An Algorithm to Classify the Asymptotic Set Associated to a Polynomial Mapping

http://www.scirp.org/journal/ajcm ISSN Online: 2161-1211 ISSN Print: 2161-1203

An Algorithm to Classify the Asymptotic Set

Associated to a Polynomial Mapping

Nguyen Thi Bich Thuy

IBILCE, UNESP, Universidade Estadual Paulista, "Júlio de Mesquita", São José do Rio Preto, São Paulo, Brazil

Abstract

We provide an algorithm to classify the asymptotic sets of the dominant po-lynomial mappings 3 3

:

F  → of degree 2, using the definition of the

so-called “façons” in [2]. We obtain a classification theorem for the asymptotic sets of dominant polynomial mappings 3 3

:

F  → of degree 2. This

algo-rithm can be generalized for the dominant polynomial mappings _F:n_→n

of degree d, with any

( )

n d, ∈

( )

∗ 2.

Keywords

Algorithms, Polynomial Mappings, Asymptotic Sets, Façons

1. Introduction

Let _F:n→n be a polynomial mapping. Let us denote by F

S the set of

points at which F is non proper, i.e.,

{ }

( )

{

n such that n, tends to infinity and tends to

}

,

F k k k k

S = a∈ ∃ ξ _∈ ⊂ ξ F ξ a

where ξk is the Euclidean norm of ξk in

n

 . The set SF is called the

asymptotic set of F. The comprehension of the structure of this set is very important by its relation with the Jacobian Conjecture. In the 90’s, Jelonek studied this set in a deep way and described the principal properties. One of the important results is that, if F is dominant, i.e.,

( )

n n

F  = , then S_F is an

empty set or a hypersurface [1].

Notice that it is sufficient to define SF by considering sequences

{ }

ξk

tending to infinity in the following sense: each coordinate of these sequences either tends to infinity or converges. In [2], the sequences tending to infinity such that their images tend to the points in SF are labeled in terms of “façons”,

as follows: We rank the coordinates of ξk into three categories: 1) the

coordinates tending to infinity (this cotegories is not empty); 2) the coordinates

How to cite this paper: Thuy, N.T.B. (2017) An Algorithm to Classify the Asymp-totic Set Associated to a Polynomial Map-ping. American Journal of Computational Mathematics, 7, 109-126.

https://doi.org/10.4236/ajcm.2017.71010

Received: January 18, 2017 Accepted: March 28, 2017 Published: March 31, 2017

Copyright © 2017 by author and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).

( )

xk i, such that limk→∞xk i, is a complex number “independant on the point a in a neighborhood a in SF”. This means that there exists the points neighbors of a

in SF and the sequences

{ }

ξk′ such that limk→∞F

( )

ξk′ =a′ and

, ,

lim_k→∞x_{k i}′ =lim_k→∞x_{k i}, 3) the coordinates

( )

xk i, such that limk→∞xk i, is a complex number “dependant on the point a”. This means that there not exist such points a′ neighbors of a in S_F . The example 2.5 illustrates these three

categories.

We define a “façon” of the point a∈SF as a

(

p q,

)

-tuple

(

i1,,ip

)

j1,,jq of integers where ,r

a k i

x tends to infinity for r=1,,p

and, for s=1,,q, the sequence _,

s

a k j

x tends to a complex number independently

on the point a when a describes locally SF (definition 2.7).

The aim of this paper is to provide an algorithm to classify the asymptotic sets of dominant polynomial mappings 3 3

:

F  → of degree 2, using the definition of

“façons” in [2], and then generalize this algorithm for the general case. One important tool of the algorithm is the notion of pertinent variables. The idea of the notion of pertinent variables is the following: Let

(

)

3 3

1, 2, 3 :

F = F F F  → be a

dominant polynomial mapping of degree 2 such that SF ≠ ∅. We fix a façon κ

of F and assume that

{ }

ξk is a sequence tending to infinity with the façon κ

such that F

( )

ξk tends to a point of SF. Since the degree of F is 2 then each

coordinate F F1, 2 and F3 of F is a linear combination of f1=x1, f2=x2,

3 3

f =x , f₄ =x x_{1 2}, f₅=x x_{2 3} and f₆=x x_{3 1}. We call a pertinent variable of F

with respect to the façon κ a minimum linear combination of f1,,f6 such that the image of the sequence

{ }

ξk by this combination does not tend to

infinity (see definition 3.1).

Moreover, if F is dominant then by Jelonek, the set SF has pure dimension 2

(see theorem 2.4). With this observation and with the idea of pertinent variables, we:

• _{Make the list}

( )

 of all possible façons for a polynomial mapping

3 3

:

F  → . This list is finite. In fact, there are 19 possible façons (see the list

(3.4)).

• _{Assume that a 2-dimensional irreductible stratum S of} S_F admits l fixed

façons in the list

( )

 , where 1≤ ≤l 19.

• _{Determine the pertinent variables of F with respect to these l façons.}

• _{Restrict the above pertinent variables using the dominancy of F and the fact}

that the dimension of S is 2. We get the form of F in terms of these pertinent variables.

• _{Determine the geometry of S in terms of the form of F.}

• _{Let l runs in the list}

( )

 for 1≤ ≤l 19. We get all the possible 2-dimen- sional irreductible strata of SF. Since the dimension of SF is 2, then we get

the list of all possible asymptotic sets SF.

With this idea, we provide the algorithm 3.10 to classify the asymptotic sets of dominant polynomial mappings 3 3

:

F  → of degree 2, and we obtain the

(algorithm 5.1).

2. Dominancy, Assymptotic Set and “Façons”

2.1. Dominant Polynomial Mapping

Definition 2.1. Let _F:n_→n be a polynomial mapping. Let

( )

n

F  be the

closure of _F

( )

_n in n

 . F is called dominant if

( )

n n

F  = , i.e., F

( )

n is

dense in n

 .

We provide here a lemma on the dominancy of a polynomial mapping : n n

F  → that we will use later on.

Lemma 2.2. Let

(

1, ,

)

:

n n

n

F = F  F  → be a dominant polynomial

mapping. Then, the coordinate polynomials F1,,Fn are independent. That

means, there does not exist any coordinate polynomial F_η, where η∈

{

1,,n

}

,

such that F_η is a polynomial mapping of the variables F₁,,F_η−1,F_η+1,,F_n.

Proof. Assume that Fη =ϕ

(

F1,,Fη−1,Fη+1,,Fn

)

where η∈

{

1,,n

}

and ϕ is a polynomial. Then, the dimension of

( )

n

F  is less than n.

Consequently, the dimension of

( )

n

F  is less than n. That provides the

contradiction with the fact F is dominant.

2.2. Asymptotic Set

Definition 2.3. Let F:n→n be a polynomial mapping. Let us denote by

F

S the set of points at which F is non-proper, i.e.,

{ }

( )

{

n such that n, and

}

,

F k k k k

S = a∈ ∃ ξ _∈ ⊂ ξ → ∞ F ξ →a

where ξk is the Euclidean norm of ξk in

n

 . The set SF is called the

asymptotic set of F.

