An Algorithm to Classify the Asymptotic Set Associated to a Polynomial Mapping ()
1. Introduction
Let
be a polynomial mapping. Let us denote by
the set of points at which F is non proper, i.e.,
where
is the Euclidean norm of
in
. The set
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.,
, then
is an empty set or a hypersurface [1] .
Notice that it is sufficient to define
by considering sequences
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
are labeled in terms of “façons”, as follows: We rank the coordinates of
into three categories: 1) the coordinates tending to infinity (this cotegories is not empty); 2) the coordinates
such that
is a complex number “independant on the point a in a neighborhood a in
”. This means that there exists the points neighbors of a in
and the sequences
such that
and
, 3) the coordinates
such that
is a complex number “dependant on the point a”. This means that there not exist such points
neighbors of
in
. The example 2.5 illustrates these three categories.
We define a “façon” of the point
as a
-tuple
of integers where
tends to infinity for
and, for
, the sequence
tends to a complex number independently on the point a when a describes locally
(definition 2.7).
The aim of this paper is to provide an algorithm to classify the asymptotic sets of dominant polynomial mappings
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
be a dominant polynomial mapping of degree 2 such that
. We fix a façon
of F and assume that
is a sequence tending to infinity with the façon
such that
tends to a point of
. Since the degree of F is 2 then each coordinate
and
of F is a linear combination of
,
,
,
,
and
. We call a pertinent variable of F with respect to the façon
a minimum linear combination of
such that the image of the sequence
by this combination does not tend to infinity (see definition 3.1).
Moreover, if F is dominant then by Jelonek, the set
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
. 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
admits l fixed façons in the list
, where
.
・ 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
. We get all the possible 2-dimen- sional irreductible strata of
. Since the dimension of
is 2, then we get the list of all possible asymptotic sets
.
With this idea, we provide the algorithm 3.10 to classify the asymptotic sets of dominant polynomial mappings
of degree 2, and we obtain the classification theorem 4.1. This algorithm can be generalized for the general case of polynomial mappings
of degree d, where
and
(algorithm 5.1).
2. Dominancy, Assymptotic Set and “Façons”
2.1. Dominant Polynomial Mapping
Definition 2.1. Let
be a polynomial mapping. Let
be the closure of
in
. F is called dominant if
, i.e.,
is dense in
.
We provide here a lemma on the dominancy of a polynomial mapping
that we will use later on.
Lemma 2.2. Let
be a dominant polynomial mapping. Then, the coordinate polynomials
are independent. That means, there does not exist any coordinate polynomial
, where
, such that
is a polynomial mapping of the variables
.
Proof. Assume that
where
and
is a polynomial. Then, the dimension of
is less than n. Consequently, the dimension of
is less than n. That provides the contradiction with the fact F is dominant.
2.2. Asymptotic Set
Definition 2.3. Let
be a polynomial mapping. Let us denote by
the set of points at which F is non-proper, i.e.,
where
is the Euclidean norm of
in
. The set
is called the asymptotic set of F.
Recall that, it is sufficient to define
by considering sequences
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
be a polynomial mapping. If F is dominant, then
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
be the polynomial mapping such that
Notice that by the notations
and
, we want to distinguish the source space and the target space. We determine now the asymptotic set
by using the definition 2.3. Assume that there exists a sequence
in the source space
tending to infinity such that image
does not tend to infinity. Then
and
cannot tend to infinity. Since the sequence
tends to infinity, then
must tend to infinity. Hence, we have the three following cases:
1)
tends to 0,
tends to a complex number
and
tends to infinity. In order to determine the biggest possible subset of
, we choose the sequences
tending to 0 and
tending to infinity in such a way that the product
tends to a complex number
. Let us choose, for example
where
, then
tends to a point
in
. We get a 2-dimensional stratum
of
, where
. We say that a “façon” of
is
. The symbol “(3)” in
the façon
means that the third coordinate
of the sequence
tends to infinity. The symbol “ [1] ” in the façon
means that the first coordinate
of the sequence
tends to 0 which is a fixed complex number which does not depend on the point
when a describes
. Notice that the second coordinate of the sequence
tends to a complex number
depending on the point
when a varies, then the indice “2” does not appear in the façon
. Moreover, all the sequences tending to infinity such that their images tend to a point of
admit only the façon
.
The two following cases are similar to the case 1):
2)
tends to a complex number
,
tends to 0 and
tends to infinity: then the façon
determines a 2-dimensional stratum
of
, where
.
3)
and
tend to 0, and
tends to infinity: then the façon
determines the 1-dimensional stratum
where
is the axis
in
.
