Degree of complex algebraic sets under bi-Lipschitz ...

DEGREE OF COMPLEX ALGEBRAIC SETS UNDER BI-LIPSCHITZ HOMEOMORPHISMS AT INFINITY ALEXANDRE FERNANDES AND J. EDSON SAMPAIO Abstract. We address a metric version of Zariski’s multiplicity conjec-

arXiv:1706.06614v1 [math.AG] 20 Jun 2017

ture at infinity that says that two complex algebraic affine sets which are bi-Lipschitz homeomorphic at infinity must have the same degree. More specifically, we prove that relative multiplicities at infinity of complex algebraic sets in Cn are invariant under bi-Lipschitz homeomorphisms at infinity, we also show that the local metric version of Zariski’s multiplicity conjecture and that one at infinity are equivalent and we get a proof that degree of complex algebraic surfaces in C3 is invariant of the bi-Lipschitz equivalence at infinity.

1. Introduction Let f : (Cn , 0) → (C, 0) be the germ of a reduced holomorphic function at the origin. Let (V (f ), 0) be the germ of the zero set of f at origin. We recall the multiplicity of V (f ) at the origin, denoted by m(V (f ), 0), is defined as following: we write f = fm + fm+1 + · · · + fk + · · · where each fk is a homogeneous polynomial of degree k and fm 6= 0. Then, m(V (f ), 0) := m. In 1971 (see [21]), O. Zariski proposed the following problem: Question A If V (f ) is topologically equivalent to V (g) as germs at the origin 0 ∈ Cn , i.e. there exists a homeomorphism ϕ : (Cn , V (f ), 0) → (Cn , V (g), 0), then is it true that m(V (f ), 0) = m(V (g), 0)? Although many authors have presented several partial results to this question, it remains open. In order to know more about Zariski’s multiplicity question see, for example, [7, 8, 11] and [20]. By looking from a metric point of view, and in a more general setting, we have the following metric local version of Zariski’s multiplicity question: 2010 Mathematics Subject Classification. 14B05; 32S50. Key words and phrases. Bi-Lipschitz, Degree, Zariski’s Conjecture. The first named author was partially supported by CNPq-Brazil grant 302764/2014-7. 1

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ALEXANDRE FERNANDES AND J. EDSON SAMPAIO

˜ Question A1(d) Let X ⊂ Cn and Y ⊂ Cm be two complex analytic sets with dim X = dim Y = d. If their germs at 0 ∈ Cn and 0 ∈ Cm , respectively, are bi-Lipschitz homeomorphic, i.e. there exists a bi-Lipschitz homeomorphism ϕ : (X, 0) → (Y, 0), then is it true that their multiplicities m(X, 0) and m(Y, 0) must be equal? In order to answer this question above, we showed in [9] that it is enough to address such a question by considering X and Y homogeneous complex algebraic sets. Actually, this result is stated in [9] for complex algebraic hypersurfaces in Cn , however, the proof works for higher codimension complex algebraic subsets. In this same paper, we proved that the multiplicity of a complex analytic surface singularities in C3 is a bi-Lipschitz (embedded) invariant. ˜ Let us remark that Question A1(d) was approached by some authors and, as far as we know, it remains still open. For instance, G. Comte, in the paper [6], proved that the multiplicity of complex analytic germs in Cn (not necessarily codimension 1 subsets) is invariant under bi-Lipschitz homeomorphism with Lipschitz constant close enough to 1. Notice that the bi-Lipschitz homeomorphisms considered by G. Comte have some restrictions. More recently, W. Neumann and A. Pichon in [15] showed that the multiplicity is a bi-Lipschitz invariant in the case of normal complex analytic surface singularities. See [5] for a definition of multiplicity for higher codimension analytic germs in Cn . At this point, we finish this overview on local Zariski’s multiplicity question from the metric point of view and we start to consider complex algebraic sets by looking for invariants of their Lipschitz geometry at infinity. Let f : Cn → C be a reduced polynomial and X = V (f ). The degree of the polynomial f is an important integer number associated to X; it is called the degree of X. According to the next example, it is hopeless that degree of X = V (f ) comes as a topological invariant of the embedded subset X ⊂ Cn . Example 1.1. Let f, g : C2 → C be the folowing polynomials given by f (x, y) = y and g(x, y) = x2 −y. The mapping ϕ : C2 → C2 given by ϕ(x, y) = (x, x2 −y) is a polynomial automorphism (in particular it is a smooth diffeomorphism) such that ϕ(V (f )) = V (g). However, deg(V (f )) = 1 and deg(V (g)) = 2. In this paper, we deal with the following metric question: Question A1(d) Let X ⊂ Cn and Y ⊂ Cm be two complex algebraic sets with dim X = dim Y = d. If X and Y are bi-Lipschitz homeomorphic at infinity, in