Recall that, it is sufficient to define SF by considering sequences

{ }

ξk

tending to infinity in the following sense: each coordinate of these sequences either tends to infinity or converges to a finite number.

Theorem 2.4. [1] Let _F:n→n be a polynomial mapping. If F is

dominant, then SF is either an empty set or a hypersurface.

2.3. “Façons”

In this section, let us recall the definition of façons as it appears in [2]. In order to a better understanding of the definition of façons, let us start by giving an example.

Example 2.5. [2] Let

(

)

(1 2 3) (1 2 3)

3 3

1, 2, 3 : x x x, , , ,

F= F F F  →_{α α α} be the polynomial

mapping such that

1: 1, 2: 2, 3: 1 2 3.

F =x F =x F =x x x

Notice that by the notations (1 2 3)

3 , ,

x x x

 and ( 1 2 3)

3 , ,

α α α

 , we want to distinguish the source space and the target space. We determine now the asymptotic set SF by using the definition 2.3. Assume that there exists a

sequence

{

ξk =

(

x1,k,x2,k,x3,k

)

}

in the source space (_{1 2 3})

3 , ,

x x x

1,k

x and x_2,k cannot tend to infinity. Since the sequence

{ }

ξ_k tends to

infinity, then x3,k must tend to infinity. Hence, we have the three following

cases:

1) x1,k tends to 0, x2,k tends to a complex number α2∈ and x3,k tends

to infinity. In order to determine the biggest possible subset of SF, we choose

the sequences x1,k tending to 0 and x3,k tending to infinity in such a way that

the product x x1,k 3,k tends to a complex number α3. Let us choose, for example

3 2 2 1 , , k k k α ξ α α   =  

  where α2 ≠0, then F

( )

ξk tends to a point a=

(

0,α α2, 3

)

in SF. We get a 2-dimensional stratum S1 of SF, where S1=

(

α1 =0 \

)

( 1 2 3) 3

3 , ,

0α ⊂_{α α α} . We say that a “façon” of S1 is

( )

3 1

[ ]

. The symbol “(3)” in the façon

( )

3 1

[ ]

means that the third coordinate x_3,k of the sequence

{ }

ξk

tends to infinity. The symbol “[1]” in the façon

( )

3 1

[ ]

means that the first coordinate x1,k of the sequence

{ }

ξk tends to 0 which is a fixed complex

number which does not depend on the point a=

(

0,α α2, 3

)

when a describes

1

S . Notice that the second coordinate of the sequence

{ }

ξk tends to a complex

number α2 depending on the point a=

(

0,α α2, 3

)

when a varies, then the indice “2” does not appear in the façon

( )

3 1

[ ]

. Moreover, all the sequences tending to infinity such that their images tend to a point of S1 admit only the façon

( )

3 1

[ ]

.

The two following cases are similar to the case 1):

2) x1,k tends to a complex number α1∈, x2,k tends to 0 and x3,k tends

to infinity: then the façon

( )

3 2

[ ]

determines a 2-dimensional stratum S₂ of

F

S , where

(

)

_{( )}

1 2 3

3 2 2 0 \ 0 3 , ,

S = α = α ⊂_{α α α} .

3) x1,k and x2,k tend to 0, and x3,k tends to infinity: then the façon

( )

3 1, 2

[ ]

determines the 1-dimensional stratum S₃ where S₃ is the axis 0α₃ in ( 1 2 3)

3 , ,

α α α

 .

In conclusion, we get

• _{the asymptotic set} S_F of the given polynomial mapping F as the union of

two planes

(

α1=0

)

and

(

α2 =0

)

in ( _{1 2 3})

3 , ,

α α α

 ,

• _{all the façons of} S_F of the given polynomial mapping F: they are three

façons

( )

3 1

[ ]

,

( )

3 2

[ ]

and

( )

3 1, 2

[ ]

.

Remark 2.6. The chosen sequence 3 2 2 1 , , k k k α ξ α α     ₌       

  in 1) of the above

example is called a generic sequence of the 2-dimensional irreductible component

(

α1=0

)

(a plane) of SF, since the image of any sequence of this

type (with differents α2 ≠0 and α3) falls to a generic point of the plane

(

α1=0

)

. That means the images of all the sequences

{ }

ξk when α2 runs in

{ }

\ 0

 and α3 runs in  cover S1=

(

α1=0 \ 0

)

α3 and S1 is dense in the plane

(

α1=0

)

. We can see easily that a generic sequence of the 2-dimensional irreductible component

(

α2 =0

)

of SF is 1 3

1

1 , ,k

k α α α    

  where α1≠0. More

generally, any sequence 3 2 2 1 , , r r k k α α α          

generic sequence of

(

α1=0

)

⊂SF. Any sequence 1 3 1 1 , ,

r r

k k

α α

α

 

 

 

 

 , where

{ }

\ 0

r∈ and α₁≠0, is a generic sequence of

(

α₂ =0

)

⊂S_F.

In the light of this example, we recall here the definition of façons in [2]. Definition 2.7. [2] Let F:n→n be a dominant polynomial mapping such that SF ≠ ∅. For each point a of SF, there exists a sequence

{ }

a n

k

ξ ⊂_{ ,}

(

,1, , ,

)

a a a

k xk xk n

ξ =  tending to infinity such that

( )

a k

F ξ tends to a. Then,

there exists at least one index i∈, 1≤ ≤i n such that a_, k i

x tends to infinity

when k tends to infinity. We define “a façon of tending to infinity of the sequence

{ }

a

k

ξ _{”, as a maximum}

(

p q,

)

-tuple κ=

(

i₁,,i_p

)

_j₁,,j_q_ of

different integers in

{

1,,n

}

, such that:

1) ,r

a k i

x tends to infinity for all r=1,,p,

2) for all s=1,,q , the sequence _,

s

a k j

x tends to a complex number

independently on the point a when a varies locally, that means:

a) either there exists in SF a subvariety Ua containing a such that for any

point a′ in U_a, there exists a sequence

{ }

ξ_ka′ ⊂n , ξka

(

xka_,1, ,xk na_,

)

′₌ ′ ′



tending to infinity such that i)

( )

a

k

F ξ ′ tends to a′,

ii) ,_r a k i

x′ tends to infinity for all r=1,,p,

iii) for all s=1,,q, lim ,_s lim ,_s

a a

k j k j

k x k x

′

→∞ = →∞ and this limit is finite.

b) or there does not exist such a subvariety, then we define

(

i1, ,ip

)

j1, ,jn p ,

κ = _ _{  −} _

where ,r

a k i

x tends to infinity for all r=1,,p and

{

i1,,ip

} {

∪ j1,,jn p−

}

=

{

1,,n

}

. In this case, the set of points a is a

subvariety of dimension 0 of SF.

We call a façon of tending to infinity of the sequence

{ }

a k

ξ also a a façon of a or a façon of SF.