In conclusion, we get
・ the asymptotic set
of the given polynomial mapping F as the union of two planes
and
in
,
・ all the façons of
of the given polynomial mapping F: they are three façons
,
and
.
Remark 2.6. The chosen sequence
in 1) of the above
example is called a generic sequence of the 2-dimensional irreductible component
(a plane) of
, since the image of any sequence of this type (with differents
and
) falls to a generic point of the plane
. That means the images of all the sequences
when
runs in
and
runs in
cover
and
is dense in the plane
. We can see easily that a generic sequence of the 2-dimensional
irreductible component
of
is
where
. More generally, any sequence
, where
and
, is a generic sequence of
. Any sequence
, where
and
, is a generic sequence of
.
In the light of this example, we recall here the definition of façons in [2] .
Definition 2.7. [2] Let
be a dominant polynomial mapping such that
. For each point a of
, there exists a sequence
,
tending to infinity such that
tends to a. Then, there exists at least one index
,
such that
tends to infinity when k tends to infinity. We define “a façon of tending to infinity of the sequence
”, as a maximum
-tuple
of different integers in
, such that:
1)
tends to infinity for all
,
2) for all
, the sequence
tends to a complex number independently on the point a when a varies locally, that means:
a) either there exists in
a subvariety
containing a such that for any point
in
, there exists a sequence
,
tending to infinity such that
i)
tends to
,
ii)
tends to infinity for all
,
iii) for all
,
and this limit is finite.
b) or there does not exist such a subvariety, then we define
where
tends to infinity for all
and
. In this case, the set of points a is a subvariety of dimension 0 of
.
We call a façon of tending to infinity of the sequence
also a a façon of a or a façon of
.
3. An Algorithm to Stratify the Asymptotic Sets of the Dominant Polynominal Mappings
of Degree 2
In this section we provide an algorithm to stratify the asymptotic sets associated to dominant polynominal mappings
of degree 2. In the last section, we show that this algorithm can be generalized in the general case for dominant polynominal mappings
of degree d where
and
. Recall that by degree of a polynomial mapping
, we mean the highest degree of the monomials
.
Let us consider now a dominant polynomial mapping
of degree 2 such that
. An important step of this section is to define the notion of “pertinent” variables of F.
3.1. Pertinent Variables
Let us explain at first the idea of the notion of pertinent variables: let
be a sequence in the source space
tending to infinity such that
tends to a point of
in the target space
. Then the image of
by any coordinate polynomial
, where
, cannot tend to infinity. Notice that
can be written as the sum of elements of the form
such that if
tends to infinity, then
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
, then this element must be “balanced” with another factor tending to zero with respect to the sequence
. For example, assume that the coordinate sequences
and
of the sequence
tend to infinity, then
cannot admit neither
nor
alone as an element of the above sum, but
can admit
,
,
as elements of this sum, where
. So we define:
Definition 3.1. Let
be a polynomial mapping of degree 2 such that
. Let us fix a façon
of
. Then there exists a sequence
tending to infinity with the façon
such that its image tend to a point in
. An element in the list
(3.2)
is called a pertinent variable of F with respect to the façon
if the image of the sequence
by this element does not tend to infinity.
Remark 3.3. From now on, we will denote
pertinent variables of F with respect to a fixed façon and we write
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
for the dominant polynomial mappings
of degree 2. In order to do that, we observe firstly that
・ The list of all the possible façons of
for a polynomial mapping
is
(3.4)
This list has 19 façons.
・ Since F dominant, then by the theorem 2.4, the set
has pure dimen- sion 2.
With these observations, we will:
・ assume that a 2-dimensional irreductible stratum S of
admits l fixed façons in the list (3.4), where
,
・ 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
. 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
. We get all the possible 2-dimen- sional irreductible strata S of
. Since the dimension of
is 2, then we get the list of all the possible asymptotic sets
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
admitting some fixed façons.
3.3. Example
Example 3.5. Let
be a dominant polynomial mapping of degree 2. Assume that a 2-dimensional stratum S of
admits the two façons
and
. 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
or the façon
. 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
and
:
・ With the façon
, 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:
,
,
,
,
,
,
,
,
,
,
,
,
,
and
(see definition 3.1).
・ With the façon
, 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:
,
,
,
,
,
,
and
.
Since S contains both of the façons
and
, then this surface S admits
,
,
,
and
as pertinent variables. Let us denote by
We can write
(3.6)
Step 2: Assume that
and
are two sequences tending to infinity with the façons
and
, respectively.