DEGREE OF COMPLEX ALGEBRAIC SETS UNDER ...

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the sense that there exist compact subsets K1 ⊂ X, K2 ⊂ Y and a bi-Lipschitz homeomorphism ϕ : X \ K1 → Y \ K2 , then is it true that deg(X) = deg(Y )? We do not know a complete answer of Question A1(d), however, we showed in [10] that degree 1 comes as a bi-Lipschitz invariant at infinity of complex algebraic subsets (see Section 2, for a definition of degree for higher codimension algebraic sets in Cn ). The aims of the present paper are to prove that, ˜ for each d ∈ N, A1(d) and A1(d) are equivalent questions and to give some partial positive answers for them. Let us describe how this paper is organised. Section 2 is dedicated to present the notions of degree of complex algebraic subsets in Cn , tangent cones at infinity and, also, bi-Lipschitz homeomorphisms at infinity of such subsets. Section 3 is dedicated to prove the main results of the paper. In the first, we prove that relative multiplicities at infinity of complex algebraic subsets are invariant under bi-Lipschitz homeomorphisms at infinity (Theorem 3.1). Still in Section 3 we prove that Question A1 has a positive answer if, and only if, Question ˜ has a positive answer for homogeneous algebraic subsets (Theorem 3.3). A1 Finally, in Subsection 3.1, we do some applications of Theorem 3.3 as Theorem 3.4 which says that Question A1 has positive answer for algebraic hypersurfaces in Cn whose all irreducible components of their tangent cones at infinity have singular locus with dimension ≤ 1 and, as an immediate corollary of it, we obtain that degree of complex algebraic surfaces in C3 is a bi-Lipschitz invariant at infinity.

2. Preliminaries 2.1. Degree. Let us begin this subsection by recalling some basic facts about degree of complex algebraic sets. Let ι : Cn ֒→ Pn be the embedding given by ι(x1 , · · · , xn ) = [1 : x1 : · · · : xn ] and let p : Cn+1 \ {0} → Pn be the projection mapping given by p(x0 , x1 , . . . , xn ) = [x0 : x1 : · · · : xn ]. Remark 2.1. Let A be an algebraic set in Pn and X be an algebraic set in e = p−1 (A) ∪ {0} is a homogeneous complex algebraic set in Cn+1 Cn . Then A and the closure ι(X) of ι(X) in Pn is an algebraic set in Pn .

Definition 2.2. Let A be an algebraic set in Pn . We define the degree of A e 0), where m(A, e 0) is the multiplicity of A e at 0 ∈ Cn+1 . by deg(A) = m(A, Definition 2.3. Let X be a complex algebraic set in Cn . We define the degree of X by deg(X) = deg(ι(X)).

