Brasselet number and Newton polygons
Thais M. Dalbelo Luiz Hartmann






Brasselet number and Newton polygons]Brasselet number and Newton polygons



Department of Mathematics, 
	Universidade Federal de São Carlos (UFSCar),
	Brazil
thaisdalbelo@dm.ufscar.br, thaisdalbelo@gmail.com


Department of Mathematics, 
	Universidade Federal de São Carlos (UFSCar),
	Brazil
hartmann@dm.ufscar.br
http://www.dm.ufscar.br/profs/hartmann




Luiz Hartmann is Partially support by FAPESP: 2016/16949-8 and both 
authors
are partially supported by CAPES/PVE:88881.068165/2014-01 

[2010]Primary: 14M25,55S35; Secondary: 14B05, 32S05; 58K45


	
	


	We present a formula to compute the Brasselet number of f:(Y,0)→ (, 
	0) where Y⊂ X is a non-degenerate complete intersection in a 
	toric variety X. As applications we establish 
	several results concerning invariance of the Brasselet number for 
	families of non-degenerate complete intersections. 
	Moreover, when (X,0) = (ℂ^n,0) we derive sufficient conditions 
	to obtain the invariance of the Euler obstruction for families 	of 
	complete intersections with an isolated singularity which are contained in 
	X.  		 


[
    Luiz Hartmann
    December 30, 2023
=====================








§ INTRODUCTION


Given a germ of an analytic function 
f:(ℂ^n,0) → (ℂ,0) with an isolated critical point at 
the 
origin, an important invariant of this germ is its Milnor number <cit.>, 
denoted by μ(f). The Milnor number is considered as a central invariant, 
since 
it provides algebraic, topological and geometric information about the germ 
f. For instance, the Milnor number coincides with the number of Morse points 
of a morsefication of f.



Initially the Milnor number was associated to germs of analytic functions f 
: (ℂ^n, 0) → (ℂ, 0) with an isolated critical point, and 
consequently it was used to study isolated hypersurface singularities. However
this invariant is well defined in many others contexts, for example curves
<cit.>, isolated complete intersection singularities (ICIS) <cit.>, 
and determinantal varieties of codimension two <cit.>, to name just a few.



The local Euler obstruction was defined by MacPherson in <cit.> for 
the construction
of characteristic classes of singular complex algebraic varieties. 
Thereafter, it has been
deeply investigated by many authors such as Brasselet and Schwartz <cit.>, 
Dutertre <cit.>, Gaffney, Grulha and Ruas <cit.>, Gon­za­lez-Sprinberg 
<cit.>, Lê and Teissier <cit.>,  Matsui and Takeuchi <cit.>, among others.
We denote by (X,0) a germ of an analytic singular space embedded in 
ℂ^n and by f:(X,0) → (ℂ,0) a germ of an analytic 
function 
with an isolated critical point at the origin. Brasselet, 
Massey, Parameswaran and Seade <cit.> introduced an invariant associated 
to f 
called the Euler obstruction of f  and denoted by Eu_f,X(0). Roughly speaking, Eu_f,X(0) is the obstruction to 
extending a lifting of the conjugate of the gradient vector field of f as a 
section of the Nash bundle of (X , 0). This invariant is closed related 
with the local Euler obstruction of X, what explains its name.  



An important consequence of the definition given by MacPherson, is that the local Euler obstruction is a constructible function,    
which means that, it is constant along the strata of a Whitney stratification 
of X. This is essentially a consequence of the topological            
triviality of X on the Whitney strata. As a consequence, the local Euler 
obstruction 
does not depend on the Whitney stratification of X. The following 
Lefschetz-type formula was proved by 
Brasselet, Lê and
Seade <cit.>.










	Let (X,0) ⊂ (ℂ^n,0) be an equidimensional complex analytic singularity
	germ with a Whitney stratification {V_i}, then
	given a generic linear form L, there exists ε_0
	such that for any ε with 0<ε<
	ε_0, we have
	
    Eu_X(0)=∑_iχ(V_i∩ B_ε∩ 
    	L^-1(δ) ) · Eu_X(V_i),

	where χ is the Euler-Poincaré
	characteristic, B_ε:=B_ε(0) is the ball 
	with center the origin and radius ε, Eu_X(V_i) is the 
	value of the local Euler
	obstruction of X at any point of the stratum V_i,
 and 0 < |δ|≪ε≪ 1.


The previous theorem says that the local Euler obstruction, as a 
constructible 
function
on X, satisfies the Euler condition relatively to a generic linear function.



For the Euler obstruction of an analytic function f:(X,0) → 
(ℂ,0) with an isolated critical point at the origin, there is also 
a 
Lefschetz-type formula. This formula was proved in <cit.>. The purpose 
of the authors was to understand what prevents the
local Euler obstruction from satisfying the local Euler condition with respect to
functions which are singular at the origin.

	Let (X,0) ⊂ (ℂ^n,0) be an equidimensional complex analytic singularity
	germ with a Whitney stratification {V_i}, and let f : (X,0)  → 
	(ℂ,0) be a function with an isolated
	singularity at 0. Then,
	
    Eu_f,X(0)= Eu_X(0)  -  (∑_iχ(V_i∩ B_ε∩
    	f^-1(δ) ) · Eu_X(V_i) ),

	where 0 < |δ|≪ε≪ 1.

The last equation presents the relation between the local 
Euler 
obstruction of X and 
the Euler obstruction of f.



Seade, Tibăr and Verjovsky continued the study of the properties of  
Eu_f,X(0) in <cit.>. The authors proved that the Euler obstruction of f is
closely related to the number of Morse points of a morsefication of f, as 
follows.


	Let (X,0) be an equidimensional complex analytic singularity
	germ of dimension d and f: (X,0) → (ℂ,0) a germ of an analytic function 
	with an isolated critical point at the origin. Then 
	
    Eu_f,X(0) = (-1)^d n_ reg,
 
	where n_ reg is the number of Morse points 
	in the regular part of X appearing in a stratified morsefication of f.

Therefore, the Euler obstruction of f is the number of Morse points 
of a morsefication of f on the regular part of X, up to the sign. Hence 
this 
 invariant can be seen as a generalization of the Milnor number of f. 
 
Another invariant associated with a germ of an analytic function f:(X,0) → (ℂ,0) is the Brasselet number introduced by Dutertre 
and Grulha in <cit.>. We will denote this number by B_f,X(0). If 
f 
has an isolated critical point, then the Brasselet number satisfies the 
equality 

    B_f,X(0) =  Eu_X(0) -  Eu_f,X(0).

If f is linear and generic, then B_f,X(0)= 
Eu_X(0), hence it can be viewed as a 
generalization of the local Euler obstruction. Moreover, 
even if f has a non-isolated singularity it provides interesting results. 
For example, the Brasselet number has a Lê-Greuel type formula 
(see <cit.> or Theorem <ref> below), the difference of the Brasselet numbers B_f,X(0) and B_f,X^g(0) is measure by the number of Morse critical points on the top 
stratum of the Milnor fiber of f appearing in a morsefication of g, where 
g: (X,0) → (ℂ,0) is a prepolar function (see Definition 
<ref>) and X^g=X ∩ g^-1(0).



Although they are important, the invariants mentioned above are not easily 
computed using their definition. In the literature there are formulas which 
make the computation easier, see <cit.>. Some authors worked 
on 
more specific situations. In the special case of toric surfaces,
an interesting formula for the local Euler obstruction was proved by 
Gonzalez-Sprinberg <cit.>. This formula was generalized by Matsui 
and Takeuchi <cit.> for normal toric varieties of any dimension. 



Toric varieties are particularly interesting objects, since they have a 
strong relation with elementary convex geometry. On these varieties we have 
an action of the algebraic torus (ℂ^*)^n that induces a finite 
decomposition of the variety into orbits,
all of which are homeomorphic to a torus.



In <cit.>, Varchenko described the topology of the Milnor fiber of a 
function f:(ℂ^n,0) → (ℂ,0) using the geometry of the 
Newton polygon of f, and consequently, the Milnor number can be expressed 
by volumes of polytopes related to the Newton polygon of f. In his proof, 
he constructed a toric modification of ℂ^n on which the pull-back 
of f defines a hypersurface with only normal crossing singularities. While 
ℂ^n is a very special smooth toric variety, it seems natural to 
generalize his formula to Milnor fibers over general singular toric 
varieties. This was done by Matsui and Takeuchi in <cit.>.



We use <cit.> to establish several combinatorial formulas for the 
computation of the Brasselet number of f:(Y,0)→ (, 0) where Y⊂ X 
is a non-degenerate complete intersection in a toric variety X. These 
formulas will be given in terms of volumes of Newton polygons associated to 
f.



This paper is organized as follows. In Section 2, we present some background 
material
concerning the Brasselet number and toric varieties, which will be used in the 
entire work. In Section 3, we compute the Brasselet number of a polynomial 
function f : (X, 0) → (ℂ, 0), where X ⊂ℂ^n is a 
toric variety. Moreover, we compute this invariant for functions defined on 
X^g = X∩ g^-1(0), where g: X →ℂ^k is a non-degenerate 
complete intersection. As a consequence, assuming that g has an isolated 
critical point on X and on X^f, we also obtain a formula for the number of 
stratified Morse critical points on the top stratum of the Milnor fiber of f 
appearing in a morsefication of g: X ∩ f^-1(δ) ∩ 
B_ε→ℂ. As applications we establish several 
results concerning constancy of these invariants. In Section 4, we 
consider the case where (X,0) = (ℂ^n,0) and we derive sufficient 
conditions to obtain the constancy of the Euler obstruction for families of 
complete intersections with an isolated singularity which are contained on X. 
We 
use this result to study the invariance of the Bruce-Roberts' Milnor number 
for 
families of functions defined on hypersurfaces. In Section 5, we work in the 
case of surfaces,in the case where X is a 2-dimensional
toric variety. In this situation, we present a 
characterization of a polynomial function g: X →ℂ which 
has a stratified isolated singularity at the origin. We use  
this characterization to present some examples for a class of toric surface 
that is also determinantal. In Section <ref>, we 
study the GSV-index on non-degenerated complete intersection in toric varieties.





§ GENERALITIES: STRATIFICATIONS, BRASSELET NUMBER AND TORIC VARIETIES



For the convenience of the reader and to fix the notation we present some 
general facts  in order
to establish our results.




 §.§ Stratifications and Brasselet number





In order to introduce the definition and the properties of the Brasselet 
number, 
we need some notions about stratifications. For more 
details, we refer to Massey <cit.>. Let A ⊂^n, 
B⊂^m and f 
: A → 
B be a function with n, m∈. We will 
fix the notation, A^f:=A∩ f^-1(0). 

