A remark on a polynomial mapping from $\C^n$ to $\C^n$

We provide relations of the results obtained in the articles \cite{ThuyCidinha} and \cite{VuiThang}. Moreover, we provides some examples to illustrate these relations, using the software {\it Maple} to complete the complicate calculations of the examples. We give some discussions on these relations.


INTRODUCTION
In [NT-R], we construct singular varieties V G associated to a polynomial mapping G : C n Ñ C n´1 where n ě 2 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 V G are not trivial. In [H-N], the authors prove that if there exists a so-called very good projection with respect to the regular value t 0 of a polynomial mapping G : C n Ñ C n´1 , 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 [NT-R] and [H-N]. Moreover, we provides some examples to illustrate these relations, using the software Maple to complete the calculations of the examples. We provide some discussions on these relations.

PRELIMINARIES
2.1. Intersection homology. We briefly recall the definition of intersection homology; for details, we refer to the fundamental work of M. Goresky and R. MacPherson [GM] (see also [B]).
Definition 2.1. Let X be a m-dimensional variety. A stratification of X is the data of a finite filtration X " X m Ą X m´1 Ą¨¨¨Ą X 0 Ą X´1 " H, such that for every i, the set S i " X i zX i´1 is either an emptyset or a manifold of dimension i. A connected component of S i is called a stratum of X.
Definition 2.2. A stratification of X is said to be locally topologically trivial if for every x P X i zX i´1 , i ě 0, there is an open neighborhood U x of x in X, a stratified set L and a homeomorphism h : U x Ñ p0; 1q iˆc L, such that h maps the strata of U x (induced stratification) onto the strata of p0; 1q iˆc L (product stratification).
The definition of perversities has originally been given by Goresky and MacPherson: Definition 2.3. A perversity is an pm`1q-uple of integersp " pp 0 , p 1 , p 2 , p 3 , . . . , p m q such that p 0 " p 1 " p 2 " 0 and p k`1 P tp k , p k`1 u, for k ě 2.
Let X be a variety such that X admits a locally topologically trivial stratification. We say that an Define IC p i pXq to be the R-vector subspace of C i pXq consisting in the chains ξ such that |ξ| is pp, iq-allowable and |Bξ| is pp, i´1q-allowable.
Definition 2.4. The i th intersection homology group with perversity p, denoted by IH p i pXq, is the i th homology group of the chain complex IC p pXq.
The notation IH p,c pXq will refer to the intersection homology with compact supports, the notation IH p,cl pXq will refer to the intersection homology with closed supports. In the compact case, they coincide and will be denoted by IH p pXq. In general, when we write H˚pXq (resp., IH p pXq), we mean the homology (resp., the intersection homology) with both compact supports and closed supports.
Goresky and MacPherson proved that the intersection homology is independent on the choice of the stratification satisfying the locally topologically trivial conditions [GM, ?].
The Poincaré duality holds for the intersection homology of a (singular) variety: Theorem 2.5 (Goresky, MacPherson [GM]). For any orientable compact stratified semialgebraic m-dimensional variety X, the generalized Poincaré duality holds: IH p k pXq » IH q m´k pXq, where p and q are complementary perversities.
For the non-compact case, we have: 2.2. The bifurcation set, the set of asymptotic critical values and the asymptotic set. Let G : C n Ñ C m where n ě m be a polynomial mapping.
i) The bifurcation set of G, denoted by BpGq is the smallest set in C m such that G is not C 8 -fibration on this set (see, for example, [KOS]).
ii) When n " m, we denote by S G the set of points at which the mapping G is not proper, i.e. S G :" tα P C m : Dtz k u Ă C n , |z k | Ñ 8 such that Gpz k q Ñ αu, and call it the asymptotic variety (see [J]). The following holds: BpGq " S G ( [J]).