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Remark 2.4. Let f : Cn → C is a reduced polynomial and X = V (f ). Then, deg(X) = deg(f ). 2.2. Tangent cones. In this subsection, we set the exact notion of tangent cone that we will use along the paper and we list some of its properties. Definition 2.5. Let A ⊂ Rn be a unbounded subset. We say that v ∈ Rn is a tangent vector of A at infinity if there is a sequence of points {xi }i∈N ⊂ A such that lim kxi k = +∞ and there is a sequence of positive numbers {ti }i∈N ⊂ i→∞

R+ such that

1 xi = v. i→∞ ti Let C∞ (A) denote the set of all tangent vectors of A at infinity. This subset lim

C∞ (A) ⊂ Rn is called the tangent cone of A at infinity. Proposition 2.6 (Proposition 4.4 in [10]). Let Z ⊂ Rn be an unbounded semialgebraic set. A vector v ∈ Rn belongs to C∞ (Z) if, and only if, there exists a continuous semialgebraic curve γ : (ε, +∞) → Z such that lim |γ(t)| = +∞ t→+∞

and γ(t) = tv + o∞ (t), where g(t) = o∞ (t) means

lim g(t) t→+∞ t

= 0.

Let X ⊂ Cn be a complex algebraic subset. Let I(X) be the ideal of C[x1 , · · · , xn ] given by the polynomials which vanishes on X. For each f ∈ C[x1 , · · · , xn ], let us denote by f ∗ the homogeneous polynomial composed of the monomials in f of maximum degree. Proposition 2.7 (Theorem 1.1 in [14]). Let X ⊂ Cn be a complex algebraic subset. Then, C∞ (X) is the affine algebraic set V (hf ∗ ; f ∈ I(X)i). Among other things, this result above says that tangent cones at infinity of complex algebraic sets in Cn are complex algebraic subsets as well. 2.3. Bi-Lipschitz homeomorphism at infinity. Definition 2.8. Let X ⊂ Rn and Y ⊂ Rm be two subsets. We say that X and Y are bi-Lipschitz homeomorphic at infinity, if there exist compact e ⊂ Rm and a bi-Lipschitz homeomorphism φ : X \ K → subsets K ⊂ Rn and K e Y \ K.

We finish this subsection reminding the invariance of the tangent cone at infinity under bi-Lipschitz homeomorphisms at infinity. Proposition 2.9 (Theorem 4.5 in [10]). Let X ⊂ Rn and Y ⊂ Rm be unbounded semialgebraic subsets. If X and Y are bi-Lipschitz homeomorphic at


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infinity, then there is a bi-Lipschitz homeomorphism dϕ : C∞ (X) → C∞ (Y ) with dϕ(0) = 0. 2.4. Relative multiplicities at infinity. Let X ⊂ Cn be a complex algebraic set with p = dim X ≥ 1. Let X1 , · · · , Xr be the irreducible components of C∞ (X). Let π : Cn → Cp be a linear projection such that π −1 (0) ∩ (C∞ (X)) = {0}. Therefore, π|X : X → Cp (resp. π|C∞ (X) : C∞ (X) → Cp ) is a ramified cover with degree equal to deg(X) (resp. deg(C∞ (X))) (see [5], Corollary 1 in the page 126). In particular, π|Xj : C∞ (X) → Cp is a ramified cover with degree equal to deg(Xj ), for each j = 1, · · · , r. Moreover, the ramification locus of π|X (resp. π|C∞ (X) ) is a codimension ≥ 1 complex algebraic subset σ(X) (resp. σ(C∞ (X))) of Cp . Let us denote Σ = π −1 (σ(X)). Fix j ∈ {1, · · · , r}. For a point v ∈ Xj \ (C∞ (Σ) ∪ σ(C∞ (X))), let η, R > 0 such that Cη,R (v ′ ) = {w ∈ Cp | ∃t > 0; ktv ′ − wk ≤ ηt, ∀ t ≥ R} ⊂ Cp \ σ(X) ∪ σ(C(X)), ′ where v ′ = π(v). Thus, the number of connected components of π|−1 X (Cη,R (v )) ′ (resp. B = π|−1 Xj (Cη,R (v ))) is equal to deg(X) (resp. deg(Xj )). Moreover,

there exist a connected component V of B such that v ∈ V and a compact ′ subset K ⊂ Cn such that for each connected component Ai of π|−1 X (Cη,R (v )), ′ ∞ we have C∞ (Ai ) ∩ (Cn \ K) ⊂ π|−1 C∞ (X) (Cη,R (v )). Then, we denote by kX (v) to

be the number of connected components A′i s such that C∞ (Ai ) ∩(Cn \ K) ⊂ V . ∞ By definition, we can see that kX is locally constant and as Xj \ (C∞ (Σ) ∪ ∞ σ(C∞ (X))) is connected, kX is constant on Xj \(C∞ (Σ)∪σ(C∞ (X))). Thus, we ∞ ∞ ∞ ∞ define kX (Xj ) = kX (v). In particular, kX (w) = kX (v) for all w ∈ π −1 (v ′ ) ∩Xj

and, therefore, we obtain (1)

deg(X) =

r X

∞ kX (Xj ) · deg(Xj ).