Consider X⊂^n a 
reduced complex analytic set of dimension d which is included in an open 
set U. Let F : U →ℂ be a holomorphic function and f : X →ℂ be the restriction 
of F to X, f:=F|_X.




	A good stratification of X relative to f is a stratification 
	𝒱 of X which is adapted
	to X^f, such that {V_i ∈𝒱;   V_i ⊄X^f} is a Whitney 
	stratification
	of X ∖ X^f, and for any pair of strata 
	(V_α,V_β) such that V_α⊄X^f and 
	V_β⊂ X^f, the (a_f)-Thom condition is satisfied. We 
	call the strata included in X^f the good strata.


By <cit.>, given a stratification 𝒮 of X one can refine 
𝒮 to obtain a Whitney stratification 𝒱 of X which 
is adapted to X^f. By <cit.> the refinement 
𝒱 satisfies the  (a_f)-Thom condition, good 
stratifications always exist. 



Consider X and f as before. Let 𝒱={V_i} be a stratification 
of X. The critical locus of f 
relative to 𝒱, denoted by Σ_𝒱 f, is the union 
of the 
critical locus of
f restricted to each of the strata,Σ_𝒱 f = ⋃_iΣ(f|_V_i).



A critical point of f relative to 𝒱 is a point p ∈Σ_𝒱 f. If the stratification 𝒱 is clear, we
will refer to the elements of Σ_𝒱 f simply as stratified 
critical points of f. If p is an isolated point of Σ_𝒱 
f, we call p a  stratified isolated critical point of f (with
respect to 𝒱). If 𝒱 is a Whitney stratification of 
X and f : X →ℂ has a stratified isolated
critical point at the origin, then 

    {V_α∖ X^f, 
    V_α∩ X^f∖{0 }, { 0};   
    V_α∈𝒱},

is a good stratification for f. We call it the good 
stratification induced by f.


	Suppose that X is equidimensional. Let 𝒱 = {V_i 
	}_i=0^q be a good stratification of X relative
	to f. The Brasselet number is defined by 
	
    B_f,X(0) := ∑_i 
    	=1^qχ(V_i∩ B_ε(0) ∩ f^-1(δ) ) · Eu_X(V_i),

	where 0< | δ| ≪ε≪ 1.


If f has a stratified isolated critical point at the origin and X 
is equidimensional, Theorem <ref> implies that

    B_f,X(0) =  Eu_X(0) -  Eu_f,X(0).

The Brasselet number has many interesting properties,it satisfies
several
multiplicity formulas, which enable the authors to establish in <cit.> a relative version of the local
index formula and a Gauss-Bonnet formula for B_f,X(0). However, one 
of the most important properties 
of this invariant is the Lê-Greuel type formula (Theorem <ref>). To present this result we need 
to impose some conditions on the functions to ensure that X^g meets 
X^f in a nice way. So it is necessary to define:

 



	Let 𝒱 be a good stratification of X relative to f. We say 
	that g : (X,0) → (ℂ, 0)
	is prepolar with respect to 𝒱 at the origin if the origin is 
	an isolated critical point of g.




The condition that g is prepolar means that g has an isolated critical 
point (in the stratified sense), both on X and on X^f, and that X^g 
intersects transversely each stratum of 𝒱 in a neighborhood of 
the origin, except
perhaps at the origin itself. However, it is important to note that, while 
X^g meets X^f in a nice way, X^f may have
arbitrarily bad singularities when restricted to X^g. The (a_f)-Thom 
condition in Definition 
<ref> together with the hypothesis that g is prepolar ensure 
that g:X ∩ f^-1(δ) ∩ B_ε→ℂ has no 
critical points on g^-1(0) <cit.>. 
Therefore, the number of stratified Morse critical points on the top 
stratum V_q ∩ f^-1(δ)∩ B_ε(0), in a	morsefication 
of g : X ∩ f^-1(δ) ∩ B_ε(0) →ℂ, does not depend on the morsefication.


The next theorem shows that the Brasselet number satisfies a Lê-Greuel 
type formula <cit.>. 


	Suppose that X is equidimensional and that g is prepolar with respect 
	to 𝒱 at the
	origin. Then,
	
    B_f,X(0) -  B_f,X^g(0)  = (-1)^d-1n_q,

	where n_q is the number of stratified Morse critical points on the top 
	stratum V_q ∩ f^-1(δ)∩ B_ε(0) appearing in a 
	morsefication of g : X ∩ f^-1(δ) ∩ B_ε(0) →ℂ, and 0 < |δ| ≪ε≪ 1. In particular, this number is independent on the morsefication.
   







 §.§ Toric varieties



The theory of toric varieties can be seen as a
cornerstone for the interaction between combinatorics and algebraic geometry, 
which relates the combinatorial study of convex polytopes to algebraic 
torus actions. Moreover, for polynomial functions defined on such varieties, 
it is possible to obtain a combinatorial description of the topology of their 
Milnor fibers in terms of Newton polygons (see <cit.>). The reader 
may consult <cit.> for an overview about toric 
varieties.



Let N ≅ℤ^d be a ℤ-lattice of rank d and σ 
a strongly convex rational polyhedral cone in N_ℝ = ℝ⊗_ℤ N. We denote by M the dual lattice of N and 
the polar cone σ̌ of σ in M_ℝ = ℝ⊗_ℤ M by 

    σ̌ = {v ∈ M_ℝ;   ⟨ u,v⟩≥ 0   for any  u ∈σ},
 where ⟨· , ·⟩ is the usual inner product in ^d.
Then the dimension of σ̌ is d and we obtain 
a semigroup S_σ:= σ̌∩ M.


	A d-dimensional affine toric variety X_σ is defined by the 
	spectrum of ℂ[S_σ],X=Spec(ℂ[S_σ]).


The algebraic torus T = Spec(ℂ[M])≅ 
(ℂ^*)^d acts naturally on X_σ and the T-orbits in 
X_σ are indexed by the faces Δ of 
σ̌ (Δ≺σ̌). We denote by 
𝕃(Δ) the smallest linear 
subspace of M_ℝ containing Δ. For a face Δ of 
σ̌, denote by T_Δ the T-orbit in 
Spec(ℂ[M ∩𝕃(Δ)]) which corresponds to 
Δ. We observe that the 
d-dimensional affine toric varieties are 
	exactly those d-dimensional affine, normal varieties admitting a 
	(ℂ^*)^d-action with an open, dense orbit homeomorphic to 
	(ℂ^*)^d. Moreover, each T-orbit T_Δ is 
	homeomorphic to (ℂ^*)^r, where r is the dimension of 
	𝕃(Δ).






Therefore we obtain a decomposition X_σ = _Δ≺σ̌ T_Δ into T-orbits, which are homeomorphic to 
algebraic torus (ℂ^*)^r. Due to this fact, and also the 
informations coming from the combinatorial residing in these varieties, many 
questions that were originally studied for functions defined on 
ℂ^d 
can be extended to functions defined on toric varieties. 




Consider f: X_σ→ℂ a polynomial function on 
X_σ,a function that corresponds to an element f=∑_v∈ 
S_σa_v · v of ℂ[S_σ], where a_v ∈ℂ.


	Let f=∑_v∈ S_σa_v · v be a polynomial function on 
	X_σ.
	
	
  * (a) The set {v ∈ S_σ;   a_v ≠ 0 }⊂ S_σ is called the support of f and we denote it by 
	f;

	
  * (b) The Newton polygon Γ_+(f) of f is the convex hull of 
	
    ⋃_v ∈ f (v+ σ̌) ⊂σ̌.
	
                

Now let us fix a function f ∈ℂ[S_σ] such that 0 ∉ f,f: X_σ→ℂ vanishes at the T-fixed point 
0. Considering M(S_σ) the ℤ-sublattice of rank d in 
M generated by S_σ, we have that each element v of S_σ⊂ M(S_σ) is identified with a ℤ-vector 
v=(v_1,…,v_d). Then for g = ∑_v ∈ S_σ b_v · v ∈ℂ[S_σ] we can associate a Laurent polynomial L(g) = ∑_v 
∈ S_σ b_v · x^v on T = (ℂ^*)^d, where 
x^v:=x_1^v_1· x_2^v_2…  x_d^v_d. 



  We say that f=∑_v∈ S_σa_v· v∈ℂ[S_σ] is non-degenerate if for any compact face γ of 
  Γ_+(f) the complex hypersurface 
  
    x=(x_1,…,x_d) ∈ 
      (ℂ^*)^dL(f_γ)(x) = 0

  in (ℂ^*)^d 
  is smooth and reduced, where f_γ := ∑_v ∈γ∩ 
  S_σ a_v · v.


We can also study non-degeneracy in case of complete intersections defined on 
X_σ. Let f_1,f_2, …, f_k ∈ℂ[S_σ] (1 ≤ k 
≤ d =  X_σ) and consider the following subvarieties of 
X_σ: 

    V:= {f_1=…=f_k-1 = f_k =0 }⊂ W:= 
    {f_1=…=f_k-1 =0 }.
 Assume that 0 ∈ V. For each face 
Δ≺σ̌ such that Γ_+(f_k) 
∩Δ≠∅, we set 

    I(Δ)=  j=1,2,…, k-1Γ_+(f_j) ∩Δ≠∅⊂{ 
    1,2,…,k-1 }
 
and m(Δ)= # I(Δ) + 1.

Let 𝕃(Δ) and M(S_σ∩Δ) be as before and 
𝕃(Δ)^* the dual vector space of 𝕃(Δ). Then 
M(S_σ∩Δ)^* is naturally identified with a subset of 
𝕃(Δ)^* and the polar cone Δ̌=  u 
∈𝕃(Δ)^*⟨ u, v⟩≥ 0     v∈Δ of Δ in 
𝕃(Δ)^* is a rational polyhedral convex cone with respect to 
the lattice M(S_σ∩Δ)^* in 𝕃(Δ)^*.

 
	(i) For a function f = ∑_v∈Γ_+(f) a_v · 
	v 
	∈ℂ[S_σ] on X_σ and u ∈Δ̌, 
	we set f|_Δ = ∑_v∈Γ_+(f) ∩Δ a_v · v ∈ℂ[S_σ∩Δ] and 
	
    Γ(f|_Δ;u) = 
    	v ∈Γ_+(f)∩Δ⟨ u, v⟩ 
    	=  min⟨ u,w⟩,   for  w ∈Γ_+(f) ∩Δ.
 