3. Varieties V G associated to a polynomial mapping G : C n Ñ C n´1 In [NT-R], we construct singular varieties associated to a polynomial mapping G : C n Ñ C n´1 as follows: let G : C n Ñ C n´1 such that K 0 pGq " H, where K 0 pGq is the set of critical values of G. Let ρ : C n Ñ R be a real function such that ρ " a 1 |z 1 | 2`¨¨¨`a n |z n | 2 , where ř n i"1 a 2 i ‰ 0, a i ě 0 and a i P R. Let us denote ϕ " 1 1`ρ and consider pG, ϕq as a real mapping from R 2n to R 2n´1 . Let us define M G :" SingpG, ϕq " tx P R 2n such that Rank R DpG, ϕqpxq ď 2n´2u, where DpG, ϕqpxq is the (real) Jacobian matrix of pG, ϕq : R 2n Ñ R 2n´1 at x. Notice that SingpG, ϕq " SingpG, ρq, so we have M G " SingpG, ρq.
Proposition 3.1. [NT-R] For an open and dense set of polynomial mappings G : C n Ñ C n´1 such that K 0 pGq " H, the variety M G is a smooth manifold of dimension 2n´2. Now, let us consider: Since the dimension of M G is 2n´2 (Proposition 3.1), then locally, in a neighbourhood of any point x 0 in M G , we get a mapping F : R 2n´2 Ñ R 2n´2 . Then there exists a covering tU 1 , . . . , U p u of N G by open semi-algebraic subsets (in R 2n ) such that on every element of this covering, the mapping F induces a diffeomorphism onto its image (see Lemma 2.1 of [V-V]). We can find semi-algebraic closed subsets V i Ă U i (in N G ) which cover N G as well. Thanks to Mostowski's Separation Lemma (see Separation Lemma in [M], page 246), for each i " 1, . . . , p, there exists a Nash function ψ i : N G Ñ R, such that ψ i is positive on V i and negative on N G zU i . We can choose the Nash functions ψ i such that ψ i px k q tends to zero when tx k u Ă N G tends to infinity. Let the Nash functions ψ i and ρ be such that ψ i px k q tends to zero and ρpx k q tends to infinity when x k Ă N G tends to infinity. Define a variety V G associated to pG, ρq as V G :" pF, ψ 1 , . . . , ψ p qpN G q, that means V G is the closure of N G by pF, ψ 1 , . . . , ψ p q.
In order to understand better the construction of the variety V G , see the example 4.13 in [NT-R].
Proposition 3.2. [NT-R] Let G : C n Ñ C n´1 be a polynomial mapping such that K 0 pGq " H and let ρ : C n Ñ R be a real function such that ρ " a 1 |z 1 | 2`¨¨¨`a n |z n | 2 , where ř n i"1 a 2 i ‰ 0, a i ě 0 and a i P R for i " 1, . . . , n. Then, there exists a real algebraic variety 4. The Bifurcation set BpGq and the homology, intersection homology of varieties V G associated to a polynomial mapping G : C n Ñ C n´1 We have the two following theorems dealing with the homology and intersection homology of the variety V G .
Remark 4.3. The singular set at infinity of V G depends on the choice of the function ρ, since when ρ changes, the set S G also changes. However, we have alway the property BpGq Ă S G pρq (see [DRT]).
Remark 4.4. The variety V G depends on the choice of the function ρ and the functions ψ i , but the theorems 4.1 and 4.2 do not depend on the varieties V G . Form now, we denote by V G pρq any variety V G associated to pG, ρq. If we refer to V G , that means a variety V G associated to pG, ρq for any ρ.

The Bifurcation set
BpGq and the Euler characteristic of the fibers of a polynomial mapping G : C n Ñ C n´1 Let G " pG 1 , G 2 , . . . , G n´1 q : C n Ñ C n´1 be a non-constant polynomial mapping and t 0 " pt 0 1 , t 0 2 , . . . , t 0 n´1 q P C n´1 be a regular value of G.
[H-N] A linear function L : C n Ñ C is said to be a very good projection with respect to the value t 0 if there exists a positive number δ such that for all t P D δ pt 0 q " Theorem 5.2. [H-N] Let t 0 be a regular value of G. Assume that there exists a very good projection with respect to the value t 0 . Then, t 0 is an atypical value of G if and only if the Euler characteristic of G´1pt 0 q is bigger than that of the generic fiber.
Theorem 5.3. [H-N] Assume that the zero set tz P C n :Ĝ i pzq " 0, i " 1, . . . , n´1u, whereĜ i is the leading form of G i , has complex dimension one. Then any generic linear mapping L is a very good projection with respect to any regular value t 0 of G.