j=0 ∞ Notice that kX does not depend of π.

3. Bi-Lipschitz Invariants at Infinity Before starting a proof of the bi-Lipschitz invariance of relative multiplicities at infinity, we do a slight digression to remind the notion of inner distance on connected Euclidean subsets.

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Let Z ⊂ Rℓ be a path connected subset. Given two points q, q˜ ∈ Z, we define the inner distance in Z between q and q˜ by the number dZ (q, q˜) below: dZ (q, q˜) := inf{length(γ) | γ is an arc on Z connecting q to q˜}. Theorem 3.1. Let X ⊂ Cn and Y ⊂ Cm be complex algebraic subsets, with pure dimension p = dim X = dim Y , and let X1 , . . . , Xr and Y1 , . . . , Ys be the irreducible components of the tangent cones at infinity C∞ (X) and C∞ (Y ) respectively. If X and Y are bi-Lipschitz homeomorphic at infinity, then r = s ∞ and, up to a re-ordering of indices, kX (Xj ) = kY∞ (Yj ), ∀ j.

e ⊂ Cm and Proof. By hypotheses there are compact subsets K ⊂ Cn and K e Let S = {nk }k∈N be a a bi-Lipschitz homeomorphism ϕ : X \ K → Y \ K. sequence of positive real numbers such that nk → +∞ and

ϕ(nk v) → dϕ(v), nk

where dϕ is a tangent map at infinity of ϕ like in Theorem 2.9 (for more details, see [10], Theorem 4.5). Since, dϕ is a bi-Lipschitz homeomorphism, we get r = s and there is a permutation P : {1, . . . , r} → {1, . . . , s} such that dϕ(Xj ) = YP (j) ∀ j. This is why we can suppose dϕ(Xj ) = Yj ∀ j up to a re-ordering of indices. Let π : Cn → Cp be a linear projection such that π −1 (0) ∩ (C∞ (X) ∪ C∞ (Y )) = {0}. Let us denote the ramification locus of π|X : X → Cp

and π|C∞ (X) : C∞ (X) → Cp

by σ(X) and σ(C∞ (X)) respectively. By similar way, we define σ(Y ) and e = π −1 (σ(Y )). σ(C∞ (Y )). Let us denote Σ = π −1 (σ(X)) and Σ ∞ Let us suppose that there is j ∈ {1, . . . , r} such that kX (Xj ) > kY∞ (Yj ).

Thus, given a unitary point v ∈ Xj \ (C∞ (Σ) ∪ σ(C∞ (X))) such that w = e ∪ σ(C∞ (Y ))), let η, R > 0 such that dϕ(v) ∈ Yj \ (C∞ (Σ) Cη,R (v ′ ) = {w ∈ Cp | ∃t > 0; ktv ′ − wk < ηt, ∀r ≥ R} ⊂ Cp \ σ(X) ∪ σ(C(X)) and Cη,R (w ′) = {w ∈ Cp | ∃t > 0; ktv ′ − wk < ηt, ∀r ≥ R} ⊂ Cp \ σ(Y ) ∪ σ(C(Y )), where v ′ = π(v) and w ′ = π(dϕ(v)). Therefore, there are at least two different connected components Vji and Vjl of π −1 (Cη,R (v ′ )) ∩ X and sequences {zk }k∈N ⊂ Vji and {wk }k∈N ⊂ Vjl such that tk = kzk k = kwk k ∈ S, lim t1k zk =


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lim t1k wk = v and ϕ(zk ), ϕ(wk ) ∈ Vejm , where Vejm is a connected component of π −1 (Cη,R (w ′)) ∩ Y .