A) Let us consider the façon
and its corresponding generic sequence
:
・ Assume that
tends to a non-zero complex number. Since
then all three coordinate sequences
and
tend to infinity. Hence
,
and
tend to infinity. In this case,
,
and
cannot be pertinent variables of F anymore. Then F admits only two pertinent variables
and
, or
. 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,
tends to 0.
・ Assume that
tends to a non-zero complex number. Then
tend to infinity, hence
cannot be a pertinent variable of
anymore, then
. We choose a generic sequence
satisfying the conditions:
tends to zero and
tends to a non-zero complex number, for example,
. Then
,
and
tend to the same complex number
. Combining with the fact
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,
tends to 0.
Then, with the façon
, we have
and
tend to 0. Hence
also tends to 0. Let us choose a generic sequence
satisfying these conditions, for example, the sequence
. We see that
,
and
tend to a same complex number
. Moreover,
tends to
. So we have
(3.7)
B) Let us consider now the façon
and its corresponding generic sequence
, we have two cases:
・ If
tends to 0: So
tends to 0. We have
and
tend to a same complex number
and
tends to an arbit- rary complex number
. Then in this case, we have
(3.8)
・ If
tends to a non-zero complex number
: So
and
tend to infinity, thus
and
cannot be pertinent variables of F anymore. Moreover,
tends to 0 and
tends to an arbitrary complex number
. Then in this case, we have
(3.9)
In conclusion, we have two cases:
1) From (3.6), (3.7) and (3.8), we have
(*)
2) From (3.6), (3.7) and (3.9), we have
(**)
Step 3: We restrict the pertinent variables in the step 2 by using the three following facts:
・
and
are two façons of the same stratum S,
・
,
・ 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
and
are two façons of the same stratum S, then if
is a pertinent variable of F then both
and
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
and
such that the images of the sequences
and
by
and
, respectively, tend independently to two complex numbers. In this case:
+ F must admit either
or
as a pertinent variable,
+ F must admit
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
as a pertinent variable. We see that
and
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
is dense in the target space
.
2) F is of the form (**): Similarly to the case 1, we can see easily that F can admit only
as a pertinent variable. Then the dimension of S is 1, which is a contradiction with the fact that the dimension of S is 2.
In conclusion, F has the following form:
Step 4: Describe the geometry of the 2-dimensional stratum S: On the one hand, the pertinent variables
(or
) and
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
and
(or
and
) 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
of degree 2.
3.4. Algorithm
Algorithm 3.10. We have the five following steps:
Step 1:
・ Fix l façons
in the list (3.4), where
.
・ 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
admitting only the l façons
in step 1.
・ Take generic sequences
corresponding to
, respectively.
・ Compute the limit of the images of the sequences
by the pertinent variables defined in step 1.
・ Restrict the pertinent variables in step 1 using the fact
.
Step 3: Restrict again the pertinent variables in step 2 using the three following facts:
・ the façons
belongs to S: then the images of the generic sequences
by the pertinent variables defined in the step 2 must tend to either an arbitrary complex number or zero,
・
: then there are at least two pertinent variables
and
such that the images of the sequences
and
by
and
, 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
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
of non-proper dominant polynomial mappings
of degree 2.
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
. Since the dimension of
is 2 (see theorem 2.4), we get all the possible asymptotic sets
of non-proper dominant polynomial mappings
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
of degree 2 is one of the five elements in the following list
. Moreover, any element of this list can be realized as the asymptotic set of a dominant polynomial mapping
of degree 2.
The list
:
1) A plane.
2) A paraboloid.
3) The union of a plane
and a plane of the form
where we can choose two of the three coefficients
, then the third of them and the fourth coefficient
are determined.
4) The union of a plane
and a paraboloid of the form
where we can choose two of the three coefficients
, then the third of them and the fourth coefficient
are determined.
5) The union of three planes
where:
a) for
, we can choose two of the three coefficients
, then the third of them and the fourth coefficient
are determined,
b) for
, we can choose two of the three coefficients
, then the third of them and the fourth coefficient
are determined.
In order to prove this theorem, we need the two following lemmas.
Lemma 4.2. Let
be a non-proper dominant polynomial mapping of degree 2. If
contains a surface of degree higher than 1, then either
is a paraboloid, or
is the union of a paraboloid and a plane.
Proof. Assume that
contains a surface
. Since
then
.