Relations between [NT-R] and [H-N]
Let G " pG 1 , . . . , G n´1 q : C n Ñ C n´1 pn ě 3q be a polynomial mapping such that K 0 pGq " H. Then any t 0 P C n´1 is a regular value of G. Let ρ : C n Ñ R be a real function such that ρ " a 1 |z 1 | 2`¨¨¨`a n |z n | 2 , where ř n i"1 a 2 i ‰ 0, a i ě 0 and a i P R for i " 1, . . . , n. From theorems 4.1 and 5.2, we have the following corollary.
Corollary 6.1. Let G " pG 1 , G 2 q : C 3 Ñ C 2 be a polynomial mapping such that K 0 pGq " H. Assume that there exists a very good projection with respect to t 0 P C 2 . If the Euler characteristic of G´1pt 0 q is bigger than that of the generic fiber, then 1) H 2 pV G pρq, Rq ‰ 0, for any ρ, 2) IH t 2 pV G pρq, Rq ‰ 0, for any ρ, where t is the total perversity.
Proof. Let G " pG 1 , G 2 q : C 3 Ñ C 2 be a polynomial mapping such that K 0 pGq " H. Then every point t 0 P C 2 is a regular point of G. Assume that there exists a very good projection with respect to t 0 P C 2 . If the Euler characteristic of G´1pt 0 q is bigger than that of the generic fiber, then by the theorem 5.2, the bifurcation set BpGq is not empty.
Then by the theorem 4.1, we have H 2 pV G pρq, Rq ‰ 0, for any ρ and IH t 2 pV G pρq, Rq ‰ 0, for any ρ, where t is the total perversity. From theorems 4.2 and 5.2, we have the following corollary.
Corollary 6.2. Let G " pG 1 , . . . , G n´1 q : C n Ñ C n´1 , where n ě 4, be a polynomial mapping such that K 0 pGq " H and Rank C pDĜ i q i"1,...,n´1 ě n´2, whereĜ i is the leading form of G i . Assume that there exists a very good projection with respect to t 0 P C n´1 . If the Euler characteristic of G´1pt 0 q is bigger than that of the generic fiber, then 1) H 2 pV G pρq, Rq ‰ 0, for any ρ, 2) H 2n´4 pV G pρq, Rq ‰ 0, for any ρ, 3) IH t 2 pV G , Rq ‰ 0, for any ρ, where t is the total perversity.
Proof. Let G " pG 1 , . . . , G n´1 q : C n Ñ C n´1 , where n ě 4, be a polynomial mapping such that K 0 pGq " H. Then every point t 0 P C n´1 is a regular point of G. Assume that there exists a very good projection with respect to t 0 P C n´1 . By the theorem 5.2, the bifurcation set BpGq is not empty. If Rank C pDĜ i q i"1,...,n´1 ě n´2, then by the theorem 4.2, we have 1) H 2 pV G pρq, Rq ‰ 0, for any ρ, 2) H 2n´4 pV G pρq, Rq ‰ 0, for any ρ, 3) IH t 2 pV G , Rq ‰ 0, for any ρ, where t is the total perversity.
We have also the following corollary.
Corollary 6.3. Let G " pG 1 , . . . , G n´1 q : C n Ñ C n´1 , where n ě 4, be a polynomial mapping such that K 0 pGq " H. Assume that the zero set tz P C n :Ĝ i pzq " 0, i " 1, . . . , n´1u has complex dimension one, whereĜ i is the leading form of G i . If the Euler characteristic of G´1pt 0 q is bigger than that of the generic fiber, where t 0 P C n´1 , then 1) H 2 pV G pρq, Rq ‰ 0, for any ρ, 2) H 2n´4 pV G pρq, Rq ‰ 0, for any ρ, 3) IH t 2 pV G pρq, Rq ‰ 0, for any ρ, where t is the total perversity.
Proof. At first, since the zero set tz P C n :Ĝ i pzq " 0, i " 1, . . . , n´1u has complex dimension one, then by the theorem 5.3, any generic linear mapping L is a very good projection with respect to any regular value t 0 of G. Moreover, the complex dimension of the set tz P C n :Ĝ i pzq " 0, i " 1, . . . , n´1u is the complex corank of pDĜ i q i"1,...,n´1 . Then Rank C pDĜ i q i"1,...,n´1 " n´2. By the corollary 6.2, we get the proof of the corollary 6.3.