Let us choose linear coordinates (x, y) in Cn such that π(x, y) = x. Claim. There exist a compact subset K ⊂ Cn and a constant C > 0 such that kyk ≤ Ckxk for all (x, y) ∈ Y \ K. If this Claim is not true, there exists a sequence {(xk , yk )} ⊂ Y such that lim k(xk , yk )k = +∞ and kyk k > kkxk k. Thus, up to a subsequence, one

k→+∞

yk k→+∞ kyk k

can suppose that lim

= y0 . Since

kxk k kyk k

0 such that dX (zk , wk ) ≥ Ctk , which is a contradiction. ∞ We have proved that kX (Xj ) ≤ kY∞ (Yj ), j = 1, · · · , r. By similar arguments, ∞ using that ϕ−1 is a bi-Lipschitz map, we also can prove kY∞ (Yj ) ≤ kX (Xj ),

j = 1, · · · , r.

3.1. Degree as a bi-Lipschitz invariant at infinity. The first application of our main results proved in the previous section is the bi-Lipshitz invariance of degree of complex algebraic curves in Cn . Theorem 3.2. Let X ⊂ Cn and Y ⊂ Cm be complex algebraic subsets, with pure dimension dim X = dim Y = 1. If X and Y are bi-Lipschitz homeomorphic at infinity, then deg(X) = deg(Y ). Proof. Let X1 , . . . , Xr and Y1 , . . . , Ys be the irreducible components of the tangent cones at infinity C∞ (X) and C∞ (Y ) respectively. Since dim X = dim Y = 1, we have that X1 , . . . , Xr and Y1 , . . . , Ys are complex lines. Thus, deg(X1 ) = · · · = deg(Xr ) = deg(Y1 ) = · · · = deg(Ys ) = 1 and using the Eq. 1, we get deg(X) =

r P

j=0

∞ kX (Xj ) and deg(Y ) =

Therefore, by Theorem 3.1, deg(X) = deg(Y ).

r P

j=0

∞ kX (Yj ).

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In the next, let us fix d ∈ N. Theorem 3.3. The statements below are equivalent. ˜ A1(d) Let X ⊂ Cn and Y ⊂ Cm be two complex analytic sets with dim X = dim Y = d. If their germs at 0 ∈ Cn and 0 ∈ Cm , respectively, are bi-Lipschitz homeomorphic, then m(X, 0) = m(Y, 0). A1(d) Let X ⊂ Cn and Y ⊂ Cm be two complex algebraic sets with dim X = dim Y = d. If X and Y are bi-Lipschitz homeomorphic at infinity, then deg(X) = deg(Y ). Proof. As we pointed out in the introduction of the paper; we know from ˜ [9] that statement A1(d) holds true if and only if it is true by considering just homogeneous complex algebraic sets. Let us remark that, if A ⊂ Cn is a homogeneous complex algebraic set, then deg(A) = m(A, 0). Indeed, let ι : Cn ֒→ Pn be the embedding given by ι(x1 , · · · , xn ) = [1 : x1 : · · · : xn ], let p : Cn+1 \ {0} → Pn be the projection mapping given by p(x0 , x1 , . . . , xn ) = e = p−1 (X) ∪ {0}. We know that deg(A) = [x0 : x1 : · · · : xn ], X = ι(A) and X e 0). However, since A is homogeneous, we get X e = C × A. Then, m(X, deg(A) = m(C × A, 0) = m(C, 0) · m(A, 0) = m(A, 0).