A) We prove firstly that if
contains a surface
where
then
is a paraboloid. Since
and
then
admits one façon
in such a way that among the pertinent variables of F with respect to the façon
, there exists only one free pertinent variable. That means, one of
,
and
is a pertinent variable of F with respect to
(cf. Definition 3.1). Without loose of generality, we assume that
is a pertinent variable of F with respect to the façon
. Assume that
is a generic sequence tending to infinity with the façon
and
i)
tends to infinity,
tends to 0 in such a way that
tends to an arbitrary complex number
,
ii)
tends to an arbitrary complex number
.
We see that
and
tend to
. Since
and
, then
i) one coordinate polynomial
, where
, must contain
as an element of degree 1,
ii) the another coordinate polynomial
, where
and
, must contain
or
as a pertinent variable.
Assume that the equation of the surface
is
Since
and
tend to the same complex number
, and
, then there exists an unique index
such that
and
. If
or
for all
,
, then
is the union of two lines. That provides the contradiction with the fact that
. So, there exists
,
such that
and
. Consequently, the surface
is a paraboloid.
B) We prove now that if
contains a paraboloid then the biggest possible
is the union of this paraboloid and a plane. Since
contains a paraboloid then with the same choice of the façon
as in A), the mapping F must be considered as a dominant polynomial mapping of pertinent variables
and
, that means:
We can see easily that if
is a pertinent variable of F, then
admits only the façon
and
is a paraboloid. Assume that
contains another irreductible surface
which is different from
. Then F must be considered as a polynomial mapping of pertinent variables
and
, that means:
(4.3)
Let us consider now one façon
of
such that
and let
be a corresponding generic sequence of
. Notice that one coordinate of F admits
as a pertinent variable. Let us show that
tends to 0. Assume that
tends to a non-zero complex number. As one coordinate of F admits
as a pertinent variable, then
does not tend to infinity. We have two cases:
+ If
tends to 0, then in order to
tending to infinity,
must tend to infinity. Hence, the façon
is
. That provides the contradiction with the fact
.
+ If
tends to a non-zero finite complex number, since one coordinate of F admits
as factor, then
does not tend to infinity. That provides the contradiction with the fact that
tends to infinity.
Therefore,
tends to 0. We have the following possible cases:
1)
: then F is a polynomial mapping of the form
. Combining with (4.3), then
. Therefore, F is not dominant, which provides the contradiction.
2)
: then F is a polynomial mapping of the form
. Combining with (4.3), then
. Therefore, F is not dominant, which provides the contradiction.
3)
: then F is a polynomial mapping of the form
. Combining with (4.3), then
Therefore, F is not dominant, which provides the contradiction.
4)
: then
. Combining with (4.3), we have
. Since
and
tend to 0, then
, that provides the contradiction.
5)
: in this case, F is a polynomial mapping admitting the form
Combining with (4.3), then
We know that
tends to 0. Assume that
tends to a complex number
and
tends to a complex number
, we have
where
. Since
, then the degree of
with respect to the variables
and
must be 1, for all
. Consequently, the surface
is a plane.
Lemma 4.4. Let
be a non-proper dominant polynomial mapping of degree 2. Assume that S is a 2-dimensional irreductible stratum of
. Then S admits at most two façons. Moreover, if S admits two façons, then
is a plane.
Proof. Let
be a non-proper dominant polynomial mapping of degree 2. Assume that S is a 2-dimensional irreductible stratum of
.
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
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
run in the list (3.4), we get the following possiblilities:
1)
, where
and
2)
, where
and
3)
, where
, and
4)
and
where
, for
such that
, or
5)
, where
, and
6)
and
where
, for
such that
, or
and
.
7)
, where
, and
8)
and
where
, for
such that
or
.
9)
, where
, and
where
et
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:
, where
. The other cases are proved similarly. For example, assume that S admits two façons
and
. We prove that S cannot admit the third façon
different from
and
.
Let
be a façon of
. Let us denote by
a generic sequence corresponding to
. By the example 3.5, the mapping F admits
,
(or
), and
as the pertinent variables, where
Without loose the generality, we can assume that
is a pertinent variable of F. We prove that
tends to 0. Assume that
tends to a non-zero complex number. Then:
+ If
tends to infinity, then
tends to infinity, that provides a contradiction with the fact that
is a pertinent variable of F.
+ If
tends to infinity, then
also tends to infinity since
tends to a non-zero complex number. That implies
tends to infinity and this provides a contradiction with the fact that
is a pertinent variable of F.
Hence,
and
cannot tend to infinity. Consequently,
must tend to infinity. Therefore,
tends to infinity, that provides the contradiction with the fact that
is a pertinent variable of F. We conclude that
tends to 0.