	We call Γ(f|_Δ;u) the 
	supporting face of u in Γ_+(f) ∩Δ.
	
	(ii) For j ∈ I(Δ) ∪{k } and u ∈Δ̌, we define the u-part f_j^u∈ℂ[S_σ∩Δ] of f_j by 
	
    f_j^u = 
    	∑_v∈Γ(f_j|_Δ;u) a_v · v∈ℂ[S_σ∩Δ],
 
	where f_j = 
	∑_v∈Γ_+(f_j) a_v · v∈ℂ[S_σ].


By taking a ℤ-basis of M(S_σ) and identifying the 
u-parts f_j^u with Laurent polynomials L(f_j^u) on T = 
(ℂ^*)^d as before, we have that the following definition does 
not depend on the choice of the ℤ-basis of M(S_σ).


	We say that (f_1,…,f_k) is non-degenerate if for any face Δ≺σ̌ such that Γ_+(f_k) ∩Δ≠∅ (including the case where Δ = σ̌) and any 
	u ∈ Int(Δ̌) ∩ M(S_σ∩Δ)^* the 
	following two subvarieties of (ℂ^*)^d are non-degenerate 
	complete intersections 
	
    x ∈ (ℂ^*)^dL(f_j^u)(x)=0,  ∀  j ∈ I(Δ) ; x ∈ (ℂ^*)^dL(f_j^u)(x)=0,   ∀ j ∈ 
    	I(Δ) ∪{k},

i.e., if they are reduced smooth complete intersections varieties in the torus (ℂ^*)^d.

This last definition is presented in <cit.> and it is a 
generalization of 
the one in <cit.>.


For these non-degenerate singularities, it is possible to describe 
their geometrical and topological properties using the combinatorics of the 
Newton polygon. This is 
done in <cit.> using mixed volume as follows. For each face Δ≺σ̌ of σ̌ such that 
Γ_+(f_k) ∩Δ≠∅, we set 

    f_Δ = 
    ∏_j∈ I(Δ) f_j· f_k ∈ℂ[S_σ]
 
and consider its Newton polygon Γ_+(f_Δ) = {∑_j ∈ 
	I(Δ)Γ_+(f_j) } + Γ_+(f_k) ⊂σ̌. Let γ_1^Δ,…, 
γ_ν(Δ)^Δ be the compact faces of 
Γ_+(f_Δ) ∩Δ (≠∅) such that 
γ_i^Δ = Δ -1. Then for each 1≤ i ≤ν(Δ) there exists a unique primitive vector u_i^Δ∈ 
	Int(Δ̌) ∩ M(S_σ∩Δ)^* which takes its 
minimal value in Γ_+(f_Δ) ∩Δ exactly on 
γ_i^Δ.

For j ∈ I(Δ) ∪{k }, set γ(f_j)_i^Δ 
:= Γ(f_j|_Δ;u_i^Δ) and d_i^Δ:=  
	min_w∈Γ_+(f_k) ∩Δ⟨ 
u_i^Δ,w⟩. Note that we have 
    γ_i^Δ = 
    ∑_j ∈ I(Δ) ∪{k }γ 
    (f_j)_i^Δ
 for any face Δ≺σ̌ such that 
Γ_+(f_k) ∩Δ≠∅ and 1 ≤ i ≤ν(Δ). 
For each face Δ≺σ̌ such that Γ_+(f_k) ∩Δ≠∅, Δ≥ m(Δ) and 1 ≤ i 
≤ν(Δ), we set I(Δ) ∪{k} = 
{j_1,j_2,…,j_ m(Δ)-1,k=j_ m(Δ)} 
and 
    K_i^Δ := ∑_α_1+…+α_ m(Δ)  =  
    			Δ -1α_q ≥ 1   for  q≤ 
    				m(Δ)-1α_ m(Δ)≥ 0^ 
    	Vol_ℤ(γ(f_j_1)_i^Δ,…,
    	γ(f_j_1)_i^Δ_α_1-times,…,γ(f_j_
    				m(Δ))_i^Δ,…,γ(f_j_ 
    				m(Δ))_i^Δ_α_ 
    				m(Δ)-times).
 Here 

    Vol_ℤ(γ(f_j_1)_i^Δ,…,
    	γ(f_j_1)_i^Δ_α_1-times,…,γ(f_j_
    				m(Δ))_i^Δ,…,γ(f_j_ 
    				m(Δ))_i^Δ_α_ 
    				m(Δ)-times)
 is the 
normalized (Δ -1)-dimensional mixed volume with respect to 
the lattice M(S_σ∩Δ) ∩ L(γ_i^Δ) (see 
Definition 2.6, pg 205 from <cit.>). For Δ such that 
Δ -1 =0, we set 
    K_i^Δ =  
    	Vol_ℤ(γ(f_k)_i^Δ,…,
    	γ(f_k)_i^Δ_0-times)
    := 1
 (in this case γ(f_k)_i^Δ is a point). 




§ THE BRASSELET NUMBER AND TORUS ACTIONS
 



We present formulas for the computation of the Brasselet number of 
a function defined on a non-degenerate complete intersection contained in 
toric variety using Newton 
polygons.  As applications we establish several results concerning 
its invariance for families of non-degenerate complete 
intersections.

Let X_σ⊂ℂ^n be a d-dimensional toric variety and 
(f_1,…,f_k): (X_σ,0) → (ℂ^k,0) a non-degenerate 
complete intersection, with 1≤ k ≤ d. From now on we will denote by g 
the complete intersection (f_1,…,f_k-1) and by f the function f_k. 
Let 𝒯 be the decomposition of X_σ = 
_Δ≺σ̌ T_Δ  into T-orbits, and 
𝒯_g the decomposition of X_σ^g = 
_Δ≺σ̌ T_Δ∩ X_σ^g. Since 
g is a non-degenerate complete intersection, 𝒯_g is a decomposition of X_σ^g into smooth subvarieties <cit.>.  We are interest in  the situation that 
𝒯_g is a Whitney stratification. 


	Let S_σ = ℤ_+^3 and let X_σ = ℂ^3 
	be the smooth 3-dimensional toric variety. Consider 
	(g,f):(X_σ,0) → (ℂ^2,0) a non-degenerate complete 
	intersection, where
					
    g(z_1,z_2,z_3) = z_2^2 - z_1^3 -z_1^2z_3^2.

		

	The decomposition 𝒱 = {V_0={0},   					
	V_1={axis - z_3}∖{0},   	V_2  = 	
	(X_σ^g)_reg} is a Whitney stratification of the 
	non-degenerate 	complete intersection X_σ^g, where 			
	(X_σ^g)_reg is the regular part of  X_σ^g 
	(see Figure (<ref>)).	For a subset I 
	⊂{1,2,3} we denote the face ∑_i ∈ Iℝ_+e_i^* of σ̌ = 
	(ℝ_+^3)^* 		by Δ_I and 		by Δ_0 
	the face {0}, where e_i^* 				denotes the 	
	elements of the canonical basis 
	of 																		
						(ℝ^3)^*.
		 The Whitney stratification 				𝒯 of 
	X_σ is given by the  following  T-orbits:	T_Δ_0 
	= 	{0}, T_Δ_1 = 								
	ℂ^*×{0}×{0}, T_Δ_2 = {0}×ℂ^*×{0}, 
	T_Δ_3 = {0}×{0}×ℂ^*, 
	T_Δ_12 =		ℂ^*×ℂ^*×{0}, T_Δ_13 =	ℂ^*×{0}×ℂ^*, T_Δ_23 = {0}×ℂ^*×ℂ^*, 
	T_Δ_123 = ℂ^*×ℂ^*×ℂ^*. Now, observe that 				
	T_Δ_3∩ X_σ^g = T_Δ_3 = 
	V_1, then the stratification 	
			
    𝒯_g = { T_Δ_I∩ X_σ^g,   Δ_0}_Δ_I≺σ̌

	is a Whitney stratification of X_σ^g.	


In Section <ref> we will provide others examples where 
the stratification 𝒯_g is Whitney (see <cit.> 
for more examples).
We observe that in general 𝒯_g is not a Whitney 
stratification see <cit.>.


	Let X_σ⊂ℂ^n be a d-dimensional toric variety 
	and (g,f): (X_σ,0) → (ℂ^k,0) a non-degenerate complete 
	intersection. If 𝒯_g is a Whitney stratification, then
	
    B_f,X_σ^g(0)   =  ∑_Γ_+ 
    	(f) ∩Δ≠∅Δ ≥ m(Δ) 
    	(-1)^Δ -  m(Δ)( ∑_i=1^ν(Δ) 
    	d_i^Δ· K_i^Δ)·Eu_X_σ^g(T_Δ∩ X_σ^g).





	 Let (X_σ^g)^f := X^g_σ∩ X^f_σ be the zero set of f 
	 in X^g_σ. By <cit.> or <cit.>, there exists a refinement 
	 of 𝒯_g, denoted by 𝒯_(g,f) = {W_i}, such that 
	 𝒯_(g,f) is 
	 adapted to (X_σ^g)^f and it is a Whitney stratification. 
	 Moreover, this stratification satisfies the (a_f)-Thom condition (see 
	 <cit.>), and then it 
	 is a good stratification of X^g_σ with relation to f. By 
	 definition, the Brasselet number reads
	 
    B_f,X_σ^g(0) = ∑_iχ(W_i∩ 
    	 B_ε(0) 
    	 ∩ f^-1(δ) ) · Eu_X_σ^g(W_i),

	 where 0< | δ| ≪ε≪ 1.
	 
	 As 𝒯_(g,f) is a refinement of 𝒯_g, each 
	 T_Δ∩ X^g_σ is a disjoint union of W_i, 
    T_Δ∩ X^g_σ = _W_i ∩ T_Δ∩ X^g_σ≠∅ W_i,

	 and since 𝒯_g is a Whitney stratification,
	 
    Eu_X_σ^g(T_Δ∩ X^g_σ) =  
    	 Eu_X_σ^g(W_i),

	 for any W_i that meets T_Δ∩ X^g_σ. Then
	 
    B_f,X_σ^g(0) = ∑_Δχ(T_Δ∩ X^g_σ∩ 
    	 B_ε∩ f^-1(δ) ) · Eu_X_σ^g(T_Δ∩ X^g_σ).