3. An Algorithm to Stratify the Asymptotic Sets of the

Dominant Polynominal Mappings

F

:



3

→



3

of Degree

2

In this section we provide an algorithm to stratify the asymptotic sets associated to dominant polynominal mappings _F:3_→3 of degree 2. In the last section, we show that this algorithm can be generalized in the general case for dominant polynominal mappings _F:_n_→n of degree d where

3 n≥ and 2

d≥ . Recall that by degree of a polynomial mapping F =

(

F1,,Fn

)

: n_→ n

  , we mean the highest degree of the monomials F1,,Fn.

Let us consider now a dominant polynomial mapping F =

(

F F F1, 2, 3

)

:

(1 2 3) ( 1 2 3)

3 3

, , , ,

x x x → α α α

  of degree 2 such that SF ≠ ∅. An important step of this

section is to define the notion of “pertinent” variables of F.

3.1. Pertinent Variables

(

)

{

x1,k,x2,k,x3,k

}

be a sequence in the source space (_{1 2 3})

3 , ,

x x x

 tending to infinity such that F

( )

ξk tends to a point of SF in the target space ( _{1 2 3})

3 , ,

α α α

 . Then the image of ξk by any coordinate polynomial Fη, where η =1, 2, 3, cannot

tend to infinity. Notice that F_η can be written as the sum of elements of the

form 1 2

F Fη η such that if Fη1

( )

ξk tends to infinity, then

( )

2

k

Fη ξ must tend to

0. In other words, if one element of the above sum has a factor tending to infinity with respect to the sequence

{ }

ξk , then this element must be

“balanced” with another factor tending to zero with respect to the sequence

{ }

ξk . For example, assume that the coordinate sequences x1,k and x2,k of

the sequence

{ }

ξk tend to infinity, then Fη cannot admit neither x1 nor x2 alone as an element of the above sum, but F_η can admit

(

x₁−νx₂

)

,

(

x1−νx2

)

x1,

(

x1−νx2

)

x2 as elements of this sum, where ν∈\ 0

{ }

. So we define:

Definition 3.1. Let

(

)

(1 2 3) (1 2 3)

3 3

1, 2, 3 : x x x, , , ,

F= F F F  →_{α α α} be a polynomial

mapping of degree 2 such that SF ≠ ∅. Let us fix a façon κ of SF. Then

there exists a sequence

{ }

(1 2 3)

3 , ,

k x x x

ξ ⊂_{ tending to infinity with the façon} κ

such that its image tend to a point in SF. An element in the list

(

)

, where 1, 2, 3,

, where and , 1, 2, 3, , where and , , 1, 2, 3,

, where , , and , , 1, 2, 3,

i

j j

r r

s s

h i

h i h j

h i h j l

h i h j l

X x i

X x x i j i j

X x x x i j i j l

X x x x i j j k k i i j k

ν ν ν = =   _{= + ≠ =}   _{= + ≠ =}   _{= + ≠ ≠ ≠ =}  (3.2)

{ }

(

, , , \ 0

)

i j r s

h h h h

ν ν ν ν ∈ is called a pertinent variable of F with respect to the façon κ _{if the image of the sequence}

{ }

ξk by this element does not tend to

infinity.

Remark 3.3. From now on, we will denote X1,,Xh pertinent variables of

F with respect to a fixed façon and we write

(

1, , h

)

. F =F X  X

Notice that we can also determine the pertinent variables of F with respect to a set of façons in the case we have more than one façon.

3.2. Idea of the Algorithm

The aim of the algorithm that we present in this section is to describe the list of all possible asymptotic sets SF for the dominant polynomial mappings

3 3

:

F  → of degree 2. In order to do that, we observe firstly that

• _{The list of all the possible façons of} S_F for a polynomial mapping

3 3

:

F  → is

)

(

)

)

( ) ( )

( )

)

( ) ( )

( )

)

( )

[ ]

( )

[ ]

( )

[ ]

)

( )

[ ]

( )

[ ]

( )

[ ] [ ]

( )

[ ]

( )

[ ]

( )

)

( )

[ ]

( )

[ ]

( )

[ ]

1 GroupI : 1, 2, 3 ,

2 GroupII : 1, 2 , 2, 3 and 3,1 , 3 GroupIII : 1 , 2 and 3 ,

4 GroupIV : 1, 2 3 , 1, 3 2 and 2, 3 1 ,

5 GroupV : 1 2 , 1 3 , 2 1 , 2 3 , 3 1 and 3 2 , 6 GroupVI : 1 2, 3 , 2 1, 3 and 3 1, 2 .

This list has 19 façons.

• _{Since F dominant, then by the theorem 2.4, the set} S_F has pure dimen- sion

2.

With these observations, we will:

• _{assume that a 2-dimensional irreductible stratum S of} S_F admits l fixed

façons in the list (3.4), where 1≤ ≤l 19,

• _{determine the pertinent variables of F with respect to these l façons,}

• _{restrict the above pertinent variables by using the dominancy of F and the}

fact dimS=2. We get the form of F in terms of these pertinent variables,

• _{determine the geometry of S in terms of the form of F,}

• _{let l run in the list (3.4) for} 1≤ ≤l 19. We get all the possible 2-dimen- sional irreductible strata S of SF. Since the dimension of SF is 2, then we get the

list of all the possible asymptotic sets SF of F.

The following example explains the process of the algorithm, i.e. how we can determine the geometry of a 2-dimensional irreductible stratum S of SF

admitting some fixed façons.

3.3. Example

Example 3.5. Let 3 3 :

F  → be a dominant polynomial mapping of degree

2. Assume that a 2-dimensional stratum S of SF admits the two façons

(

1, 2, 3

)

κ = _and κ′ =

( )

1, 2 3

[ ]

. That means that all the sequences tending to infinity in the source space such that their images tend to the points of S admit either the façon κ =

(

1, 2, 3

)

or the façon κ′ =

( )

1, 2 3

[ ]

. In order to describe the geometry of S, we perform the following steps:

Step 1: Determine the pertinent variable of F with respect to the façons

(

1, 2, 3

)

κ = _and κ′ =

( )

1, 2 3

[ ]

:

• _{With the façon} κ=

(

1, 2, 3

)

, all the three coordinate sequences of the corresponding sequence tend to infinity (cf. Definition 2.7). Up to a suiable linear change of coordinates, the mapping F admits the pertinent variables:

1 2

x −x ,

(

x₁−x₂

)

x₁ ,

(

x₁−x₂

)

x₂ ,

(

x₁−x₂

)

x₃ , x₁−x₃ ,

(

x₁−x₃

)

x₁ ,

(

x1−x3

)

x2,

(

x1−x3

)

x3, x2−x3,

(

x2−x3

)

x1,

(

x2−x3

)

x2,

(

x2−x3

)

x3,

1 2 3

x −x x , x₂−x x_{1 3} and x₃−x x_{1 2} (see definition 3.1).

