Γ(dA). Definition 2.3. Let f : R n, 0 R, 0 be a real analytic function-germ defined by f(x) = ν c ν x ν, where ν = (ν,..., ν n ). We say that f is an A-form of degree d if ν Γ(ρ d ) for all c ν 0 in the taylor series of f. Definition 2.4. Let f : R n, 0 R, 0 be a real analytic function-germ defined by f(x) = H d (x) + + H l (x) + where H i are A-forms of degree i. We say that 0 is an Γ + (ρ d )-isolated point of f if, for each compact face γ of Γ(ρ d ), the equation has not solution in (R {0}) n. f γ (x) = 0 Lemma 2. ([0], p. 525). Let f : R n, 0 R, 0 be a real analytic function-germ defined by f(x) = H d (x) + H l (x) +. If f admits 0 as an Γ + (ρ d )-isolated point, then there exists a positive real number c 2 such that f(x) c 2 ρ(x) d Sob a supervisão CPq/ICMC

4 20 A. FERNANDES AND H. SOARES in a neighbourhood of the origin in R n. Let Γ + (A) be a Newton polyhedron. We denote by R n the dual space of R n and for each a = (a,..., a n ) R n +, α = (α,..., α n ) R n + we denote by a, α = a α + + a n α n, l(α) = min{ a, α / a Γ + (A)}, γ(α) = {a Γ + (A) / a, α = l(α)}. The vector α is called a primitive integer vector if α is the vector with minimum length in C(α) (Z n + {0}), where C(α) is the half ray emanating from 0 and passing through α. Let λ : R, 0 R, 0 a real analytic funtion. We can write λ(t) = a k t k + a k+ t k+ +, with a k 0. We say that λ(t) is equivalent to t k and denote by λ(t) t k. Lemma 2.2. Let f : R n, 0 R, 0 be an analytic function-germ. If supp(f) Γ + (ρ d ) there exists c > 0 such that, in a neighbourhood of the origin, that is, f(x) ρ(x) d is bounded. f(x) c ρ(x) d, Proof. It is suficient to prove that for all x a Γ + (ρ d ) there exists a real number c > 0 such that ρ d c x a in a neighbourhood of the origin. Let us make this by contradiction. Suppose that for all c > 0, ρ d < c x a. Then, 0 is in the hull of the set X := {(x, c) / ρ(x) d < c x a }. As X is semi-analytic, by the curve selection lemma, there exists an analytic curve γ : (0, ɛ] X, with γ(0) = 0, γ(t) = (λ (t),..., λ n (t), λ n+ (t)) and λ (t) t α,..., λ n (t) t α n e λ n+ (t) t β. Then, ρ d (γ(t)) < t β λ(t) a, with λ(t) = (λ (t),..., λ n (t)), give us j t 2pα a j t 2pα n a j n d 2p < t β.t α a t α na n < t α a t α na n Sob a supervisão da CPq/ICMC

5 BILIPSCHITZ TRIVIALITY 2 therefore j t 2p α,aj d 2p < t α,a and we obtain that α, a < inf j { α, daj } what it contradicts the fact that γ(α) is a face of the polihedron Γ + (ρ d ) and therefore contain some da j as a vertex. Definition 2.5. Let f : R n, 0 R, 0 be a real analytic function-germ and v k = (v k,..., v k n), k =,..., r, the primitive vectors of the compact faces γ (of dimension n ) of Γ(A). We define the Newton filtration of f with respect to Γ + (A) by fil(f) := inf{ϕ(α) / α Supp(f)} where and M = l.c.m.{l(v k )}. ϕ(α) = { } min r M k= l(v k ) α, vk 3. BILIPSCHITZ TRIVIALITY The aim of this section is to give a sufficient condition for the bilipschitz triviality of families of map-germs. Definition 3.. Let λ R be a positive number. A mapping φ : U R n R p is called λ-lipschitz if it satisfies: φ(x) φ(y) λ x y, x, y U. Let f t : R n, 0 R p, 0 t I (an interval in R) be a smooth family of smooth map-germs, that is, there exist a neighbourhood U of 0 in R n and a smooth map F : U I R p such that F (0, t) = 0 and f t (x) = F (x, t), t I, x U. Definition 3.2. We call f t bilipschitz trivial when there exist t 0 I, λ > 0 and ϕ : U I U I of type ϕ(x, t) = (φ t (x), t), with φ t (x) λ-lipschitz and its inverse λ -Lipschitz, such that f t φ t = f t0 for all t I. We fix a Newton filtration associated to Γ + (A). We define the numbers { } { } M M R = max max j i l(v j ) vj i and r = min min j i l(v j ) vj i Sob a supervisão CPq/ICMC