	 Note that f:X_σ→ℂ has the isolated 
	 stratified critical value 0 ∈ℂ by 
	 <cit.>. Thus the integers χ (T_Δ∩ 
	 X^g_σ∩	 B_ε∩ f^-1(δ) ) can be calculated by the nearby cycle functor 
	 ψ_f <cit.>,
    χ(	T_Δ∩ X^g_σ∩	 B_ε∩ 
    	 f^-1(δ) )	= χ(ψ_f(ℂ_T_Δ∩ 	
    	 X_σ^g)_0),

	 since X_σ^g∩ B_ε∩ f^-1(δ)  = 
	 _Δ≺σ̌ T_Δ∩ B_ε∩ X_σ^g∩ 
	 f^-1(δ) is a 
	 Whitney stratification of the Milnor fiber X_σ^g∩ 
	 B_ε∩ 
	 f^-1(δ). By the proof of <cit.> equations (52), (77), (78), we have that
	 
    χ(ψ_f(ℂ_T_Δ∩ X_σ^g)_0) =  
    	(-1)^Δ -  m(Δ)( ∑_i=1^ν(Δ) 
    	d_i^Δ· K_i^Δ).

	 To complete the proof, we observe that f≡ 0 on T_Δ for any 
	 face Δ such that Γ_+(f) ∩Δ = ∅, 
	 then we can neglect those faces. 




If f: X_σ^g→ℂ has a stratified isolated
critical point, then Equation (<ref>) holds. Altogether, we have: 


	Let X_σ⊂ℂ^n be a d-dimensional toric 
	variety and (g,f): (X_σ,0) → (ℂ^k,0) a non-degenerate 
	complete intersection. If 𝒯_g is a Whitney 
	stratification and f: X_σ^g→ℂ 
	has an isolated  singularity at the origin, then
	
    Eu_f,X_σ^g(0)   =   Eu_X_σ^g(0) - 
    	∑_Γ_+ (f) ∩Δ≠∅Δ ≥ m(Δ) (-1)^Δ -  m(Δ)( ∑_i=1^ν(Δ) d_i^Δ· K_i^Δ)·Eu_X_σ^g(T_Δ∩ X_σ^g).





When k=1, using a similar argument used in Theorem <ref>, we 
obtain B_f,X_σ(0). In fact, using again <cit.> we can 
pass to a refinement of 𝒯 to obtain a good stratification of 
X_σ relative to f. Thus we can use the same argument as 
in the proof of Theorem <ref> to obtain  

    B_f,X_σ(0) = ∑_Δχ(T_Δ∩ 
    B_ε(0) 
    ∩ f^-1(δ) ) · Eu_X_σ(T_Δ).

Now for each face Δ≺σ̌ such that 
Γ_+(f) ∩Δ≠∅, let 
β_1^Δ, β_2^Δ, 
…,β_μ(Δ)^Δ be 
the compact faces of Γ_+(f) ∩Δ such that 
dimβ_i^Δ = Δ -1. Let Γ_i^Δ 
be the convex hull of β_i^Δ⊔{0} in 
𝕃(Δ) and consider the normalized (Δ)-dimensional 
volume of 
Γ_i^Δ,
Vol_ℤ(Γ_i^Δ) ∈ℤ,  with respect to 
the lattice M(S_σ∩Δ). Here M(S_σ∩Δ) denotes the sublattice of 
M(S_σ) generated by S_σ∩Δ. Then we have the 
following result.


	Assume that f = ∑_v∈ S_σ a_v · v ∈ℂ[S_σ] is non-degenerate. Then 
    B_f,X_σ(0)= ∑_Γ_+ (f) ∩Δ≠∅(-1)^Δ - 1( 
    	∑_i=1^μ(Δ)Vol_ℤ(Γ_i^Δ) 
    	)·Eu_X_σ(T_Δ).
  


 
	For each (Δ -1)-compact face β_i^Δ of 
	Γ_+(f) ∩Δ≠∅, with 1≤ i ≤μ(Δ) we have 
	
    K_i^Δ :=  
    	Vol_ℤ(β_i^Δ,…,β_i^Δ_(
    	 dimΔ -1)-times),

 	and from 
	<cit.> we now that 	
	
    Vol_ℤ(β_i^Δ,…,
    	β_i^Δ_(Δ - 1)-times) =  
    	Vol_ℤ(β_i^Δ).

	Therefore, the result follows 
	from the fact that 
	
    Vol_ℤ(Γ_i^Δ) = 
    	d_i^Δ· K_i^Δ,

	for 1≤ i ≤μ(Δ).


We will apply Theorem <ref> in order to show that the Brasselet 
number is invariant for some families of complete intersections.  We need 
some new concepts and notations.


A familia é para todo t∈?  Ok

A deformation of a map germ f : (X, 0) → (ℂ^k, 0) is another
map germ F : (ℂ× X) → (ℂ^k, 0) such that F(0, 
x) = f(x), for all x ∈ X. 

We assume
that F is origin preserving, that is, F(t, 0) = 0 for all t ∈ℂ, so we have a 1-parameter family of
map germs f_t : (X, 0) → (ℂ^k, 0) given by f_t(x) = F(t, x). 
Moreover, associated to the family f_t : (X, 0) → (ℂ^k, 0) we 
have the family X^f_t = X ∩ f_t^-1(0) of subvarieties of X.
In the particular case of a polynomial function f : (X, 0) → (ℂ, 
0), any polynomial deformation f_t can be written as:
			
    f_t(x) = f(x)+ ∑_i=1^rθ_i(t) ·  h_i(x)

for some polynomials h_i: (X,0) → (ℂ,0) and θ_i:(ℂ,0) → (ℂ,0), where θ_i(0)=0, for all i=1,…,r.

Suppose that f: X_σ→ℂ is a polynomial function  
defined on a toric variety X_σ. Consider a family as in 
family1 with Γ_+(h_i) ⊂Γ_+(f), for all 
i=1,…,r. Let
γ_i_1^Δ,γ_i_2^Δ,…,
γ_i_ν(Δ)^Δ 
and
β_1^Δ,β_2^Δ,…,β_μ(Δ)^Δ 
be the compact faces of Γ_+(h_i) ∩Δ and of Γ_+(f) ∩Δ, respectively, such that 
γ_l^Δ=β_j^Δ = Δ -1.
If for each face Δ≺σ̌, with Γ_+(h_i) ∩Δ≠∅, 
we have that γ_i_l^Δ∩β_j^Δ = ∅, for 
all 
l=1,…,i_ν(Δ) and j=1,…,μ(Δ), then

    Γ_+(f_t) = Γ_+(f),   for all  t∈ℂ.

In this case, we fix the notation  			

    Γ_+(h_i) ⫋Γ_+(f),

for all i=1,…,r.


In the sequel, we present some applications of Theorem <ref>.


	Let X_σ⊂ℂ^n be a d-dimensional toric variety.
	Suppose that
	(g,f_t): (X_σ,0) → (ℂ^k,0) is a 
	non-de­ge­ne­ra­te 
	complete intersection for each t∈ℂ with
	
    f_t(x) = f(x)+∑_i=1^rθ_i(t) · h_i(x)

	being a polynomial function on X_σ with h_i satisfying the 
	condition (<ref>) for all i=1,…,r. If 
	𝒯_g is a Whitney 
	stratification and (g,f_t) is a family of
	non-degenerate 					complete	intersections, then 
	B_f_t,X_σ^g(0) is constant for the family. 



	As we have already noted, for each face Δ≺σ̌ 
	such that Γ_+(f) ∩Δ≠∅, 
	the Newton polygon Γ_+(f_Δ) of the function 
	
    f_Δ = ∏_j∈ I(Δ) f_j· f ∈ℂ[S_σ]
 
	is {∑_j ∈ I(Δ)Γ_+(f_j) } + Γ_+(f) ⊂σ̌. 
	Therefore, Γ_+(f_Δ) = Γ_+((f_t)_Δ), for all 
	t ∈ℂ, where 
	
    (f_t)_Δ = ∏_j∈ I(Δ) f_j· 
    	f_t,
 since 
	Γ_+(f) = Γ_+(f_t). Then the result follows by Theorem 
	<ref>.


Roughly speaking the Brasselet number depends only on the 
monomials of smallest degree in each variable.



Given f: (X_σ,0) → (ℂ,0) a non-degenerate function, as 
before, we denote by 𝒯_f the decomposition 

    X_σ^f= _Δ≺σ̌ T_Δ∩ 
    X_σ^f
 of X_σ^f, and by 𝒯^f the 
decomposition { T_Δ∩ X_σ^f,    T_Δ∖ X_σ^f,   {0}}_Δ≺σ̌ of X_σ. If 𝒯_f 
is a Whitney stratification of X_σ^f, then 𝒯^f is a 
Whitney stratification of X_σ which is adapted to X_σ^f. 
In fact, this follows from the fact that 𝒯 is a Whitney 
stratification of X_σ. Consequently, 𝒯^f is 
a 
good 		
stratification of X_σ relative to f.



Let g and f be non-degenerate polynomial functions on X_σ. In 
general we have no way to relate 
the local Euler obstructions  Eu_X_σ^g(T_Δ∩ 
X_σ^g) 
to  Eu_X_σ(T_Δ). However, if we assume 
the 
additional hypothesis that g has an isolated critical point at 0 both in 
X_σ and in X_σ^f (in stratified sense), the 
following result holds.




	Let X_σ⊂ℂ^n be a d-dimensional toric variety 
	and (g,f): (X_σ,0) → (ℂ^2,0) a non-degenerate complete intersection. Suppose that 𝒯^f is a good stratification   		of X_σ relative to f and that g is prepolar with respect to 𝒯^f, then
	
    B_f,X_σ^g(0)   =  ∑_Γ_+ 
    	(f) ∩Δ≠∅Δ ≥ 2 
    	(-1)^Δ -  2( ∑_i=1^ν(Δ) 
    	d_i^Δ· K_i^Δ)·Eu_X_σ(T_Δ).



 Consider 𝒯_g = { T_Δ∩ 
X_σ^g}_Δ≺σ̌ the stratification 
of 	
X_σ^g. Using the same argument as in the proof of Theorem 
<ref> we can 
	pass to a refinement 𝒯_(g,f) and we obtain a good 
	stratification of 
	X_σ^g relative to f. Then 
	
    B_f,X_σ^g(0)   =  ∑_Γ_+(f)∩Δ≠∅χ(W_Δ∩ B_ε∩ 										f^-1(δ) 
    				) 
    				· Eu_X_σ^g(W_Δ),

	where W_Δ are the strata of 𝒯_(g,f) which are not 
	contained in {f = 
	0 }, and 0 < |δ| ≪ε≪ 1. Moreover,
	for Δ≺σ̌, we have Eu_X_σ(T_Δ) =  Eu_X_σ^g(W_Δ), 
	since X_σ^g intersects the strata of 𝒯^f transversally 
	(see <cit.>). Hence,
		
    B_f,X_σ^g(0)      =     ∑_Γ_+(f)∩Δ≠∅χ(X_σ^g∩ 						T_Δ∩ 
    		B_ε∩ f^-1(δ) ) · Eu_X_σ(T_Δ).
 