• _{With the façon} κ′ =

( )

1, 2 3

[ ]

, the first and second coordinate sequences of the corresponding sequence tend to infinity, the third coordinate sequence of the corresponding sequence tends to a fixed complex number. As we refer to the same mapping F, then up to the same suiable linear change of coor- dinates, the mapping F admits the pertinent variables: x3, x x1 3, x x2 3,

1 2

x −x ,

(

x₁−x₂

)

x₁,

(

x1−x2

)

x2,

(

x1−x2

)

x3 and

(

x1−x3

)

x3.

Since S contains both of the façons κ _and κ′_{, then this surface S admits}

1 2

x −x ,

(

x₁−x₂

)

x₁,

(

x1−x2

)

x2,

(

x1−x2

)

x3 and

(

x1−x3

)

x3 as pertinent variables. Let us denote by

(

)

(

)

(

)

(

)

1 1 2 2 1 2 1 3 1 2 2

4 1 2 3 5 1 3 3

, , ,

, .

X x x X x x x X x x x

X x x x X x x x

= − = − = −

We can write

(

1, 2, 3, 4, 5

)

.

F=F X X X X X (3.6)

Step 2: Assume that

{

ξ =k

(

x1,k,x2,k,x3,k

)

}

and

{

ξ′k =

(

x1,′k,x2,′k,x3,′k

)

}

are two

sequences tending to infinity with the façons κ and κ′_{, respectively.}

A) Let us consider the façon κ=

(

1, 2, 3

)

and its corresponding generic sequence

{

ξ =k

(

x1,k,x2,k,x3,k

)

}

:

• _{Assume that} X₁

_{( )}

ξ =_k

(

x_1,k−x_2,k

)

tends to a non-zero complex number.

Since κ =

(

1, 2, 3

)

then all three coordinate sequences x_1,k,x_2,k and x3,k

tend to infinity. Hence X2

( )

ξk =

(

x1,k−x2,k

)

x1,k, X3

( )

ξk =

(

x1,k−x2,k

)

x2,k

and X4

( )

ξk =

(

x1,k−x2,k

)

x3,k tend to infinity. In this case, X2, X3 and

4

X cannot be pertinent variables of F anymore. Then F admits only two

pertinent variables X1 and X5, or F=F X X

(

1, 5

)

. We can see that the dimension of S in this case is 1, that provides a contradiction with the fact that the dimension of S is 2. Consequently,

(

x1,k −x2,k

)

tends to 0.

• _{Assume that}

(

x_1,k−x_3,k

)

tends to a non-zero complex number. Then

( )

(

)

5 k 1,k 3,k 3,k

X ξ = x −x x tend to infinity, hence X5 cannot be a pertinent variable of F anymore, then F=F X X

(

1, 2,X3,X4

)

. We choose a generic sequence

{ }

ξk satisfying the conditions: X1,k =

(

x1,k −x2,k

)

tends to zero

and

(

x1,k−x3,k

)

tends to a non-zero complex number, for example, ξk =

(

k+α k k, +β k k, +γ

)

. Then X₂

( )

ξ_k =

(

x_1,k−x_2,k

)

x_1,k , X3

( )

ξk =

(

x1,k−x2,k

)

x2,k and X4

( )

ξk =

(

x1,k−x2,k

)

x3,k tend to the same complex

number λ µ− _{. Combining with the fact} X_1,k =

(

x_1,k−x_2,k

)

tends to zero,

we conclude that the dimension of S in this case is 1, that provides a contradiction with the fact that the dimension of S is 2. Consequently,

(

x1,k−x3,k

)

tends to 0.

Then, with the façon κ_{, we have}

(

x_1,k−x_2,k

)

and

(

x_1,k−x_3,k

)

tend to 0.

Hence

(

x2,k−x3,k

)

also tends to 0. Let us choose a generic sequence

{ }

ξk

satisfying these conditions, for example, the sequence

(

)

{

ξ_k = k+α k k, +β k k, +γ k

}

. We see that X2,k =

(

x1,k−x2,k

)

x1,k,

(

)

3,k 1,k 2,k 2,k

X = x −x x and X_4,k =

(

x_1,k −x_2,k

)

x_3,k tend to a same complex

number λ α β= − _{. Moreover,} X_5,k =

(

x_1,k−x_3,k

)

x_3,k tends to µ α γ= − . So

we have

( )

(

)

lim _k 0, , , ,

k→∞F ξ =F λ λ λ µ

 _(3.7)

B) Let us consider now the façon κ′ =

( )

1, 2 3

[ ]

and its corresponding generic sequence

{

ξ′k =

(

x1,′k,x2,′k,x3,′k

)

}

, we have two cases:

• _If X₁

_{( )}

ξ ′_k =

(

x_1,′_k−x_2,′_k

)

tends to 0: So X4

( )

ξ ′k =

(

x1,′k−x2,′k

)

x3,′k tends to 0.

We have X2

( )

ξ ′k =

(

x1,′k−x2,′k

)

x1,′k and X2

( )

ξ ′k =

(

x1,′k−x2,′k

)

x2,′k tend to a

same complex number λ′ and X5

( )

ξ ′k =

(

x1,′k−x3,′k

)

x′3,k tends to an arbit-

rary complex number µ′_{. Then in this case, we have}

( )

_k

(

0, , , 0,

)

.

F ξ′ =F λ λ′ ′ µ′ (3.8)

• If X1

( )

ξ ′k =

(

x1,′k−x2,′k

)

tends to a non-zero complex number λ′∈: So

( )

2 k

pertinent variables of F anymore. Moreover, X4

( )

ξk′ tends to 0 and

( )

5 k

X ξ′ tends to an arbitrary complex number µ′. Then in this case, we

have

(

)

( )

(

)

1, 4, 5 , 0, . k

F F X X X

F ξ F λ µ

=

′ = ′ ′



 (3.9)

In conclusion, we have two cases: 1) From (3.6), (3.7) and (3.8), we have

(

)

( )

(

)

( )

(

)

1, 2, 3, 4, 5 lim 0, , , ,

lim 0, , , 0, . k

k

k k

F F X X X X X

F F

F F

ξ λ λ λ µ

ξ λ λ µ

→∞

→∞

= =

′ = ′ ′ ′

  

(*)

2) From (3.6), (3.7) and (3.9), we have

(

)

( )

(

)

( )

(

)

1, 4, 5

lim 0, ,

lim , 0, .

k k

k k

F F X X X

F F

F F

ξ λ µ

ξ λ µ

→∞

→∞

= =

′ = ′ ′

 



(**)

Step 3: We restrict the pertinent variables in the step 2 by using the three following facts:

• κ _and κ′ _{are two façons of the same stratum S,} • dimS=2,

• F is dominant.

Let us consider the two cases (*) and (**) determined in the step 2: 1) F is of the form (*):

• _{At first, we use the fact that} κ _and κ′ _{are two façons of the same stratum}

S, then if Xi is a pertinent variable of F then both Xi

( )

ξk and Xi

( )

ξk′

must tend to either an arbitrary complex number or zero.

