American Journal of Computational Mathematics
Vol.07 No.01(2017), Article ID:75289,18 pages
10.4236/ajcm.2017.71010
An Algorithm to Classify the Asymptotic Set Associated to a Polynomial Mapping
Nguyen Thi Bich Thuy
IBILCE, UNESP, Universidade Estadual Paulista, "Júlio deMesquita", São José do Rio Preto, São Paulo, Brazil
Copyright © 2017 by author and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: January 18, 2017; Accepted: March 28, 2017; Published: March 31, 2017
ABSTRACT
We provide an algorithm to classify the asymptotic sets of the dominant polynomial mappings 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 of degree 2. This algorithm can be generalized for the dominant polynomial mappings of degree , with any .
Keywords:
Algorithms, Polynomial Mappings, Asymptotic Sets, Façons
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.
Cite this paper
Thuy, N.T.B. (2017) An Algorithm to Classify the Asymptotic Set Associated to a Polynomial Mapping. American Journal of Computational Mathematics, 7, 109-126. https://doi.org/10.4236/ajcm.2017.71010
References
- 1. Jelonek, Z. (1993) The Set of Points at Which Polynomial Map Is Not Proper. Annales Polonici Mathematici, 58, 259-266.
- 2. Nguyen, T B.T. (2016) La méthode des façons. https://arxiv.org/pdf/1407.5329.pdf