6 22 A. FERNANDES AND H. SOARES where M = l.c.m.{l(v j )} and v j = (v j,..., vj n) are the associated primitive vectors of the compact faces (of dimension n ) of Γ(A). Let h : R n, 0 R, 0 be a function-germ such that supp(h) Γ + (ρ d ), it follows from Lemma 2.2 that h(x) is bounded. In fact, if the filtration of h is sufficiently higher that the ρ(x) d ( ) filtration of ρ d h, it is possible to prove that Grad is bounded, hence h is Lipschitz. ρ d ρ d Therefore we have the following result Proposition 3.. If fil(h) d.m + R then h(x) ρ(x) d is Lipschitz. Proof. Let f(x) = h(x) ρ(x) d, then Grad(f(x)) = ρ(x) 2d (ρd.grad(h) h.grad(ρ d )), fil(ρ d.grad(h) h.grad(ρ d )) fil(h) + fil(ρ d ) R d.m + fil(ρ d ) = fil(ρ 2d ). It follows by Lemma 2.2 that Grad(f) is bounded, therefore f is Lipschitz. Let f : R n, 0 R p, 0 be an analytic map-germ and let NR f := I M 2α I I where M I denotes a p p minor of df, I = {i,..., i p }, α I = α/s I, with s I := fil(m I ) and α := l.c.m.{s I }. Let suppose that NR f = H D + + H L, where H i are A-forms of degree i, with 0 an Γ + (ρ D )-isolated point of NR f. It follows by Lemmas 2.2 and 2. that and N Rf H D + + H L c D ρ D + + c L ρ L (c D + + c L )ρ D Hence, there exist c, c 2 > 0 such that N Rf cρ D. c ρ D N Rf c 2 ρ D. Suppose that f t = f + tθ is a deformation of f with fil(θ i ) > fil(f i ). Then, defining NR f t := I M 2α I t I we have N Rf t = N Rf + tθ, with fil(θ) > fil(n Rf). Therefore N R f N R f t + Θ, 0 t. Sob a supervisão da CPq/ICMC

7 BILIPSCHITZ TRIVIALITY 23 Now, there exists c > 0 such that c ρ D NR f N R f t + Θ and since fil(θ) > fil(nr f) we obtain lim x 0 Θ/ρD = 0. Hence Now, c ρ D N Rf t, N Rf t N Rf + Θ c 2 ρ D + Θ (c 2 + c 3 )ρ D. From this, we obtain the following result. Lemma 3.. Let f : R n, 0 R p, 0 be a polynomial map-germ. Suppose that NR f := I M 2α I I = H D + + H L admits 0 as an Γ + (ρ D )-isolated point. Then, if f t = f + tθ is a deformation of f with fil(θ i ) > fil(f i ), there exist c, c 2 > 0 such that c ρ D N Rf t c 2 ρ D. Theorem 3.. Let f : R n, 0 R p, 0 be a polynomial map-germ. Suppose that NR f := I M 2α I I = H D + + H L admits 0 as an Γ + (ρ D )-isolated point. If f t = f + tθ is a deformation of f with fil(θ i ) fil(f i ) + R r, then f t is bilipschitz trivial. Proof. Let be M ti a p p minor of df t, I = (i,..., i p ) (,..., n). Then, there exists a vector field W I associated to M ti, such that f t t M t I = df(w I ), where W I = n i= w i, with x i { wi = 0, if i I w im = p j= N ji m ( f t t ) j, if i m I f j x i m and N jim is the (p ) (p ) minor cofactor of in df. (See [7] for more details). Let W R := I M 2α I I W I. We have NR f t f t t = df t (W R ) and fil(w R ) = min{fil(m 2α I I ) + fil(w I )} min{2α fil(m I ) + fil(n jim ) + fil(θ j )} min{2α fil(m I ) + fil(m I ) fil( f j x im ) + fil(θ j )} min{2α (fil(f j ) r) + fil(θ j )} 2α + R. Let V : R n R, 0 R n R, 0 be the vector field V (x) = W R NR f. t Sob a supervisão CPq/ICMC