	Finally, as Γ_+(g) ∩Δ≠∅ for any 
	face 0⪵Δ≺σ̌, then m(Δ) = 2, for 
	all face Δ≺σ̌ such that Γ_+(f) 
	∩Δ≠∅. Since g is prepolar, 
	
    X_σ^g∩ 
    	B_ε∩ f^-1(δ)  = 
    	_Δ≺σ̌ T_Δ∩ B_ε∩ X_σ^g∩ 
    	f^-1(δ)
 
	is a 
	Whitney stratification of the Milnor fiber X_σ^g∩ 
	B_ε∩ 
	f^-1(δ). Therefore, we obtain the result by nearbycicle1 
	and (<ref>).


Therefore, if 𝒯^f is a good stratification of X_σ relative to f and g is prepolar with respect to 𝒯^f, we can obtain a more general version of Corollary <ref>, since we can relate 
the local Euler obstructions  Eu_X_σ^g(T_Δ∩ 
X_σ^g) to the 
Euler local obstructions Eu_X_σ(T_Δ). 




	Let X_σ⊂ℂ^n be a d-dimensional toric variety 
	and (g,f): (X_σ,0) → (ℂ^2,0) a non-degenerate 
	complete intersection. Suppose that 
	
    g_s(x),f_t(x)  =  g(x) + 
    	∑_i=1^mξ_i(s)· l_i(x), f(x)+∑_j=1^rθ_j(t)· h_j(x)

	is a family of non-degenerate complete 
	intersections with l_i and h_j satisfying the condition 
	(<ref>) for all i=1,…,m and j=1,…,r. If 
	𝒯^f_t is a good stratification   		of X_σ 
	relative to f_t and g_s is prepolar with respect to 
	𝒯^f_t at the origin, for all s,t ∈ℂ,
	then, B_f_t,X_σ^g_s(0) is constant for all 
	t,s ∈ℂ.



  Since Γ_+(l_i) ⊂Γ_+(g)   and  Γ_+(h_j) ⊂Γ_+(f), for each face Δ≺σ̌   such that Γ_+(f) ∩Δ≠∅, the 
  Newton polygon Γ_+(f_Δ) of the function 
    f_Δ = 
      ∏_Γ_+(f) ∩Δ≠∅ 
      g · f ∈ℂ[S_σ]
 is equals to   
  Γ_+((f_t)^s_Δ ), where 
    (f_t)^s_Δ = ∏_Γ_+(f_t) ∩Δ≠∅ g_s · f_t ∈ℂ[S_σ].
 Then by nearbycicle1 and 
  (<ref>), we can conclude 
  that the Euler characteristic 
 	
    χ X_σ^g_s∩ T_Δ∩ 
    	B_ε∩ f_t^-1(δ)
 
	is constant for all s,t ∈ℂ. Moreover, as g_s is prepolar 
	with respect to 𝒯^f_t, then
	we can proceed exactly in the same way as in Theorem 
	<ref> to obtain
	
    Eu_X_σ(T_Δ) =  
    	Eu_X_σ^g_s(T_Δ∩ X_σ^g_s),
 
	and this concludes the proof.


As a consequence of Proposition 
<ref> and Theorem <ref>, we have that if 
𝒯^f 
is a good stratification of X_σ relative to f and if g: 
X_σ→ℂ is prepolar with respect to 𝒯^f, we can express the 
number of stratified Morse critical points on the stratum of maximum dimension 
appearing in a morsefication of g : X_σ∩ f^-1(δ) ∩ 
B_ε→ℂ in terms of volumes of convex polytopes. 
More precisely, by Theorem <ref>, we have 


    [                                                                                                                               (-1)^d-1n_d                                                                    =     ∑_Γ_+ (f) ∩Δ≠∅(-1)^Δ - 1( 
    ∑_i=1^μ(Δ)Vol_ℤ(Γ_i^Δ) 
    )·Eu_X_σ(T_Δ)                                                                     ;                                                                                                                                                                                                              - ∑_Γ_+ (f) ∩Δ≠∅Δ≥ 2 (-1)^Δ - 2( 
    ∑_i=1^ν(Δ) d_i^Δ· K_i^Δ)·Eu_X_σ(T_Δ)                                                                     ;                                                                      ]


where n_d is the number of stratified Morse critical points on the top 
stratum T_Δ_d∩ f^-1(δ)∩ B_ε appearing in a 
morsefication of g : X_σ∩ f^-1(δ) ∩ B_ε→ℂ.
Therefore, if f_t(x)= f(x)+∑_j=1^rθ_j(t) · h_j(x) is a 
family of non-degenerate polynomial functions on X_σ and if 
(g_s,f_t): (X_σ,0) → (ℂ^2,0) is a family of non-degenerate 
complete intersections which satisfy the same hypotheses as Corollary 
<ref>, then 							
	
    (-1)^d-1n_d =  
    	B_f_t,X_σ(0) -  B_f_t,X_σ^g_s(0)
 
	is constant for all s,t ∈ℂ. More precisely, we 
	can state the following result.


	Let X_σ⊂ℂ^n be a d-dimensional toric variety 
	and (g,f):(X_σ,0) → (ℂ^2,0) a non-degenerate complete 
	intersection. Suppose that 

    g_s(x),f_t(x)  =  g(x) + 
    	∑_i=1^mξ_i(s)· l_i(x), f(x)+∑_j=1^rθ_j(t)· h_j(x)

	is a family of non-degenerate complete 
	intersections with l_i and h_j satisfying the condition 
	(<ref>) for all i=1,…,m and j=1,…,r. If  
	𝒯^f_t is a good stratification of X_σ 
	relative to f_t and g_s is prepolar with respect to 
	𝒯^f_t at the origin, for all s,t ∈ℂ, 
	then 
	(-1)^d-1n_d is constant for all s, t ∈ℂ.


We will present some applications of the results presented before in the next 
section and some examples in Section <ref>. 














§ THE LOCAL EULER OBSTRUCTION AND BRUCE-ROBERTS' MILNOR NUMBER


GSV index vai para a seção 5 junto com a parte final

A introducao da secao esta estranha, talvez colocar na ordem em que 
é a secao.

+ -

In this section, we derive sufficient 
conditions to obtain the invariance of the local Euler obstruction for families 
of 
complete intersections  which are 
contained in X_σ⊂^n. As an application, we study the 
invariance of the Bruce-Roberts' Milnor number for families of functions 
defined on hypersurfaces. 






 §.§ Local Euler obstruction of non-degenerate complete intersections


 
As observed in <cit.> the local Euler obstruction is not a 
topological 
invariant. However, for non-degenerate complete intersections we have the following result.


	Let  X_σ⊂ℂ^n be a d-dimensional toric variety 
	and k a positive integer in {2,…,d}. Consider 
	g=(f_1,…,f_k-1): 
	(X_σ,0) →	(ℂ^k-1,0) a non-degenerate complete 
	intersection, such that 𝒯_g is a Whitney stratification. 
	Suppose that
	
    g_s(x) = 		 f_1(x) + ∑_i_1=1^m_1θ_i_1(s) 
    			· h_i_1(x),…,f_k-1(x) + 
    			∑_i_k-1=1^m_k-1θ_i_k-1(s) 
    			· h_i_k-1(x)

	is a family of non-degenerate complete intersections, such that 
	𝒯_g_s is a Whitney stratification and h_i_p satisfies 
	the condition (<ref>)
	for all p ∈{1,…,k-1 } and i_p ∈{1, …, 
	m_p}. 
	If L: ℂ^n 
	→ℂ is a linear form  which is generic with respect to 
	X_σ^g_s 
	for all s ∈ℂ and (g_s,L) is a non-degenerate complete 
	intersection, then Eu_X_σ^g_s(0) is 
	invariant for the family {g_s}_s∈. 



		As L: ℂ^n →ℂ is a linear form  which is generic with respect to 
		X_σ^g_s, for all s ∈ℂ,
		
    Eu_X_σ^g_s(0) =  B_L,X_σ^g_s(0).

		Moreover, T_Δ∩ X^g_s_σ∩	 B_ε∩ L^-1(δ) induces a Whitney
		stratification on the Milnor fiber X^g_s_σ∩	 
		B_ε∩ L^-1(δ). Therefore, the result 
		follows from Theorem <ref>, and from the fact that for all 
		face 
		Δ≠{0} of σ̌, the Newton 
		polygon Γ_+(L_Δ) of the function 
		
    L_Δ = ∏_j∈ I(Δ) f_j 
    		· L ∈ℂ[S_σ]

is equal to Γ_+(L_Δ^s). Here, 
		
    L_Δ^s = ∏_p∈ I(Δ) f_p + 
    		∑_i_p=1^m_pθ_i_p· h_i_p·  L 	
    		∈ℂ[S_σ].








	Let S_σ = ℤ_+^n and let X_σ = ℂ^n 
	be 
	the smooth n-dimensional toric variety. Consider
	g=(f_1,…,f_k-1): 					ℂ^n →ℂ^k-1 a non-degenerate complete intersection with an isolated 
	singularity at 0, where 2≤ k ≤ n. 
	Suppose that
			
    g_s(x) = 		 f_1(x) + ∑_i_1=1^m_1θ_i_1(s) 
    			· h_i_1(x),…,f_k-1(x) + 
    			∑_i_k-1=1^m_k-1θ_i_k-1(s) 
    			· h_i_k-1(x)
 
	is a family of non-degenerate complete intersections with an isolated 
	singularity at 0, where h_i_p satisfies the condition 
	(<ref>)
	for all p ∈{1,…,k-1 } and i_p ∈{1, …, 
	m_p}. If L: ℂ^n 
	→ℂ is a linear form  which is generic with respect to 
	X_σ^g_s, 
	for all s ∈ℂ and (g_s,L) is a non-degenerate complete 
	intersection, then Eu_X_σ^g_s(0) is 
	invariant for the family {g_s}_s∈. 
	




		As g_s is a family of complete intersections with an isolated 
		singularity at 0, the decomposition 𝒯_g_s of 
		X_σ^g_s = 
		_Δ≺σ̌ T_Δ∩ 
		X_σ^g_s is a Whitney stratification, once 𝒯 
		is a Whitney stratification of X_σ. Therefore, the result 
		follows from Theorem <ref>.