From now, we are ready to start the proof of the theorem. First, let us suppose that statement A1(d) is true. Since cones which are bi-Lipschitz homeomorphic as germs at their vertices are globally bi-Lipschitz homeomorphic, ˜ as was remarked in [16], it follows from observations above that A1(d) holds ˜ true as well. Second, let us suppose that A1(d) holds true. So, we are going to prove that A1(d) is true as well. Let X ⊂ Cn and Y ⊂ Cm be two complex algebraic sets with d = dim X = dim Y . Let us suppose that X and Y are e ⊂ Cm bi-Lipschitz homeomorphic at infinity. Then, there exist K ⊂ Cn and K e two compact subsets and a bi-Lipschitz homeomorphism ϕ : X \ K → Y \ K. Let us denote by X1 , . . . , Xr and Y1 , . . . , Ys the irreducible components of the

cones C∞ (X) and C∞ (Y ) respectively. It comes from Theorem 3.1 that r = s and the bi-Lipschitz homeomorphism dϕ : C∞ (X) → C∞ (Y ), up to re-ordering ∞ of indices, sends Xi onto Yi and kX (Xi ) = kY∞ (Yi ) ∀ i. Furthemore, dϕ(0) = 0.

By Proposition 2.7, the tangent cones at infinity C∞ (X) and C∞ (Y ) are homogeneous complex algebraic subsets. Thus, the irreducible components X1 , . . . , Xr and Y1 , . . . , Ys are homogeneous complex algebraic subsets as well. ˜ Since A1(d) is true, we have m(Xi , 0) = m(Yi , 0) ∀ i, hence deg(Xi ) = deg(Yi ) ∀ i. Finally, by using Eq. 1, we get deg(X) = deg(Y ) which give us that A1(d) is true.


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Let us denote by C1,∞ the set of all complex algebraic sets X ⊂ Cn , such that each component Xj of C∞ (X) satisfies dim Sing(Xj ) ≤ 1. Theorem 3.4. Let f, g : Cn → C be two polynomials. Suppose that V (f ) ∈ e ⊂ Cn and a bi-Lipschitz homeC1,∞ . Suppose there exist compact subsets K, K e such that ϕ(V (f ) \ K) = V (g) \ K. e Then omorphism ϕ : Cn \ K → Cn \ K V (g) ∈ C1,∞ and deg(V (f )) = deg(V (g)).

Proof. By the proof of Theorem 3.1, we can suppose that f and g are irreducible homogeneous polynomials. By Theorem 5.4 in [17] and by using observations in the very beginner of the proof of Theorem 3.1, it follows that deg(V (f ))) = deg(V (g)).

As a corollary of the theorem above, we have that degree is a bi-Lipschitz invariant at infinity in the case of complex algebraic surfaces in C3 . Theorem 3.5. Let f, g : C3 → C be two polynomials. Suppose there exist e ⊂ C3 and a bi-Lipschitz homeomorphism ϕ : C3 \ K → compact subsets K, K e such that ϕ(V (f ) \ K) = V (g) \ K. e Then deg(V (f )) = deg((V (g)). C3 \ K

Let us finish the paper showing that degree of polynomial functions and mappings are invariant for special bi-Lipschitz equivalences of them. Definition 3.6 (See [18]). We say that two polynomials f, g : Cn → Cm are e ⊂ Cn , rugose equivalent at infinity, if there are compact subsets K, K e such that constants C1 , C2 > 0 and a bijection ϕ : Cn \ K → Cn \ K (1)

1 kx C1

− yk ≤ kϕ(x) − ϕ(y)k ≤ C1 kx − yk, for all x ∈ Cn \ K and

y ∈ f −1 (0) \ K; (2)

1 kf (x)k C2

≤ kg ◦ ϕ(x)k ≤ C2 kf (x)k,

∀x ∈ Cn \ K.

Theorem 3.7. Let f, g : Cn → C be two polynomials. If f and g are rugose equivalent at infinity, then deg(f ) = deg(g). Proof. As it was pointed out by S. Ji, J. Kollar and B. Shiffman ([12], Remark 10), we know that if h : Cn → C is a polynomial non-constant, then there is a constant C > 0 such that (2)

dist(x, V (h))deg(h) ≤ C|h(x)|,

∀x ∈ Cn .