• _{Since the dimension of S is 2 then} F must have at least two pertinent

variables Xi and Xj such that the images of the sequences ξk and ξk′

by Xi and Xj, respectively, tend independently to two complex numbers.

In this case:

+ F must admit either X2 =

(

x1−x2

)

x1 or X3=

(

x1−x2

)

x2 as a pertinent variable,

+ F must admit X5 =

(

x1−x3

)

x3 as a pertinent variable.

• _{Since F is dominant then F must admit at least 3 independent pertinent}

variables (see lemma 2.2). Then in this case, F must also admit X1= x1−x2 as a pertinent variable. We see that X1

( )

ξk and X1

( )

ξk′ tend to 0. We can

say that this variable is a “free” pertinent variable. The role of this variable is to guarantee the fact that _F

( )

_3 is dense in the target space _3_.

In conclusion, F has the following form:

(

)

( )

(

)

( )

(

)

1, 2, 3, 5

lim 0, , ,

lim 0, , , .

k k

k k

F F X X X X

F F

F F

ξ λ λ µ

ξ λ λ µ

→∞

→∞

=

=

′ = ′ ′ ′



 

Step 4: Describe the geometry of the 2-dimensional stratum S: On the one hand, the pertinent variables X2 (or X3) and X5 tending independently to two complex numbers have degree 2; on the other hand, the degree of F is 2, then the degree of the surface S with respect to the variables λ and µ (or

λ′ and µ′_{) is 1 (notice that by degree of S, we mean the degree of the}

equation defining S). We conclude that S is a plane.

In light of the example 3.5, we explicit now the algorithm for classifying the asymptotic sets of the non-proper dominant polynomial mappings

(

)

3 3

1, 2, 3 :

F = F F F  → of degree 2. 3.4. Algorithm

Algorithm 3.10. We have the five following steps: Step 1:

• _{Fix l façons} κ1,,κl in the list (3.4), where 1≤ ≤l 19.

• _{Determine the pertinent variables with respect to these l façons (knowing}

that they must be refered to a same mapping F). Step 2:

• _{Assume that S is a 2-dimensional stratum of} S_F admitting only the l façons

1, , l

κ κ in step 1.

• _{Take generic sequences} 1 , , l

k k

ξ  ξ corresponding to κ1,,κl, respectively.

• _{Compute the limit of the images of the sequences} 1 , , l

k k

ξ  ξ by the pertinent variables defined in step 1.

• _{Restrict the pertinent variables in step 1 using the fact} dimS=2.

Step 3: Restrict again the pertinent variables in step 2 using the three following facts:

• _{the façons} κ₁,,κ_l belongs to S: then the images of the generic sequences 1_{, ,} l

k k

ξ ξ by the pertinent variables defined in the step 2 must tend to either an arbitrary complex number or zero,

• dimS=2: then there are at least two pertinent variables Xi and Xj such

that the images of the sequences ξk and ξk′ by Xi and Xj, respectively,

tend independently to two complex numbers,

• _{F is dominant: then there are at least 3 independent pertinent variables (see}

lemma 2.2).

Step 4: Describe the geometry of the 2-dimensional irreductible stratum S of

F

S in terms of the pertinent variables obtained in the step 3.

Step 5: Letting l run from 1 to 19 in the list (3.4).

Theorem 3.11. With the algorithm 3.10, we obtain the list of all possible asymptotic sets SF of non-proper dominant polynomial mappings

3 3

:

Proof. On the one hand, the process of the algorithm 3.10 is possible, since the number of the façons in the list (3.4) is finite (19 façons). On the other hand, by the step 2, step 4 and step 5, we consider all the possible cases for all 2-dimensional irreductible strata of SF. Since the dimension of SF is 2 (see

theorem 2.4), we get all the possible asymptotic sets SF of non-proper

dominant polynomial mappings 3 3 :

F  → of degree 2.

4. Results

In this section, we use the algorithm 3.10 to prove the following theorem.

Theorem 4.1. The asymptotic set of a non-proper dominant polynomial mapping _F:3→_3 of degree 2 is one of the five elements in the following list ( )3,2

F S

 . Moreover, any element of this list can be realized as the asymptotic set of a dominant polynomial mapping _F:3→_3 of degree 2.

The list ( )3,2

F S

 : 1) A plane. 2) A paraboloid.

3) The union of a plane

( )

P :r x1 1+r x2 2+r x3 3+ =r4 0, and a plane of the form

( )

P′ :r x1 1′ +r x2′ 2+r x3 3′ + =r4′ 0, where we can choose two of the three coefficients r r r1′ ′ ′, ,2 3, then the third of them and the fourth coefficient r4′ are determined.

4) The union of a plane

( )

P :r x1 1+r x2 2+r x3 3+ =r4 0 and a paraboloid of the

form

( )

2

{

} {

}

4

:r x_i′_i +r x_j′ _j+r x_l′_l+ =r′ 0, , ,i j l = 1, 2, 3 ,

H where we can choose two

of the three coefficients r r r1′ ′ ′, ,2 3 , then the third of them and the fourth coefficient r4′ are determined.

5) The union of three planes

( )

P :r x1 1+r x2 2+r x3 3+ =r4 0,

( )

P′ :r x1 1′ +r x2′ 2+r x3 3′ + =r4′ 0,

( )

P′′ :r x1 1′′ +r x2′′2+r x3 3′′ + =r4′′ 0, where:

a) for

( )

P′ _{, we can choose two of the three coefficients} r r r₁′ ′ ′, ,_{2 3}, then the third of them and the fourth coefficient r4′ are determined,

b) for

( )

P′′ , we can choose two of the three coefficients r r r1′′ ′′ ′′, ,2 3 , then the third of them and the fourth coefficient r4′′ are determined.

In order to prove this theorem, we need the two following lemmas. Lemma 4.2. Let

(

)

3 3

1, 2, 3 :

F = F F F  → be a non-proper dominant

polynomial mapping of degree 2. If SF contains a surface of degree higher than

1, then either SF is a paraboloid, or SF is the union of a paraboloid and a

plane.

Proof. Assume that SF contains a surface

( )

H . Since degF =2 then

( )

1≤deg H ≤2.

A) We prove firstly that if SF contains a surface

( )

H where deg

( )

H =2

then

( )

H is a paraboloid. Since deg

( )

H =2 and degF =2 then S_F

respect to the façon κ , there exists only one free pertinent variable. That means, one of x1, x2 and x3 is a pertinent variable of F with respect to κ (cf. Definition 3.1). Without loose of generality, we assume that x1 is a pertinent variable of F with respect to the façon κ=

( )

3 2

[ ]

. Assume that

(

)

{

ξk = x1,k,x2,k,x3,k

}

is a generic sequence tending to infinity with the façon κ

and

i) x3,k tends to infinity, x2,k tends to 0 in such a way that x x2,k 3,k tends to

an arbitrary complex number λ,

ii) x1,k tends to an arbitrary complex number µ.