8 24 A. FERNANDES AND H. SOARES It follows from Lemma 3. and Proposition 3. that V is Lipschitz. The equation f t t (x, t) = (df t) x (V (x, t)) implies the bilipschitz triviality of the family f t in a neighbourhood of t = 0. As the same argument holds in a neighbourhood of t = t 0, t 0 [0, ], the proof is completed. 4. EXAMPLES Example Let f : R 2 R 2 be f(x, y) = (xy, x 2b+2 y 2b x 2r y 2s ), where r + s = b, br + (b + )s < (b + )b, r + 2s = b + and r > s. Then, the 2 2 minor of df is M = 2((b + )x 2b+2 + by 2b + (r s)x 2r y 2s ) Let A be the Newton matrix A = b 0 r (b+)b 0 b+ s (b+)b The control is ρ(x, y) = (x 2b+2 + y 2b + x 2r y 2s ) 2b(b+) Observe that M = H 2b(b+) where H 2b(b+) = 2((b + )x 2b+2 + by 2b + (r s)x 2r y 2s ) is an A-form of degree 2b(b + ). We have that v = (, ), v 2 = (, 2), and l(v ) = 2b, l(v 2 ) = 2b + 2. Therefore m.m.c{l(v ), l(v 2 )} = 2b(b + ) and fil(xy) = min{(b + ) (, ), (, ), b (, 2), (, ) } = 2b + 2, fil(y 2b ) = min{(b + ) (, ), (0, 2b), b (, 2), (0, 2b) } = 2b(b + ), fil(x 2r y 2s ) = min{(b + ) (, ), (2r, 2s), b (, 2), (2r, 2s) } = 2b(b + ), fil(x 2b+2 ) = min{(b + ) (, ), (2b + 2, 0), b (, 2), (2b + 2, 0) } = 2b(b + ). Hence, fil(xy) = 2b + 2, fil(x 2b+2 y 2b + x 2r y 2s ) = 2b(b + ), and we have R = 2b, r = b. Now, let be θ = x b y and θ 2 = x 2b+ y. Then, fil(θ ) = b(b + ) fil(f ) + R r = 3b + 2 fil(θ 2 ) = b(2b + 3) = fil(f 2 ) + R r = b(2b + 3) Sob a supervisão da CPq/ICMC