Remark 6.4. We can construct the variety V G pLq, where L is a very good projection defined in 5.2 as the following: Let G " pG 1 , . . . , G n´1 q : C n Ñ C n´1 , where n ě 2, be a polynomial mapping such that K 0 pGq " H. Assume that there exists a very good projection L : C n Ñ C with respect to t 0 P C n´1 . Then L is a linear function. Assume that L " ř n i"1 a i z i . Then the variety V G pLq is defined as the variety V G pρq, where ρ " with |a i |, |z i | are the modules of the complex numbers a i and z i , respectively. With this variety V G pLq, all the results in the corollaries 6.1, 6.2 and 6.3 hold. Moreover, the varieties V G pLq makes the corollaries 6.1, 6.2 and 6.3 simplier.
Remark 6.5. In the construction of the variety V G [NT-R] (see section 3), if we replace F by the restriction of pG, ϕq to M G , that means F :" pG, ϕq |M G , then we have the same results than in [NT-R]. In fact, in this case, since the dimension of M G is 2n´2, then locally, in a neighbourhood of any point x 0 in M G , we get a mapping F : R 2n´2 Ñ R 2n´1 . There exists also a covering tU 1 , . . . , U p u of N G by open semi-algebraic subsets (in R 2n ) such that on every element of this covering, the mapping F induces a diffeomorphism onto its image. We can find semi-algebraic closed subsets V i Ă U i (in N G ) which cover N G as well. Thanks to Mostowski's Separation Lemma, for each i " 1, . . . , p, there exists a Nash function ψ i : N G Ñ R, such that ψ i is positive on V i and negative on N G zU i . Let the Nash functions ψ i and ρ be such that ψ i pz k q and ϕpz k q " 1 1`ρpz k q tend to zero where tz k u is a sequence in N G tending to infinity. Define a variety V G associated to pG, ρq as V G :" pF, ψ 1 , . . . , ψ p qpN G q " pG, ϕ, ψ 1 , . . . , ψ p qpM G q.
We get the p2n´2q-dimensional singular variety V G in R 2n´1`p , the singular set at infinity of which is S Gˆt 0 R p`1 u.
With this construction of the set V G , the corrolaries 6.1, 6.2 and 6.3 also hold.

Some discussions
A natural question is to know if the converses of the corollaries 6.1 and 6.2 hold. That means, let G " pG 1 , . . . , G n´1 q : C n Ñ C n´1 pn ě 3q be a polynomial mapping such that K 0 pGq " H then Question 7.1. If there exists a very good projection with respect to t 0 P C n´1 and if either IH t 2 pV G , Rq ‰ 0 or H 2 pV G , Rq ‰ 0, then is the Euler characteristic of G´1pt 0 q bigger than the one of the generic fiber?
By the theorem 5.2, the above question is equivalent to the following question: Question 7.2. If BpGq " H then are IH t 2 pV G , Rq " 0 and IH t 2 pV G , Rq " 0?
This question is equivalent to the converse of the theorems 4.1 and 4.2. Note that by the proposition 3.2, the singular set at infinity of the variety V G is contained in S G pρqˆt0 R 1`p u. Moreover, in the proofs of the theorems 4.1 and 4.2, we see that the characteristics of the homology and intersection homology of the variety V G pρq depend on the set S G pρq. In [NT-R], we provided an example to show that the answer to the question 7.2 is negative. In fact, let G : C 3 Ñ C 2 , Gpz, w, ζq " pz, zζ 2`w q, then K 0 pGq " H and BpGq " H. if we choose the function ρ " |ζ| 2 , then S G pρq " H and IH t 2 pV G pρq, Rq " 0; if we choose the function ρ 1 " |w| 2 , then S G pρ 1 q ‰ H and IH t 2 pV G pρ 1 q, Rq ‰ 0. Then, we suggest the two following conjectures.
Conjecture 7.3. Does there exist a function ρ such that if BpGq " H then S G pρq " H?