We see that 2 1,k

x and

(

x_1,k +x_2,k

)

x_1,k tend to µ2. Since _degF=₂ and

( )

deg H =2, then

i) one coordinate polynomial Fη, where η∈

{

1, 2, 3

}

, must contain x1 as an element of degree 1,

ii) the another coordinate polynomial F_η′, where η′∈

{

1, 2, 3

}

and η η′ ≠ , must contain 2

1

x or

(

x₁+x₂

)

x₁ as a pertinent variable.

Assume that the equation of the surface

( )

H is 1 2

1 1 2 2

p p

rα +rα + 3

3 3 4 0.

p

rα + =r Since 2 1,k

x and

(

x_1,k+x_2,k

)

x_1,k tend to the same complex

number µ2, and _deg

( )

_H =₂, then there exists an unique index _i∈

{

_{1, 2, 3}

}

such that r_i≠0 and p_i=2. If r_j=0 or p_j =0 for all j≠i, j∈

{

1, 2, 3

}

, then

( )

H is the union of two lines. That provides the contradiction with the fact that deg

( )

H =2. So, there exists j≠i, j∈

{

1, 2, 3

}

such that r_j≠0 and

1 j

p = . Consequently, the surface

( )

H is a paraboloid.

B) We prove now that if SF contains a paraboloid then the biggest possible

F

S is the union of this paraboloid and a plane. Since S_F contains a paraboloid

then with the same choice of the façon κ=

( )

3 2

[ ]

as in A), the mapping F must be considered as a dominant polynomial mapping of pertinent variables

1, 2, 1 2

x x x x and x x_{2 3}, that means:

(

1, 2, 1 2, 2 3

)

. F=F x x x x x x

We can see easily that if x2 is a pertinent variable of F, then SF admits

only the façon κ and SF is a paraboloid. Assume that SF contains another

irreductible surface

( )

H′ _{which is different from}

( )

H _{. Then F must be}

considered as a polynomial mapping of pertinent variables x x x1, 1 2 and x x2 3, that means:

(

1, 1 2, 2 3

)

.

F=F x x x x x (4.3)

Let us consider now one façon κ′ _of

_{( )}

H ′ such that κ′ ≠κ and let

(

)

{

ξk′= x1,′k,x2,′k,x3,′k

}

be a corresponding generic sequence of κ′. Notice that

one coordinate of F admits x1 as a pertinent variable. Let us show that x1,′k

tends to 0. Assume that x1,′k tends to a non-zero complex number. As one

coordinate of F admits x x1 2 as a pertinent variable, then x2,′k does not tend to

infinity. We have two cases:

+ If x2,′k tends to 0, then in order to ξk′ tending to infinity, x3,′k must tend

+ If x2,′k tends to a non-zero finite complex number, since one coordinate of

F admits x x2 3 as factor, then x3,′k does not tend to infinity. That provides the

contradiction with the fact that ξk′ tends to infinity.

Therefore, x1,′k tends to 0. We have the following possible cases:

1) κ′ =

( )

2 1

[ ]

: then F is a polynomial mapping of the form

(

1, 3, 1 2, 1 3

)

F =F x x x x x x . Combining with (4.3), then F=F x x x

(

1, 1 2

)

. Therefore, F is not dominant, which provides the contradiction.

2) κ′ =

( )

3 1

[ ]

: then F is a polynomial mapping of the form

(

1, 2, 1 2, 1 3

)

F =F x x x x x x . Combining with (4.3), then F=F x x x

(

₁, _{1 2}

)

. Therefore,

F is not dominant, which provides the contradiction.

3) κ′ =

( )

2, 3 1

[ ]

: then F is a polynomial mapping of the form

(

1, 1 2, 1 3

)

F =F x x x x x . Combining with (4.3), then F=F x x x

(

1, 1 2

)

. Therefore, F is not dominant, which provides the contradiction.

4) κ′ =

( )

3 1, 2

[ ]

: then F =F x x x x x x x x

(

₁, ₂, _{1 2}, _{2 3}, _{3 1}

)

. Combining with (4.3), we have F =F x x x x x

(

1, 1 2, 2 3

)

. Since x x1,′ ′k 2,k and x1,′k tend to 0, then

( )

dim H ′ ≤1, that provides the contradiction.

5) κ′ =

( )

2 1, 3

[ ]

: in this case, F is a polynomial mapping admitting the form

(

1, 3, 1 2, 2 3, 3 1

)

. F =F x x x x x x x x

Combining with (4.3), then

(

1, 1 2, 2 3

)

. F=F x x x x x

We know that x1,′k tends to 0. Assume that x x1,′ ′k 2,k tends to a complex

number λ and x x2,′ ′k 3,k tends to a complex number µ, we have

( )

H′ =

{

(

F1

(

0, ,λ µ

) (

,F2 0, ,λ µ

) (

,F3 0, ,λ µ

)

)

: ,λ µ∈

}

,

where F=

(

F F F  1, 2, 3

)

. Since degF=2, then the degree of Fi with respect to

the variables λ and µ _{must be 1, for all} i∈

{

1, 2, 3

}

. Consequently, the

surface

( )

H′ _{is a plane.}

Lemma 4.4. Let 3 3 :

F  → be a non-proper dominant polynomial

mapping of degree 2. Assume that S is a 2-dimensional irreductible stratum of

F

S . Then S admits at most two façons. Moreover, if S admits two façons, then F

S is a plane.

Proof. Let 3 3 :

F  → be a non-proper dominant polynomial mapping of

degree 2. Assume that S is a 2-dimensional irreductible stratum of SF.

A) We provide firstly the list of pairs of façons that S can admit and we write F in terms of pertinent variables in each of these cases. Let us fix a pair of façons

(

κ κ, ′

)

in the list (3.4) and assume that S admits these two façons. We use the

steps 1, 2, 3 and 4 of the algorithm 3.10. In the same way than the example 3.5, we can determine the form of F in terms of its pertinent variables with respect to two fixed façons after using the conditions of dimension of S and the dominancy of F. Letting two façons κ κ′, run in the list (3.4), we get the following

possiblilities:

1)

(

κ κ′ =,

) (

(

1, 2, 3 ,

) (

i i1,2

)

[ ]

j

)

, where

{

i i1, ,2 j

} {

= 1, 2, 3

}

and

(

) (

) (

) (

)

(

i1 i2, i1 i2 i1, i1 i2 i2, i1 i2 j, i1 j j

)

.

2)

(

κ κ′ =,

) (

(

1, 2, 3 ,

) ( )

i

[

j1,j2

]

)

, where

{

i j j, 1, 2

} {

= 1, 2, 3

}

and

(

) (

) (

) (

) (

) (

)

(

i j1 j1, i j1 j2, i j2 j1, i j2 j2, j1 j2 , j1 j2 i

)

.

F =F x −x x x −x x x −x x x −x x x −x x −x x

3)

(

κ κ′ =,

) ( )

(

1, 2 3 ,

[ ]

( )

i

[ ]

3,j

)

, where

{ } { }

i j, = 1, 2 , and

(

)

(

3, j 3, i 3, i j j

)

.