With the same assumptions as in Theorem <ref>, suppose that f_t(x) = 
f_k(x)+∑_i_k=1^m_kθ_i_k(t)· h_i_k(x) is a family of 
polynomial functions such that (g_s,f_t) is a family of non-degenerate 
complete 
intersections where h_i_k satisfies the condition (<ref>),
    Γ_+(h_i_k) ⫋Γ_+(f_k),   for all  i_k=1,…,m_k.

Moreover, assume that f_t: X_σ^g_s→ℂ has a stratified 
isolated critical point at 0. For each face Δ≺σ̌ 
satisfying Γ_+(f_k) ∩Δ≠∅, the Newton polygon 
Γ_+(f_Δ) of the function 
	
    f_Δ = (∏_j∈ I(Δ) f_j    
    		)· f_k ∈ℂ[S_σ]

is equals to Γ_+((f_t)^s_Δ ), where 
****  conferir as trocas de indices e ξ por θ
	
    (f_t)^s_Δ = ∏_p∈ I(Δ) f_p 
    		+ 
    		∑_i_p=1^m_pθ_i_p· h_i_p· f_t 
    		= 						f_k+∑_i_k=1^m_kθ_i_k· 
    		h_i_k∈ℂ[S_σ].

By Theorem <ref> we conclude that the 
Euler characteristic of the Milnor fiber of f_t: X_σ^g_s→ℂ is invariant for all s,t∈. Therefore, Eu_f_t, 
X_σ^g_s(0) is invariant for the family.








 §.§ Bruce-Roberts' Milnor number



In <cit.>, Bruce and Roberts introduced a Milnor number for 
functions germs on singular varieties. 

Let X be a 
sufficiently small representative of the germ (X,0) and let I(X) denote 
the ideal in 𝒪_n,0 consisting of the germs of functions 
vanishing on X. We say that two germs f and g in 𝒪_n,0 
are ℛ_X- equivalent if there exists a germ of diffeomorphism 
ϕ: (ℂ^n,0)→ (ℂ^n,0) such that ϕ(X)=X 
and f∘ϕ=g. Let θ_n denote the 𝒪_n,0- module of 
germs of vector fields 
on (ℂ^n,0). Each vector field ξ∈θ_n can be seen as a 
derivation ξ:𝒪_n,0→𝒪_n,0. We denote by 
θ_X those vector fields that are tangent to X,
    θ_X:=ξ∈θ_ndg(ξ)=ξ g∈ I(X), ∀ g∈ 
    I(X).



	Let f be a function in 𝒪_n,0 Acho que aqui está faltando um be 
	alguma coisa and let df(θ_X) be the ideal {ξ f: 
	ξ∈θ_X} in 𝒪_n,0. The number 
	
    μ_BR(X,f)=_ℂ𝒪_n,0/df(θ_X)

	is called the Bruce-Roberts number of f with respect to X.


We refer to <cit.> for more details about μ_BR(X,f). 
In particular, μ_BR(X,f) is finite
if and only if f is ℛ_X-finitely determined.

An interesting open problem is to know whether the Bruce-Roberts number is a 
topological invariant or not. In <cit.> 
Grulha 
gave a partial answer. The author proved that, if (X, 
0) is a hypersurface whose logarithmic characteristic variety LC(X) 
<cit.>colocar onde está a definicao no artigo ok, is 
Cohen-Macaulay and if 
f_t is a C^0- R_X-trivial deformation of f, then μ_BR(f_t,X) is 
constant. 

For any hypersurface X the problem of LC(X) being Cohen-Macaulay 
remains open. When X is a weighted homogeneous hypersurface with an isolated 
singularity, LC(X) is Cohen-Macaulay by <cit.>.

Let us recall that μ(f) denotes the Milnor number <cit.> of a 
germ of an analytic function f : (ℂ^n,0) → (ℂ, 0) with 
an isolated critical point at the origin and it is defined as 

    μ(f) = _ℂ𝒪_n,0/J (f ),
 where 𝒪_n,0 is the ring of germs of 
analytic functions at the origin, and J(f ) is the Jacobian ideal of f.


Using <cit.>, and assuming that L satisfies the same 
hypotheses as in Corollary <ref> we prove the following result.


	Let S_σ = ℤ_+^n and let X_σ = ℂ^n 
	be the smooth n-dimensional toric variety. Consider 
	(g,f):(X_σ,0) → (ℂ^2,0) a non-degenerate complete 
	intersection, and 
			
    g_s(x),f_t(x)  =  g(x) + 
    				∑_i=1^mξ_i(s)· l_i(x), f(x)+∑_j=1^rθ_j(t)· h_j(x)

	a family of non-degenerate complete intersections with h_j 
	and l_i satisfying the	condition (<ref>). Suppose 
	that, 
	for all s,t ∈ℂ,  X_σ^g_s⊂ℂ^n is 
	a weighted homogeneous hypersurface with 	an		isolated singularity 
	at 
	the origin. If f_t: ℂ^n →ℂ is a polynomial 
	function 
	with an isolated singularity at the origin such that f_t: 
	X_σ^g_s→ℂ has also a stratified isolated 
	singularity at the origin, then μ_BR(f_t,X_σ^g_s) is 
	constant to 		all s,t ∈ℂ.


	From <cit.> we have 
				
    μ_BR(f_t,X_σ^g_s) = μ(f_t) +  Eu_X_σ^g_s(0) + (-1)^n-1( Eu_f_t,X_σ^g_s(0) + 1).

By the hypothesis X_σ = ^n, then L(g^μ)(x) = g_μ(x), for all 
μ∈ Int(Δ̌) ∩ M(S_σ∩Δ)^*.
From definitions <ref> and <ref>, we can conclude that	
f_t:ℂ^n →ℂ is a family of non-degenerate polynomial 
functions. Furthermore,
	
    Γ_+(f) = Γ_+(f_t),   for all  t ∈ℂ,
 
	since h_j satisfies the	condition (<ref>). Then,	by 
	<cit.> 
	
    χ(f^-1(δ) ∩ B_ε) = χ(f_t^-1(δ) ∩ 
    	B_ε),   for all  t ∈ℂ,

	where 0 < | δ| ≪ε≪ 1. Consequently, μ(f_t) is constant, once 
	χ(f^-1(δ) ∩ 
	B_ε)= 1 + (-1)^n-1μ(f).  	Therefore the result follows 
	from 
	Corollary <ref> and the remark that follows Corollary 
	<ref>. 


We observe that this result is a kind of generalization of<cit.>.








§ TORIC SURFACES





Let f be a polynomial function defined on a 2-dimensional toric variety 
X_σ⊂ℂ^n. In this section, we present a characterization 
of the polynomial functions g: X_σ→ℂ which are 
prepolar with respect to 𝒯^f at the origin. Using this characterization and the results of the last sections we 
present some examples of computation of the Brasselet number B_f,X_σ, for  
a class of toric surfaces X_σ that are also determinantal.

Let us remember that a strongly convex cone in ℝ^2 has the 
following normal form.


	Let σ⊂ℝ^2 be a strongly convex cone, then 
	σ is isomorphic to the cone generated by the vectors v_1 = pe_1 - 
	qe_2 and v_2 = e_2, for some integers p,q ∈ℤ_>0 such 
	that 0<q<p and p,q are coprime.


Given a cone σ⊂ℝ^2, Riemenschneider <cit.> 
proved
 that the binomials which generate the ideal I_σ are 
given by the quasiminors of a quasimatrix, where X_σ = 
V(I_σ). In the following we recall the definition of 
quasimatrix.

 
	Given A_i, B_i, C_l,l+1∈ℂ with i=1,…,n and 
	l=1,…,n-1, a quasimatrix with entries A_i, B_i, C_l,l+1 is 
	written as
	
    A=
    	[     A_1             A_2       ⋯   A_n-1             A_n;     B_1             B_2       ⋯   B_n-1             B_n;           C_1,2               ⋯         C_n-1,n         ].

	
	The quasiminors of the quasimatrix A are defined by 
	
    A_i · B_j - B_i· (C_i,i+1· C_i+1,i_2⋯  
    	C_j-1,j)· A_j
 Adicionei pontos de multiplicacao na equacao, estao certos?ok
	for 1 ≤ i < j ≤ n.


Given σ⊂ℝ^2 generated by v_1 = p e_1 - q e_2 
and v_2 = e_2, with p and q as above, let us consider the 
Hirzebruch-Jung continued fraction 
    p/p-q = a_2 - 1/a_3 - 
    1/… - 1/a_n-1 = [[a_2, a_3, …,a_n-1]]
 where 
the integers a_2, …, a_n-1 satisfies a_i ≥ 2, for 
i=2,…,n-1. By <cit.> we have: 
completar...Esse acho que não da pra colocar pq eu encontrei essa configuração em uma traducao na verdade


	The ideal I_σ is generated by the quasiminors of the quasimatrix
	
    [           z_1                         z_2                         z_3             ⋯         z_n-2                       z_n-1;           z_2                         z_3                         z_4             ⋯         z_n-1                         z_n;                 z_2^a_2 - 2                 z_3^a_3 - 2                           ⋯               z_n-1^a_n-1-2;               ],

	where the a_i are given by the Hirzebruch-Jung continued fraction of 
	p/p-q. Moreover, this set of generators is minimal.


Then, if a_i = 2 for i=3,…, n-2, we have that X_σ is a 
determinantal surface <cit.>, in particular if the minimal dimension of 
embedding of X_σ is 4,if 
    p/p-q = a_2 - 
    1/a_3
 then X_σ is always determinantal and the ideal 
I_σ is generated by the 2 × 2 minors of the matrix


    [        z_1                   z_2            z_3^a_3 -1; z_2^a_2 -1                   z_3                   z_4;            ].