Conjecture 7.4. Let G " pG 1 , . . . , G n´1 q : C n Ñ C n´1 pn ě 2q be a polynomial mapping such that K 0 pGq " H. Assume that there exists a very good projection with respect to t 0 P C n´1 . If the Euler characteristic of G´1ptq is constant, for any t P C n´1 , then there exists a real positive function ρ : C n Ñ R such that H 2 pV G pρq, Rq " 0 and IH t 2 pV G pρq, Rq " 0.
Remark 7.5. The construction of the variety V G in [NT-R] (see section 3) can be applied for any polynomial mapping G : C n Ñ C m , where 1 ď m ď n´2, such that K 0 pGq " H. In fact, if G is generic then similarly to the propositon 3.1, the variety M G :" SingpG, ϕq " tx P R 2n such that RankD R pG, ϕqpxq ď 2mu, has the real dimension 2m. Hence, if we consider F :" G |M G , that means F is the restriction of G to M G , then locally we get a real mapping F : R 2m Ñ R 2m . Moreover, in this case, we also have BpGq Ă S G pρq for any ρ (see [DRT]), where S G pρq :" tα P C m | Dtz k u Ă SingpG, ρq : z k tends to infinity, Gpz k q tends to αu.
So, we can use the same arguments in [NT-R], and we have the following results.
Proposition 7.6. Let G : C n Ñ C m be a polynomial mapping, where 1 ď m ď n´2, such that K 0 pGq " H. Let ρ : C n Ñ R be a real function such that ρ " a 1 |z 1 | 2`¨¨¨`a n |z n | 2 , where ř n i"1 a 2 i ‰ 0, a i ě 0 and a i P R for i " 1, . . . , n. Then, there exists a real variety V G in R 2m`p , where p ą 0, such that: 1) The real dimension of V G is 2m, 2) The singular set at infinity of the variety V G is contained in S G pρqˆt0 R p u.
Similarly to [NT-R], we have the two following theorems (see theorems 4.1 and 4.2).
Theorem 7.7. Let G " pG 1 , G 2 q : C n Ñ C 2 , where n ě 4, be a polynomial mapping such that K 0 pGq " H. If BpGq ‰ H then where t is the total perversity.

Examples
Example 8.1. We give here an example to illustrate the calculations of the set V G in the case of a polynomial mapping G : C 2 Ñ C where K 0 pGq " H, BpGq ‰ H and there exists a very good projection with respect to any point of BpGq. In general, the calculations of the set V G are enough complicate, but the software Maple may support us. That is what we do in the this example. Let us consider the Broughton's example [Br]: G : C 2 Ñ C, Gpz, wq " z`z 2 w.
So G´1p0q is not homeomorphic to G´1p q for any ‰ 0. Hence BpGq " t0u. We determine now all the possible very good projections of G with respect to t 0 " 0 P BpGq. In fact, for any δ ą 0 and for any t P D δ p0q, we have G´1ptq " tpz, wq P C 2 : z`z 2 w " t ‰ 0u " " pz, wq P C 2 : z ‰ 0 and w " t´z z 2 * .
Assume that tpz k , w k qu is a sequence in G´1ptq tending to infinity. If z k tends to infinity then w k tends to zero. If w k tends to infinity then z k tends to zero. If L is a very good projection with respect to t 0 " 0 then, by definition, the restriction L t :" L| G´1ptq Ñ C is proper. Then L " az`bw, where a ‰ 0 and b ‰ 0. We check now the cardinal 7L´1pλq of L´1pλq where λ is a regular value of L. Let us replace w " t´z z 2 in the equation az`bw " λ, we have the following equation az`b t´z z 2 " 0, where z ‰ 0. This equation has always three (complex) solutions. Thus, the number 7L´1pλq does not depend on λ. Hence, any linear function of the form L " az`bw, where a ‰ 0 and b ‰ 0, is a very good projection of G with respect to t 0 " 0. It is easy to see that the set of very good projections of G with respect to t 0 " 0 is dense in the set of linear functions.
We choose L " z`w and we compute the variety V G associated to pG, ρq where ρ " |z| 2`| w| 2 . Let us denote where x 1 , x 2 , x 3 , x 4 P R. Consider G as a real polynomial mapping, we have The set N G " SingpG, ρq is the set of the solutions of the determinant of the minors 3ˆ3 of the matrix D R pG, ρq "¨1`2 ‚.