F =F x x x x x x −x x

4)

(

κ κ′ =,

) ( )

(

1, 2 3 , 3 1, 2

[ ]

( )

[ ]

)

and

(

)

(

)

(

1 3, 2 3, 1 2,1 1 3 2 2 3 3 1 2 1 4 1 2 2

)

, F=F x x x x x −x r x x +r x x +r x −x x +r x −x x

where rl∈, for l=1,, 4, such that

(

r1≠0,r2≠0,

(

r r3, 4

) ( )

≠ 0, 0

)

, or

(

) ( ) (

) ( )

(

r r1, 2 ≠ 0, 0 , r r3, 4 ≠ 0, 0

)

.

5)

(

κ κ′ =,

) ( )

(

1, 3 2 ,

[ ]

( )

i

[ ]

2,j

)

, where

{ } { }

i j, = 1, 3 , and

(

)

(

2, 2 j, i 2, i j j

)

.

F =F x x x x x x −x x

6)

(

κ κ′ =,

) ( )

(

1, 3 2 , 2 1, 3

[ ]

( )

[ ]

)

and

(

)

(

)

(

1 2, 2 3, 1 3,1 1 2 2 2 3 3 1 3 1 4 1 3 3

)

, F =F x x x x x −x r x x +r x x +r x −x x +r x −x x

where rl∈, for l=1,, 4, such that

(

r1≠0,r2≠0

)

, or

(

r r1, 2

) ( )

≠ 0, 0 and

(

r r3, 4

) ( )

≠ 0, 0 .

7)

(

κ κ′ =,

) ( )

(

2, 3 1 ,

[ ]

( )

i

[ ]

1,j

)

, where

{ } { }

i j, = 2, 3 , and

(

)

(

1, 1 i, 1 j, i j j

)

.

F =F x x x x x x −x x

8)

(

κ κ′ =,

) ( )

(

2, 3 1 , 1 2, 3

[ ]

( )

[ ]

)

and

(

)

(

)

(

1 3, 1 2, 3 2,1 1 3 2 1 2 3 3 2 3 4 3 2 2

)

, F=F x x x x x −x r x x +r x x +r x −x x +r x −x x

where rl∈, for l=1,, 4, such that

(

r1≠0,r2≠0,

(

r r3, 4

) ( )

≠ 0, 0

)

or

(

) ( ) (

) ( )

(

r r1, 2 ≠ 0, 0 , r r3, 4 ≠ 0, 0

)

.

9)

(

κ κ, ′ =

) ( )

(

1 2, 3 ,

[ ]

( )

i

[ ]

1,j

)

, where

{ } { }

i j, = 2, 3 , and

(

j, 1 i,1 i j 2 1 j

)

,

F=F x x x r x x +r x x

where r1 et r2 are the non-zero complex numbers.

B) We prove now that S admits at most two façons. We prove the result for the first case of the above possibilities:

(

κ κ′ =,

) (

(

1, 2, 3 ,

) (

i i1,2

)

[ ]

j

)

, where

{

i i1, ,2 j

} {

= 1, 2, 3

}

. The other cases are proved similarly. For example, assume

that S admits two façons κ =

(

1, 2, 3

)

and κ′ =

( )

1, 2 3

[ ]

. We prove that S cannot admit the third façon κ′′ different from κ and κ′_.

Let κ′′ _{be a façon of} S_F. Let us denote by

{

ξk′′=

(

x1,′′k,x2,′′k,x3,′′k

)

}

a generic

sequence corresponding to κ′′_{. By the example 3.5, the mapping F admits} X1,

2

X (or X₃), and X₅ as the pertinent variables, where

(

)

(

)

(

)

1 1 2, 2 1 2 1, 3 1 2 2, 5 1 3 3.

X = −x x X = x −x x X = x −x x X = x −x x

Without loose the generality, we can assume that X2 is a pertinent variable of F. We prove that X1

( )

ξk′′ =

(

x1,′′k −x′′2,k

)

tends to 0. Assume that X1

( )

ξk′′ =

(

x1,′′k −x2,′′k

)

tends to a non-zero complex number. Then:

+ If x1,′′k tends to infinity, then X2

( )

ξk′′ tends to infinity, that provides a

+ If x2,′′k tends to infinity, then x1,′′k also tends to infinity since

(

x1,′′k−x2,′′k

)

tends to a non-zero complex number. That implies X2

( )

ξk′′ tends to infinity

and this provides a contradiction with the fact that X2 is a pertinent variable of F.

Hence, x1,′′k and x2,′′k cannot tend to infinity. Consequently, x3,′′k must tend

to infinity. Therefore, X5

( )

ξ ′′k =

(

x1,′′k−x3,′′k

)

x3,′′k tends to infinity, that provides

the contradiction with the fact that X5 is a pertinent variable of F. We conclude that

(

x1,′′k −x2,′′k

)

tends to 0.

Then we have two possibilities:

a) either both of x1,′′k and x2,′′k tend to 0: then X1

( )

ξ ′′k =

(

x1,′′k −x2,′′k

)

,

( )

(

)

2 k 1,k 2,k 1,k

X ξ′′ = x′′ −x′′ x′′ and X₃

( )

ξ_k′′ =

(

x_1,′′_k−x′′_2,k

)

x′′_2,k tend to 0, which pro-

vides the contradiction with the fact that the dimension of S is 2,

b) or both of x1,′′k and x2,′′k tend to infinity: Since X5 is a pertinent variable of F, then x3,′′k tends to 0 or infinity. We conclude that the façon κ′′ is

(

1, 2, 3

)

or

( )

1, 2 3

[ ]

.

In conclusion, S admits only the two façons κ =

(

1, 2, 3

)

and κ′ =

( )

1, 2 3

[ ]

. c) We prove now that if there exists a 2-dimensional irreductible stratum S of

F

S admitting two façons, then S_F is a plane. Similarly to B), we prove this

fact for the first case of the possibilities in A), that means, the case of

(

κ κ′ =,

) (

(

1, 2, 3 ,

) (

i i1,2

)

[ ]

j

)

, where

{

i i1, ,2 j

} {

= 1, 2, 3

}

. The other cases are proved similarly. For example, assume that S admits two façons κ =

(

1, 2, 3

)

and κ′ =

( )

1, 2 3

[ ]

. With the same arguments than in the example 3.5, the stratum S is a plane. By B), the asymptotic set SF admits also only two façons

(

1, 2, 3

)

κ = _and κ′ =

( )

1, 2 3

[ ]

. In other words, S_F and S concide. We con-

clude that SF is a plane.

We prove now the theorem 4.1.