We will consider σ as in Proposition <ref>. Take 
a_2,…, a_n-1 the integers coming from the Hirzebruch-Jung 
continued fraction of p/p-q, onde entram os a's? ok we will 
denote by  
    μ_1 = 
    (μ_1^1,μ_1^2)=(1,0),   μ_2 = (μ_2^1,μ_2^2)=(1,1),   μ_i+1^j=a_i ·μ_i^j - μ_i-1^j,
 the minimal set of 
generators of S_σ, with i=2,…,n-1; j=1,2. We note that it is 
possible to show that μ_n =(μ_n^1,μ_n^2)=(q,p) (see <cit.>). Thus φ: (ℂ^*)^2 × X_σ→ X_σ 
given by 

    φ((t_1,t_2),(z_1,…,z_n)) = (t_1 · z_1,t_1 · t_2 · 
    z_2,t_1^μ_3^1· t_2^μ_3^2· z_3,…, t_1^q· 
    t_2^p· z_n)

is an action of (ℂ^*)^2 in X_σ. Each orbit of φ 
is an embedding of a d-dimensional torus, 0≤ d 
≤ 2, in X_σ. The action φ has 4 orbits, that are

    [                                                                     T_Δ_0                                                                         =                                                                {(0,…,0) };                                                                     T_Δ_1                                                                         =                                                  (t_1,0,…,0)t_1 ∈ℂ^* ≅ℂ^*;                                                                     T_Δ_2                                                                         =                                      (0,…,0,t_1^q· t_2^p)t_1,t_2 ∈ℂ^*≅ℂ^*;                                                                     T_Δ_3                                                                         = (t_1,t_1 · t_2,t_1^μ_3^1· t_2^μ_3^2,…,t_1^q· 
    t_2^p)t_1,t_2 ∈ℂ^*≅ (ℂ^*)^2 ].


Moreover, as in Section 3, 
    X_σ = _Δ_i 
    ≺σ̌ T_Δ_i,
 with i=0,1,2,3, is a decomposition of 
X_σ in strata satisfying the Whitney conditions. 

The toric surfaces obtained in Proposition <ref> are normal 
toric surfaces, then they are smooth or they have isolated singularity at the 
origin. Therefore, if f =∑_v∈ S_σ a_v · v ∈ℂ[S_σ] is a 
	non-degenerate polynomial function on X_σ, then 
	
    𝒯_f=T_Δ_i∩ X_σ^fi=0,1,2,3,

	is a Whitney stratification of X_σ^f, since T_Δ_1 and T_Δ_2 are smooth subvarieties of T_Δ_3 = 		X_σ which satisfy T_Δ_1∩T_Δ_2 = { (0,…,0)}. As a consequence,
		
    𝒯^f=T_Δ_i∖ X_σ^f, 
    			  
    			T_Δ_i∩ X_σ^f,   {0 } 
    			i=0,1,2,3,

is a good stratification of X_σ relative to f. 
	


Next, we characterize the polynomial functions 
which have a stratified isolated singularity at the origin.


	Let σ⊂ℝ^2 be a strongly convex cone and 
	𝒯 the Whitney stratification of X_σ⊂ℂ^n whose the strata are T_Δ_0, T_Δ_1, 
	T_Δ_2 and T_Δ_3. Then, a non-degenerate polynomial 
	function g: (X_σ,0) → (ℂ,0) has an isolated singularity 
	at the origin (in the stratified sense) if, 
	and only if, 
    g(z_1,…,z_n) = c_1z_1^p_1 + h(z_1,…,z_n) + 
    	c_nz_n^p_n,
 where h is a polynomial function on X_σ, 
	c_1,c_n ∈ℂ^* and p_1,p_n ∈^*_+.


 
	Let us write g as follows 
    g(z_1,…,z_n) = 
    	∑_l=1^m c_lz_1^p_1^l z_2^p_2^l… 
    	z_n^p_n^l,
 where l=1,…,m, p_i^l∈^*_+ 
	and c_l ∈ℂ.
	
	Suppose that g has a stratified isolated singularity at the origin 0 
	∈ℂ^n, with respect to the stratification 𝒯, then there must be l_1,l_n ∈{1,…,m 
	} such that
	
	
    [                       c_l_1∈ℂ^*,                 p_1^l_1≠ 0   and   p_i^l_1 = 0,   for  i ∈{2,…,n};                       c_l_n∈ℂ^*,                 p_n^l_n≠ 0   and p_i^l_n = 0,   for  i ∈{1,…,n-1} ],

	
	otherwise T_Δ_1, T_Δ_2⊂Σ_𝒯 g, since
	
    T_Δ_1  =  (t_1,0,…,0)t_1 ∈ℂ^*,  
    	T_Δ_2  =   (0,…,0,t_1^q· t_2^p)t_1,t_2 
    	∈ℂ^*.

	In other words, g must contain monomials of the 
	form c_1z_1^p_1 and c_nz_n^p_n. 
	
	Now, suppose that g has the form mentioned above, then  Γ_+(g) 
	∩Δ_1 ≠∅, Γ_+(g) ∩Δ_2≠∅ and  
	Γ_+(g) ∩Δ_3 ≠∅. By the proof of 
	<cit.> we can conclude 
	that 									g|_T_Δ_1: 
	T_Δ_1→ℂ and g|_T_Δ_2: T_Δ_2→ℂ are non-degenerate polynomial 					
	functions. Moreover,
	
    [                                                                        T_Δ_1                                                                            =                                                    (t_1,0,…,0)t_1 ∈ℂ^* ≅ℂ^*,;                                                                        T_Δ_2                                                                            =                                        (0,…,0,t_1^q· t_2^p)t_1,t_2 ∈ℂ^*≅ℂ^*,;                                                                        T_Δ_3                                                                            = (t_1,t_1 · t_2,t_1^μ_3^1· t_2^μ_3^2,…,t_1^q· 
    		t_2^p)t_1,t_2 ∈ℂ^*≅ (ℂ^*)^2. ]

	 Then by <cit.> there exists an ε>0 such 
	 that g|_T_Δ_1, g|_T_Δ_2 and 
	 g|_T_Δ_3	
	have no singularities in T_Δ_1∩ B_ε, 
	T_Δ_2∩ B_ε and T_Δ_3∩ 									B_ε, respectively. 



As a consequence of Lemma <ref> we obtain information about the 
singular set of g by
just looking at its Newton polygon Γ_+(g). More precisely, a 
non-degenerate polynomial function g: (X_σ,0) → (ℂ,0) has 
an isolated 
singularity at the origin (in the stratified sense) if, and only if, Γ_+(g) intersects 
Δ_1 and Δ_2, exactly in 
the same way as the classic case,in the case where X_σ = 
ℂ^2.



Let (g,f): (X_σ,0) → (ℂ^2,0) be a non-degenerate complete intersection, such that f and g have no irreducible components in common. The polynomial function g is prepolar with respect to 𝒯^f if, and only if, 

    g(z_1,…,z_n) = c_1z_1^p_1 + h(z_1,…,z_n) + c_nz_n^p_n,
 where 
h is a polynomial function on X_σ, c_1,c_n ∈ℂ^* 
and p_1,p_n ∈_+^*.



	Consider the good stratification 
	
    𝒯^f = T_Δ_i∖ X_σ^f,   
    	T_Δ_i∩ X_σ^f,   {0 }i=0,1,2,3,

	of X_σ relative to f. The sets T_Δ_1 and T_Δ_2 
	are given by
		
    T_Δ_1  =  (t_1,0,…,0)t_1 ∈ℂ^*  
    		T_Δ_2  =   (0,…,0,t_1^q· t_2^p)t_1,t_2 
    		∈ℂ^*.

	From the fact that f is a non-degenerate polynomial function, there are 
	only two possibilities for each stratum T_Δ_i∩ X_σ^f. Either T_Δ_i∩ X_σ^f is a finite 
	set or T_Δ_i∩ X_σ^f = T_Δ_i. Therefore, if g 
	is prepolar with respect 
	to 									𝒯^f, then Γ_+(g) 
	∩Δ_1≠∅ and Γ_+(g) ∩Δ_2≠∅, otherwise 						T_Δ_i∩ 
	X_σ^f⊂Σ_𝒯^f g or T_Δ_i∖ 
	X_σ^f⊂Σ_𝒯^f g, for 			i=1,2. 
	
	
	Now, suppose that g have the form mentioned above, then the result 
	follows 
	from Lemma <ref> and from the fact that, as f and g have no 
	irreducible components in common,X_σ^g∩ X_σ^f 
	is a finite set.



	Let σ⊂ℝ^2 be the cone ge­ne­ra­ted by the 
	vectors v_1 = e_2 and v_2 = ne_1 - e_2. The toric surface associated 
	to σ is X_σ = V(I_σ) ⊂ℂ^n+1, 
	where I_σ is the ideal generated by the 2 × 2 minors of 
	the matrix
	
    [   z_1   z_2   z_3     … z_n-1   z_n;   z_2   z_3   z_4     …   z_n z_n+1 ],

	i.e., X_σ is a determinantal 
	surface with codimension n-1. Consider f: X_σ→ℂ 
	the function given by 
	f(z_1,…,z_n+1)=z_1^d + z_n+1^d + tg(z_1,…,z_n+1), where 
	
    g(z_1,…,z_n+1) = ∑_l=1^m z_1^p_1^l 
    	z_2^p_2^l… z_n+1^p_n+1^l

	is a polynomial function on X_σ satisfying p_1^l + 
	p_2^l + … + p_n+1^l > d  
	for every l=1,…,m. If f is a 
	non-degenerate polynomial function then 
	
    B_f,X_σ(0)= 2d -nd^2.

	Indeed, consider h:X_σ→ℂ the function given by h(z_1,…,z_n+1) = z_1^d + 
	z_n+1^d. The Newton polygon Γ_+(h) has an unique 
	1-dimensional compact face β_1, that is the straight line segment 
	connecting the points (d,0) and (d,nd) in σ̌. Using the same notation of Proposition <ref>, we have that 
	Γ_1^Δ_1 is the straight line segment connecting the points 
	(0,0) and (d,0), Γ_1^Δ_2 is the straight line segment 
	connecting the points (0,0) and (d,nd) and Γ_1^Δ_3 is 
	the triangle with vertices (0,0), (d,0) and (d,nd). Therefore, 
	
    Vol_ℤ(Γ_1^Δ_1)=Vol_ℤ(Γ_1^Δ_2)
    	 = d   and  Vol_ℤ(Γ_1^Δ_3)=nd^2.

	By Proposition <ref>,
	
    B_h,X_σ(0)=2d-nd^2,

	since X_σ has an isolated singularity at the origin, and 
	consequently Eu_X_σ(T_Δ_1) = 
	 													
	Eu_X_σ(T_Δ_2)
	 =  	Eu_X_σ(T_Δ_3) = 1.
	Now, S_σ is the semigroup generated by 
	
    {(1,0),(1,1),(1,2)…,(1,n) },

	and then Γ_+(g) ⊂Γ_+(h). Moreover, by Lemma	<ref>, f has an isolated 
	singularity at the origin, 
	thus 
	
    B_f,X_σ(0)=  Eu_X_σ(0) -  
    	Eu_f,X_σ(0).