9 BILIPSCHITZ TRIVIALITY 25 Therefore, the family f t (x, y) = f(x, y) + t(x b y, x 2b+ y) is bilipschitz trivial. The Briançon-Speder example [] Let f : K 3, 0 K, 0 defined by f(x, y, z) = x 5 + xy 7 + z 5, where K = R or C. When K = C, the family F (x, y, z, t) = x 5 + xy 7 + z 5 + ty 6 z is topological trivial, since the Milnor number µ(f t ) is constant for all t. Briançon and Speder showed in [] that the variety F (0) in C 4 is not equisingular along the parameter space 0 C at 0. A complete description of all equisingular deformations of f is given in [9]. The variety F (0), defined by F (x, y, z, t) = f(x, y, z) + tx a y b z c is equisingular along the parameter space at 0 if, and only if, the monomial x a y b z c is in the Newton polyhedron Γ + ({(5, 0, 0), (0, 8, 0), (0, 0, 5), (, 7, 0)}). We consider here the analogous question for the real family F : R 3 R, 0 R, 0; F (x, y, z, t) = f(x, y, z) + tx a y b z c. Fernandes and Ruas in [3] showed that if a + 2b + 3c 7 then the family is equisingular along of the parameter space at 0. Let A = Then N R f := M 2α + M 2α M 2α 3 3 admits 0 as an Γ + (ρ 84 )-isolated point. The primitive vectors of the compact faces are v = (6, 3, 2), v 2 = (2, 2, 2), and l(v ) = 84, l(v 2 ) = 28. In this case, since l.c.m.{84, 28} = 84, we have. ϕ(x a y b z c ) := min{ (a, b, c), (6, 3, 2), 3 (a, b, c), (4, 4, 7) } = min{6a + 3b + 2c, 2a + 2b + 2c}. Hence, ϕ(x 5 ) = 90, ϕ(xy 7 ) = 96 and ϕ(z 5 ) = 05. Therefore fil(f) = 90. Observe that R = 2 and r = 6. Then, for f t to be bilipschitz trivial it is sufficient that i.e. fil(x a y b z c ) fil(f) + R r, min{6a + 3b + 2c, 2a + 2b + 2c} = 05. Therefore the variety F (0) is equisingular along of the parameter space 0 R at 0. Remark: In the work [3] of Fernandes and Ruas cannot detect whether the deformations by the monomials xy 3 z 3, x 3 y 2 z 3, x 2 yz 4, xz 5, z 5, x 7 z 3 are equisingular. From the Theorem 3. we have that deformations by this monomials are Whitney equisingular. It is valuable to observe that the family f t (x, y, z) = f(x, y, z) + tx 7 is bilipschitz trivial (c.f. [3]), however here fil(x 7 ) = 02. Sob a supervisão CPq/ICMC

10 26 A. FERNANDES AND H. SOARES REFERENCES. J. Briançon & J. N. Speder, La trivialité topologique n implique pas le conditions de Whitney. C. R. Acad. Sc. Paris, t. 280 (975). 2. C. Bivià, M. J. Saia & T. Fukui, Newton filtrations, graded algebras and codimension of non-degenerate ideals. Math. Proc. Cambridge Philos. Soc., 33, no., (2002). 3. A. Fernandes & M. A. Ruas, bilipschitz determinacy of quasihomogeneous germs. Notas do ICMC-USP (2002). 4. J. -P. Henry & A. Parusinski, Existence of moduli for bilipschitz equivalence of analytic functions. Prépublicationes de l Universite d Angers, no. 37 (200). 5. T. Mostowski, Lipschitz equisingularity. Dissertationes Math., (Rozprawy Mat.) 243 (985). 6. A. Parusinski, Lipschitz properties of semi-analytic sets. Ann. Inst. Fourier (Grenoble), no. 4, 38 (988), Ruas, M. A. S. On the degree of C l -determinacy. Math. Scandinavica, 59 (986). 8. M. A. Ruas & M. J. Saia, C l -determinacy of weighted homogeneous germs. Hokkaido Math. Journal, Vol. 26 (997), M. J. Saia, The integral closure of ideals and the Newton filtration. J. Algebraic Geometry, Vol. 5 (996), O. M. A. Yacoub, Polyèdre de Newton et Trivialité en Famille. J. Math. Soc. Japan, vol. 54, no. 3 (2002).. A. N. Varchenko, Newton poliedra and estimation of oscillating integrals. Funct. Anal. Appl., Vol. 0 (976), Sob a supervisão da CPq/ICMC