Proof. (The proof of theorem 4.1). The cases 1) and 2) are easily achievable by the lemmas 4.4 and 4.2, respectively. Let us prove the cases 3), 4) and 5). In these cases, on the one hand, since SF contains at least two irreductible surfaces,

then SF admits at least two façons; on the other hand, by the lemma 4.4, each

irreductible surface of SF admits only one façon. Assume that κ , κ′ are two

different façons of SF and

{ }

ξk ,

{ }

ξk′ are two corresponding generic

sequences, respectively. We use the algorithm 3.10 and in the same way than the proofs of the lemmas 4.2 and 4.4, we can see easily that the pairs of façons

(

κ κ, ′

)

must belong to only the following pairs of groups: (I, IV), (I, V), (I, VI),

(II, VI), (IV, V), (IV, VI), (V, VI) and (VI, VI) in the list (3.4).

i) If κ belongs to the group I and κ′ _{belongs to the group IV, for example}

(

1, 2, 3

)

κ = _and κ′ =

( )

1, 2 3

[ ]

. From the example 3.5, F is a dominant poly- nomial mapping which can be written in terms of pertinent variables:

(

) (

) (

) (

)

(

)

( )

(

)

( )

(

)

1 2, 1 2 1, 1 2 2, 1 2 3, 1 3 3

lim 0, , , , ,

lim 0, , , 0, ,

k k

k k

F F x x x x x x x x x x x x x x

F F

F F

ξ λ λ λ µ

ξ λ λ µ

→∞

→∞

= − − − − −

=

′ = ′ ′ ′

  

{ }

ξk , the pertinent variables tending to an arbitrary complex numbers have the

degree 2, then the façon κ _{provides a plane, since the degree of F is 2. In the}

same way, the façon κ′ _{provides a plane. Furthermore, it is easy to check that}

these two planes must have the form of the case 3) of the theorem and SF is

the union of these two planes.

ii) If κ _{belongs to the group I and} κ′ _{belongs to the group V, for example}

(

1, 2, 3

)

κ = and κ′ =

( )

1 2

[ ]

. Then, on the one hand, F is a dominant poly- nomial mapping which can be written in terms of pertinent variables:

(

) (

) (

)

(

2 3, 2 3 2, 2 3 3, 1 2 2

)

. F=F x −x x −x x x −x x x −x x

On the other hand, with the same arguments than the example 3.5, and for suitable generic sequences

{ }

ξk and

{ }

ξk′ , we obtain:

( )

(

)

( )

(

2

)

lim 0, , , ,

lim , 0, , ,

k k

k k

F F

F F

ξ λ λ µ

ξ λ λ µ

→∞

→∞

=

′ = ′ ′ ′





where λ µ λ µ, , ′ ′∈, . With the same arguments than in the case i), we have: a) either the façons κ and κ′ _{provide two planes of the form of the case 3)}

of our theorem,

b) or the façon κ _{provides the plane}

( )

P _{and, by the lemma 4.2, the façon}

κ′ _{provides the paraboloid}

( )

H of the form of the case 4) of our theorem. By an easy calculation, we see that if SF admits another façon κ′′ which is

different from the façons κ and κ′_{, then this façon provides a 1-dimensional}

stratum contained in

( )

P _{or contained in}

( )

H _.

iii) Proceeding in the same way for the cases where

(

κ κ, ′

)

is a pair of façons belonging to the pairs of groups: (I, VI), (II, VI), (IV, V), (IV, VI) and (V, VI), we obtain the case 3) or the case 4) of the theorem.

iv) Consider now the case where κ and κ′ _{belong to the group VI, for}

example, κ =

( )

1 2, 3

[ ]

and κ′ =

( )

2 1, 3

[ ]

, then F is a dominant polynomial mapping which can be written in terms of pertinent variables:

(

3, 1 2, 2 3, 3 1

)

. F=F x x x x x x x

With the same arguments than the example 3.5, and for suitable generic sequences

{ }

ξk and

{ }

ξk′ , we obtain:

( )

(

)

( )

(

)

lim 0, , 0, , lim 0, , , 0 ,

k k

k k

F F

F F

ξ λ µ

ξ λ µ

→∞

→∞

=

′ = ′ ′





where λ µ λ µ, , ′ ′∈, . In this case, we have two possibilities:

a) either F admits x3 as a pertinent variable: This case is similar to the case i) and we have the case 3) of the theorem,

b) or F does not admit x3 as a pertinent variable, that means

(

1 2, 2 3, 3 1

)

.

F =F x x x x x x (4.4)

In this case, SF admits one more façon κ′′ =

( )

3 1, 2

[ ]

such that with a corre-

( )

(

)

( )

(

)

( )

(

)

lim , 0, ,

lim , , 0 ,

lim 0, , ,

k k

k k

k k

F F

F F

F F

ξ λ µ

ξ λ µ

ξ λ µ

→∞

→∞

→∞

=

′ = ′ ′

′′ = ′′ ′′

  

where λ µ′′ ′′∈, . In this case, S_F is the union of three planes the forms of

which are as in the case 5) of the theorem.

5. The General Case

The algorithm 3.10 can be generalized to clasify the asymptotic sets of non- proper dominant polynomial mappings _F:_n→_n of degree d where

3 n≥

and d≥2 as the following.

Algorithm 5.1. We have the six following steps: Step 1: Determine the list (n d, )

F

 of all the possible façons of SF.

Step 2: Fix l façons κ1,,κl in the list

( )n d, F

 obtained in step 1. Determine the pertinent variables with respect to these l façons (in the similar way than the definition 3.1).

Step 3:

• _{Assume that S is a}

(

n−1

)

-dimensional stratum of S_F admiting only the l

façons κ1,,κl determined in step 1.

• _{Take generic sequences} 1_{, ,} l

k k

ξ ξ corresponding to κ1,,κl, respectively.

• _{Compute the limit of the images of the sequences} 1 , , l

k k

ξ ξ by pertinent variables defined in step 1.

• _{Restrict the pertinent variables defined in step 2 using the fact} dimS= −n 1.

Step 4: Restrict again the pertinent variables in step 3 using the three following facts:

• _{all the façons} κ₁,,κ_l belong to S: then the images of the generic sequences 1

, , l

k k

ξ ξ by the pertinent variables defined in the step 2 must tend to either an arbitrary complex number or zero,

• dimS= −n 1: then there are at least n−1 pertinent variables

1, , n1

i i

X  X ₋

such that the images of the sequences

{ } { }

1 _{, ,} l

k k

ξ  ξ by Xi1,,Xin−1, re-

spectively, tend independently to

(

n−1

)

complex numbers,

• _{F is dominant: then there are at least n independent pertinent variables (see}

lemma 2.2).

Step 5: Describe the geometry of the

(

n−1

)

-dimensional irreductible stratum S in terms of the pertinent variables obtained in the step 4.

Step 6: Let l run in the list obtained in the step 1.

Theorem 5.2. With the algorithm 5.1, we obtain all possible asymptotic sets of non-proper dominant polynomial mapping _F:_n→_n of degree d.

Proof. In the one hand, by theorem 2.4, the dimension of SF is n−1. By

the step 3, step 5 and step 6, we consider all the possible cases of the all

(

n−1

)

- dimensional irreductible strata of SF. Since the dimension of SF is n−1

(see theorem 2.4), we get all the possible asymptotic sets SF of non-proper