	However, González-Sprinberg <cit.> proved that Eu_X_σ(0) = 3-(n+1). Hence, 
	
    Eu_f,X_σ(0) = 3-(n+1)-2d+nd^2.

	Therefore, a morsefication of f 		has 3-(n+1)-2d+nd^2 Morse points 
	on the regular part of X_σ. 


 colocar uma familia em g tambem e usar esse exemplo no final da secao 3 e tb no final da secao 5
	Let σ⊂ℝ^2 be the cone ge­ne­ra­ted by the 
	vectors v_1 = e_2 and v_2 = 2e_1 - e_2. The toric surface associated 
	to σ is X_σ = V(I_σ) ⊂ℂ^3, with 
	I_σ the ideal generated by z_1z_3 - z_2^2. Consider f: 
	X_σ→ℂ the function given by f(z_1,z_2,z_3)=z_2^2 - 
	z_1^3, which is a non-degenerate polynomial function, whose singular 
	set is 
	
    Σ f = (0,0,z_3)z_3 ∈ℂ⊂ X_σ.

	Moreover, Γ_+(f) has a unique 1-dimensional 
	compact face β_1, which is the straight line segment connecting the 
	points (3,0) and (2,2) in σ̌. Thus, 
	Γ_1^Δ_1 is the straight line segment connecting the points 
	(0,0) and (3,0), Γ_1^Δ_3 is the triangle with vertices 
	(0,0), (3,0) and (2,2), and Γ_1^Δ_2 = ∅. 
	Therefore, by Proposition <ref>, 
	
    B_f,X_σ(0)=3-6=-3

	since X_σ has an isolated singularity at the origin, and 
	consequently 
	
    Eu_X_σ(T_Δ_1) =  Eu_X_σ(T_Δ_2) =  	Eu_X_σ(T_Δ_3) = 1.

	Now let g: X_σ→ℂ be the non-degenerate polynomial function given by 
	
    g(z_1,z_2,z_3)= z_1 - z_3^2,

	 which is prepolar 
	with respect to 𝒯^f. Moreover, (g,f) is a non-degenerated 
	complete intersection. The Newton polygon Γ_+(g· f) has two 
	1-dimensional 
	compact faces γ_1 and γ_2, which are the straight line 
	segment connecting the points (4,0) and (3,2) and the straight line 
	segment connecting the points (3,2) and (4,6), respectively. Thus, the 
	primitive vectors 
	
    u_1^Δ_3, u_2^Δ_3∈ 
    	Int(Δ̌_̌3̌) ∩ M(S_σ∩Δ_3)^*
 
	which take their minimal value in Γ_+(g· f) ∩Δ_3 exactly on γ_1 
	and γ_2, respectively, are u_1^Δ_3=(2,1) and 
	u_2^Δ_3=(4,-1). Let us observe that
	
	
    [        γ(g)_1^Δ_3                := Γ(g|_Δ_3;u_1^Δ_3)                 =          {(1,0) };        γ(g)_2^Δ_3                := Γ(g|_Δ_3;u_2^Δ_3)                 =               α_1;           d_1^Δ_3                :=           d_2^Δ_3                :=                 6;           K_1^Δ_3                 =           K_2^Δ_3                 =                 1 ],

	
	where α_1 is the 1-dimensional compact face of 
	Γ_+(g). Applying Theorem	<ref>, we have 
	B_f,X_σ^g(0)= 12.
	Therefore, we obtain the following equality 
	
    B_f,X_σ(0) -  
    	B_f,X_σ^g(0) = -3 - 12 =-15,

	which means that the number 
	of stratified Morse critical points on the top stratum T_Δ_3∩ 
	f^-1(δ)∩ B_ε(0) appearing in a morsefication of g : 
	X_σ∩ f^-1(δ) ∩ B_ε(0) →ℂ is 
	15. Moreover, if we consider h, l: X_σ→ℂ the 
	polynomial functions given by 
				
    h(z_1,z_2,z_3)  =  -z_1^2 z_3^2,     
    					l(z_1,z_2,z_3)  =  z_3^3,

	and	observe that 
			
    Γ_+(h)  ⫋Γ_+(f),     Γ_+(l)  ⫋Γ_+(g),

	by Corollary <ref> we have 
				
    B_f_t,X_σ(0) =  B_f,X_σ(0) 
    					=  -3,      B_f_t,X_σ^g_s(0)  =   
    					B_f,X_σ^g(0) =  12,

	where f_t(x)=f(x)+t · h(x) is a deformation of the cusp 
	f_0(z_1,z_2,z_3) = z_2^2 - z_1^3 (see Figure 
	(<ref>)) and g_s(x)=g(x)+ s · l(x). 
	Consequently,
	
    B_f_t,X_σ(0) -  
    	B_f_t,X_σ^g_s(0) = -3 - 12 =-15,

 	for all t, s ∈ℂ. 











§ INDICES OF VECTOR FIELDS





A toric surface X_σ, which is a cyclic quotient singularity,
always possesses a smoothing <cit.>. Therefore, when we consider a radial continuous vector field v on
X_σ with an isolated singularity at 0, we can relate the Euler 
characteristic of a fiber of this smoothing with the
GSV index of v in X_σ. The definition of this index for 
smoothable isolated singularity can be found in <cit.>. 


In the particular, for the case of toric surfaces 
which are also isolated determinantal singularities, we have the following result concerning GSV index.

Let X_σ⊂ℂ^n be a toric surface that is also an 
isolated determinantal singularity,σ is generated by the 
vectors v_1 = p e_1 - q e_2 and v_2 = e_2, where 0 < q < p, p,q are coprime,
and whose the Hirzebruch-Jung continued fraction is 
	
    p/p-q = [[a_2, 2, 2, …, 2, a_n-1]].

Consider f_t(x) = f(x)+∑_j=1^rθ_j(t)· h_j(x) a family of 
non-degenerate polynomial 
functions on X_σ, which satisfies the conditions
		
    Γ_+(h_j) ⫋Γ_+(f),   for all  j=0,…,r.

If this family has an isolated singularity at the origin, then the 
following result holds.


	Let v_t be the vector field given by the gradient of the function 
	f_t. Then, the following are equivalent:
	
  * (a) Eu_f_t,X_σ(0) is constant for the family;
	
	
  * (b) Ind_GSV(v_t,X_σ,F) is constant for the family, 
	where F is the flat map associated to the smoothing of X_σ.



	By <cit.> the determinantal Milnor number of the 
	function f on the Isolated Determinantal Singularity X_σ is 
			
    μ(f|_X_σ) = #Σ(f̃|_X_σ_s),

	where X_σ_s is a fiber of a smoothing of X_σ, f̃|_X_σ_s is a morsefication of f and #Σ(f̃|_X_σ_s) denote the number of Morse points of 
	f̃ on X_σ_s. From the definition of the GSV 
	index in the case of smoothable varieties (see <cit.>) we have
	
    μ(f|_X_σ) =  Ind_GSV(v,X_σ,F).
 Then the 
	proof follows by <cit.>, where it is proved that Eu_f_t,X_σ(0) is constant for the 
	family if and only if μ(f_t|_X_σ) is constant for the family.


In <cit.>, the authors extended the concept of GSV index and proved a Lê-Greuel formula (see <cit.>) which holds with the same hypotheses of Theorem <ref>. 
However, in <cit.> the authors worked with the constructible function 
given by the characteristic function, while in <cit.> is considered the 
local Euler obstruction. Hence the GSV index and the Brasselet number 
are not related in general. 


Assuming that f_t is generically a submersion, for non-degenerate complete intersections, we have the following result.

 Troquei o corolario por proposicao. ok
	Let S_σ = ℤ_+^n and X_σ = ℂ^n be the smooth n-dimensional toric variety. Let (g,f): (X_σ,0) → 			(ℂ^2,0) be a non-degenerate complete intersection, such that 𝒯_g is a Whitney stratification of X_σ^g. If 
	
    g_s(x),f_t(x)  =  g(x) + 
    	∑_i=1^mξ_i(s)· l_i(x), f(x)+∑_j=1^rθ_j(t)· h_j(x)

	is a family of non-degenerate complete 
	intersections with h_j and l_i satisfying	the	condition 
	(<ref>)
	for all i=1,…,m and j=1,…,r, such that 𝒯_g_s is a Whitney stratification of X_σ^g_s and g_s is prepolar 			with respect to 𝒯^f_t at the origin. Then, Ind_GSV(g_s, 0; f_t ) is invariant to the family.



	In <cit.>, the authors used <cit.> (considering the Euler characteristic as constructible function) to provide 		the following interpretation to the GSV index,

    ∑_Δ≺σ̌( χ(T_Δ∩ X_σ∩ B_ε(0) ∩ f^-1(δ) ) - χ(T_Δ∩ X_σ^g∩ B_ε(0) ∩ f^-1(δ) ) ) =  Ind_GSV(g, 0; f ),

	where Ind_GSV(g, 0; f ) is the GSV-index of g on X^f relative to the 
	function f (see <cit.>). Moreover, f_t is a 
	family of non-degenerate polynomial functions, since (g_s,f_t) 
	is 								non-degenerate complete intersections, 
	for all s,t ∈ℂ. Then, to compute Ind_GSV(g, 0; f 
	) we apply <cit.> to the 
	first 
	term of the 			equality above and  to the 
	second term of the equality above we used nearbycicle1, 
	(<ref>). Therefore, the 		result follows from the fact 
	that 
	h_j 
	and l_i 				satisfying	the	condition 				
	(<ref>)	for all i=1,…,m and j=1,…,r.







 
	Consider the toric surface X_σ = V(I_σ) ⊂ℂ^3, with 
	I_σ the ideal generated by z_1z_3 - z_2^2. Let f_t and g_s be the same families of functions from Example <ref>, 			then
				
    Ind_GSV(g_s, 0; f_t ) = -15

	for all t, s ∈ℂ.






§ ACKNOWLEDGMENTS


The authors are grateful to Nivaldo de Góes Grulha Jr. from ICMC-USP for helpful
conversations in developing this paper and to Bruna Oréfice Okamoto from 
DM-UFSCar for helpful conversations about the Bruce-Roberts' Milnor number.
Through the project CAPES/PVE Grant  88881. 
068165/2014-01 of the program Science without borders, 
Professor Mauro Spreafico visited the DM-UFSCar in São Carlos providing 
useful discussions with the authors. Moreover, the
authors were partially supported by this project, therefore we are grateful 
to this program. We would like to thank the referee for 
many valuable suggestions which improved this paper.



amsalpha-lmp