Moreover, deg(h) is the biggest number θ satisfying e dist(x, V (h))θ ≤ C|h(x)|,

∀x ∈ Cn ,

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e > 0. In fact, if θ > deg(h) and v 6∈ C∞ (V (h)), then we have that for some C there exists a sequence {tk } ⊂ (0, +∞) such that lim tk = +∞ and k→+∞

dist(tk v, V (h)) 6= 0. k→+∞ tk lim

Thus, dist(tk v, V (h))θ = +∞. k→+∞ |h(tk v)| e ⊂ Cn , constants C1 , C2 > 0 By hypotheses, there are compact subsets K, K e such that and a bijection ϕ : Cn \ K → Cn \ K lim

(i)

1 kx C1

− yk ≤ kϕ(x) − ϕ(y)k ≤ C1 kx − yk, for all x ∈ Cn \ K and

y ∈ V (f ) \ K; (ii)

1 kf (x)k C2

≤ kg ◦ ϕ(x)k ≤ C2 kf (x)k,

∀x ∈ Cn \ K.

Thus, by (i), we get dist(x, V (f )) ≤ C1 dist(ϕ(x), V (g)). Then, dist(x, V (f ))deg(g)

deg(g)

≤ C1 dist(ϕ(x), V (g))deg(g) (2) deg(g) ≤ C1 C|g(ϕ(x))| (ii)

deg(g)

≤ C1

CC2 |f (x)|.

Therefore, by remark made just after inequality 2, we get deg(g) ≤ deg(f ). Moreover, if we change ϕ by ϕ−1 and we argument in a similar way as before, we get deg(f ) ≤ deg(g). Therefore, deg(f ) = deg(g).

Definition 3.8. We say that two polynomial mappings F, G : Cn → Cm are bi-Lipschitz contact equivalent at infinity if there are compact subsets e ⊂ Cn , a constant C > 0 and a bi-Lipschitz homeomorphism ϕ : Cn \K → K, K

e such that Cn \ K

1 kG(x)k ≤ kF ◦ ϕ(x)k ≤ CkG(x)k, ∀x ∈ Cn \ K. C The notion of local Bi-Lipschitz contact equivalence was approached, for

instance, by [1, 2, 4] and [19]. Definition 3.9. Let F = (f1 , · · · , fm ) : Cn → Cm be a polynomial mapping. We define the degree of F by deg(F ) = max{deg(f1 ), · · · , deg(fm )}. Theorem 3.10. Let F, G : Cn → Cm be two polynomial mappings. If that F and G are bi-Lipschitz contact equivalent at infinity, then deg(F ) = deg(G). Proof. Let us denote X = {x ∈ Cn ; G(x) = 0} and Y = {x ∈ Cn ; F (x) = 0}. We have that X and Y are bi-Lipschitz homeomorphic at infinity. By Theorem 2.9 and Proposition 2.7, C∞ (X) and C∞ (Y ) are closed and bi-Lipschitz


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e ⊂ Cn , a homeomorphic sets. By hypotheses, there are compact subsets K, K e positive constant C and a bi-Lipschitz homeomorphism ϕ : Cn \ K → Cn \ K such that

1 kG(x)k ≤ kF ◦ ϕ(x)k ≤ CkG(x)k, C

∀x ∈ Cn \ K.

Let us suppose that deg(G) < deg(F ) = k. Let S = {nj }j∈N ⊂ N be a sequence such that nj → +∞ and

ϕ(nj v) → dϕ(v), nj

like in the Theorem 2.9. Moreover, dϕ : Cn → Cn is a bi-Lipschitz homeomorphism. Then, there is v ∈ Cn such that dϕ(v) ∈ Cn \ {x ∈ Cn ; F ∗ (x) = 0}, where F ∗ is the homogeneous polynomial mapping composed of the monomials in F of maximum degree. Therefore, kG(nj v)k kF ◦ ϕ(nj v)k ≤C , k nj nkj

∀nj ∈ S.

By taking j → +∞, we obtain kF ∗ (dϕ(v))k ≤ 0, which is a contraction. Then, deg(G) ≥ deg(F ) = k and using ϕ−1 instead of ϕ, we obtain the other inequality. Therefore, deg(G) = deg(F ).

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