Then we have two possibilities:
a) either both of
and
tend to 0: then
,
and
tend to 0, which pro- vides the contradiction with the fact that the dimension of S is 2,
b) or both of
and
tend to infinity: Since
is a pertinent variable of F, then
tends to 0 or infinity. We conclude that the façon
is
or
.
In conclusion, S admits only the two façons
and
.
c) We prove now that if there exists a 2-dimensional irreductible stratum S of
admitting two façons, then
is a plane. Similarly to B), we prove this fact for the first case of the possibilities in A), that means, the case of
, where
. The other cases are proved similarly. For example, assume that S admits two façons
and
. With the same arguments than in the example 3.5, the stratum S is a plane. By B), the asymptotic set
admits also only two façons
and
. In other words,
and S concide. We con- clude that
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
contains at least two irreductible surfaces, then
admits at least two façons; on the other hand, by the lemma 4.4, each irreductible surface of
admits only one façon. Assume that
,
are two different façons of
and
,
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
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
belongs to the group I and
belongs to the group IV, for example
and
. From the example 3.5, F is a dominant poly- nomial mapping which can be written in terms of pertinent variables:
where
(see (3.6), (3.7) and (3.8)). We see that, with the sequence
, the pertinent variables tending to an arbitrary complex numbers have the degree 2, then the façon
provides a plane, since the degree of F is 2. In the same way, the façon
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
is the union of these two planes.
ii) If
belongs to the group I and
belongs to the group V, for example
and
. Then, on the one hand, F is a dominant poly- nomial mapping which can be written in terms of pertinent variables:
On the other hand, with the same arguments than the example 3.5, and for suitable generic sequences
and
, we obtain:
where
. With the same arguments than in the case i), we have:
a) either the façons
and
provide two planes of the form of the case 3) of our theorem,
b) or the façon
provides the plane
and, by the lemma 4.2, the façon
provides the paraboloid
of the form of the case 4) of our theorem.
By an easy calculation, we see that if
admits another façon
which is different from the façons
and
, then this façon provides a 1-dimensional stratum contained in
or contained in
.
iii) Proceeding in the same way for the cases where
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
and
belong to the group VI, for example,
and
, then F is a dominant polynomial mapping which can be written in terms of pertinent variables:
With the same arguments than the example 3.5, and for suitable generic sequences
and
, we obtain:
where
. In this case, we have two possibilities:
a) either F admits
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
as a pertinent variable, that means
(4.4)
In this case,
admits one more façon
such that with a corre- sponding suitable generic sequence
of
, we have
where
. In this case,
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
of degree d where
and
as the following.
Algorithm 5.1. We have the six following steps:
Step 1: Determine the list
of all the possible façons of
.
Step 2: Fix l façons
in the list
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
-dimensional stratum of
admiting only the l façons
determined in step 1.
・ Take generic sequences
corresponding to
, respectively.
・ Compute the limit of the images of the sequences
by pertinent variables defined in step 1.
・ Restrict the pertinent variables defined in step 2 using the fact
.
Step 4: Restrict again the pertinent variables in step 3 using the three following facts:
・ all the façons
belong to S: then the images of the generic sequences
by the pertinent variables defined in the step 2 must tend to either an arbitrary complex number or zero,
・
: then there are at least
pertinent variables
such that the images of the sequences
by
, re- spectively, tend independently to
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
-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
of degree d.
Proof. In the one hand, by theorem 2.4, the dimension of
is
. By the step 3, step 5 and step 6, we consider all the possible cases of the all
- dimensional irreductible strata of
. Since the dimension of
is
(see theorem 2.4), we get all the possible asymptotic sets
of non-proper dominant polynomial mappings
of degree d. In the other hand, the number of all the possible façons of a polynomial mapping
is finite, as the shown of the following lemma:
Lemma 5.3. Let
be a polynomial mapping such that
. Then, the number of all possible façons of
is finite. More precisely, the maximum number of façons of
is equal to
where
Proof. Assume that
is a façon of
. We have the following cases:
i) If
: we have
possible façons.
ii) If
and
: we have
possible façons.
iii) If
and
: We have
possible façons.
As the three cases are independent, then the maximum number of façons of
is equal to
Remark 5.4. In the example 3.5 and in the proofs of the lemmas 4.2 and 4.4, we use a linear change of variables to simplify the pertinent variables (so that we can work without coefficients and then we can simplify calculations). This change of variables does not modify the results of the theorem 4.1. However, in the algorithms 3.10 and 5.1, we do not need the step of linear change of variables, since the computers can work with coefficients of pertinent variables without making the problem heavier.