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In [1], we construct singular varieties associated to a polynomial mapping where such that if G is a local submersion but is not a fibration, then the 2-dimensional homology and intersection homology (with total perversity) of the variety are not trivial. In [2], the authors prove that if there exists a so-called very good projection with respect to the regular value of a polynomial mapping , then this value is an atypical value of G if and only if the Euler characteristic of the fibers is not constant. This paper provides relations of the results obtained in the articles [1] and [2]. Moreover, we provide some examples to illustrate these relations, using the software Maple to complete the calculations of the examples. We provide some discussions on these relations. This paper is an example for graduate students to apply a software that they study in the graduate program in advanced researches.

We briefly recall the definition of intersection homology; for details, we refer to the fundamental work of M. Goresky and R. MacPherson [

Definition 1.1. Let X be an m-dimensional variety. A stratification of X is the data of a finite filtration

such that for every i, the set

We denote by

Definition 1.2. A stratification of X is said to be locally topologically trivial if for every

such that h maps the strata of

The definition of perversities has originally been given by Goresky and MacPherson:

Definition 1.3. A perversity is an

Traditionally we denote the zero perversity by

Let X be a variety such that X admits a locally topologically trivial stratification. We say that an i-dimensional subset

Define

Definition 1.4. The

The notation

Goresky and MacPherson proved that the intersection homology is independent on the choice of the stratification satisfying the locally topologically trivial conditions.

The Poincaré duality holds for the intersection homology of a (singular) variety:

Theorem 1.5. (Goresky, MacPherson [

where

For the non-compact case, we have:

Let

i) The bifurcation set of G, denoted by

ii) When

and call it the asymptotic variety (see [

In [

where

where

Proposition 2.1. [

Now, let us consider:

a)

b)

Since the dimension of

that means

In order to understand better the construction of the variety

Proposition 2.2. [

where

1) The real dimension of

2) The singular set at infinity of the variety

We have the two following theorems dealing with the homology and intersection homology of the variety

Theorem 3.1. [

1)

2)

Theorem 3.2. [

1)

2)

3)

Remark 3.3. The singular set at infinity of

Remark 3.4. The variety

Let

Definition 4.1. [

i) The restriction

ii) The cardinal of

Theorem 4.2. [

Theorem 4.3. [

Let

Corollary 5.1. Let

1)

2)

Proof. Let

From theorems 3.2 and 4.2, we have the following corollary.

Corollary 5.2. Let

1)

2)

3)

Proof. Let

1)

2)

3)

We have also the following corollary.

Corollary 5.3. Let

1)

2)

3)

Proof. At first, since the zero set

Remark 5.4. We can construct the variety

with

Remark 5.5. In the construction of the variety

then we have the same results than in [

We get the

With this construction of the set

A natural question is to know if the converses of the corollaries 5.1 and 5.2 hold. That means, let

Question 6.1. If there exists a very good projection with respect to

By the theorem 4.2, the above question is equivalent to the following question:

Question 6.2. If

This question is equivalent to the converse of the theorems 3.1 and 3.2. Note that by the proposition 2.2, the singular set at infinity of the variety

then

Conjecture 6.3. Does there exist a function

Conjecture 6.4. Let

Remark 6.5. The construction of the variety

has the real dimension 2 m. Hence, if we consider

So, we can use the same arguments in [

Proposition 6.6. Let

where

1) The real dimension of

2) The singular set at infinity of the variety

Similarly to [

Theorem 6.7. Let

1)

2)

Theorem 6.8. Let

1)

2)

3)

Example 7.1. We give here an example to illustrate the calculations of the set

Let us consider the Broughton’s example [

We have

and for any

So

Assume that

where

We choose

where

and

The set

Using Maple, we:

A) Calculate the determinants of the minors

1) Calculate the determinant of the minor defined by the columns 1, 2 and 3:

2) Calculate the determinant of the minor defined by the columns 1, 2 and 4:

3) Calculate the determinant of the minor defined by the columns 1, 3 and 4:

4) Calculate the determinant of the minor defined by the columns 2, 3 and 4:

B) Solve now the system of equations of the above determinants:

We conclude that

C) In order to calculate

1) Calculate and draw

2) Calculate and draw

+ Calculate and draw

Since

Example 7.2. If we take the suspension of the Broughton’s example

then, similarly to the example 7.1, the variety

Example 7.3. If we take the Broughton example for

Nguyen, T.B.T. (2016) A Remark on Polynomial Mappings from to and an Application of the Software Maple in Research. Applied Mathematics, 7, 1868-1881. http://dx.doi.org/10.4236/am.2016.715154