Stratifications of polynomial spaces

Publicacions Matem`atiques, Vol 42 (1998), 383–410. STRATIFICATIONS OF POLYNOMIAL SPACES L. Birbrair Abstract In the paper we construct some stratifications of the space of monic polynomials in real and complex cases. These stratifications depend on properties of roots of the polynomials on some given semialgebraic subset of Ror C. We prove differential triviality of these stratifications. In the real case the proof is based on properties of the action of the group of interval exchange transformations on the set of all monic polynomials of some given degree. Finally we compare stratifications corresponding to different semialgebraic subsets. 1. Introduction In Theory of Singularities many authors use the following conception: local (or global) stability follows from transversality to some stratification of some space of polynomial (analytic or smooth) maps (see, for example, [AVG], [dPW]). Here we present some stratifications of spaces of monic polynomials on one variable connected to problems of stability of singularities of projection and singularities near boundary points of hypersurfaces. In the proof of Tarski-Seidenberg Theorem (see, for example, [BR]) appears the following partition {Qk}of the space of monic polynomials: f∈Qkif fhas exactly kroots. In fact the main step of the proof of Tarski-Seidenberg Theorem is to prove that Qkare semialgebraic subsets. The partition {Qk}is not a stratification. In [GWPL] it is mentioned that if we consider not just a number of roots but multiplicities of these roots we obtain a Whitney regular stratification of the space of monic polynomials. In [G] it is proved that if we fix multiplicities of the eigenvalues of n×n matrices and the structures of their Jordan blocks (so-called Segre symbol) we obtain a stratification of the space of n×nmatrices. C. G. Gibson also proved that this stratification is Whitney regular. 384 L. Birbrair In the present paper we consider some generalizations of the stratification considered in [GWPL]. Namely, let Nbe some closed semialgebraic subset of C. Thus we have the following decomposition N=Int(N)∪S 1∪S 2∪Sing(∂N), where Int(N) is the set of internal points of N;S1⊂∂N is a subset of the border ∂N contains only Cρ-smooth points which have the internal points of Nin every small neighbourhood; S2is a complement to S1in the set of Cρ-smooth points of ∂N; as usual Sing(∂N) is a subset of Cρ-singular points of ∂N.We define a multiplicity symbol (see sections 2.1 and 3.1) which is determined by multiplicities of the roots of given polynomial on Int(N), S1, S2and Sing(N). The main result of the paper is the theorem that all stratifications given by these multiplicity symbols are differentially trivial and thus Whitney regular (Theorems 2.4 and 3.1). Observe that the stratifications under considerations are different and depend on Nand ρ. In the case N=Cwe obtain the stratification considered in [GWPL]. The paper has the following structure. Part 2 is devoted to stratifications of the space of real monic polynomials. Since any closed semialgebraic subset of Ris or Ritself, or a finite union of closed segments, points and closed halflines we begin our consideration from the case of finite segment. The case of the union of segments and halflines can be treated in the same way. We define a multiplicity symbol for polynomials. This multiplicity symbol is connected to some fixed segment [b, c] and characterizes the number of roots a polynomial has on this segment and their multiplicities. For each multiplicity symbol we define a stratum (the set of all polynomials with the same multiplicity symbol). We prove that the stratification by multiplicity symbols gives us a semialgebraic stratification. It is known [BCR] that each semialgebraic stratification can be finitely subdivided to obtain a semialgebraic stratification satisfying “a” and “b” axioms of Whitney. The interesting property of the stratification by multiplicity symbol is the following: it is not necessary to subdivide it because it is Whitney regular itself. Paragraphs 2.3 and 2.4 are devoted to proving this fact. Section 2.3 is devoted to Interval Exchange Transformations [Ke]. Interval Exchange Transformation is a very popular object in ergodic theory. Here we use it in real algebraic geometry. We prove that all the strata defined in 2.1 are invariant by an action of group of interval exchange transformations. The action of any nontrivial interval exchange transformation on the space of polynomials is not continuous, but if the roots of some polynomial are well situated the action of a given interval exchange transformation is a difeomorphism on a neighbourhood of the polynomial. This important property of this group action helps us a lot of prove in section 2.4 that the considering Stratifications of polynomial spaces 385 stratifications satisfy the boundary axiom. In fact the boundary axiom can be obtained as a corollary of Whitney regularity, but the semiorder relation itself has a nice combinatorial and geometrical nature, which gives us a more detailed description of the stratifications. Part 3 is connected to the complex case, where Nis a closed semialgebraic subset of C. To prove a differential triviality we consider socalled disk exchange transformations, which correspond to interval exchange transformations in real case. These transformations have not such nice properties as interval exchange transformations (they do not form a group), but they are useful to prove a differential triviality of the stratifications under consideration. Part 4 is devoted to the comparing of different stratifications. In section 4.1 we compare real and complex stratifications for the same N.In section 4.2 we prove that two stratifications {Pn,µ(N1)}and {Pn,µ(N2)} (see the notations of part 3) are Cρ-equivalent if the corresponding subsets N1and N2are Cρ-equivalent. Finally I’d like to mention that these stratifications are rather natural. If we consider a partition {Qk(N)}(the similar as in [BR]) given by the number of roots of some polynomial in Nand apply the algorithm defined in [BCR] to obtain a Whitney regular stratification we will obtain exactly the stratification by multiplicity symbol. 2. Stratifications of real monic polynomials 2.1. Multiplicity symbol. Let Pnbe a space of all monic polynomials of degree nwith real coefficients. Pncan be identified to the space Rnin the standard way: f=un+an−1un−1+···+a 1u+a 0∈P n↔(a 0,a 1,... ,a n−1)∈R n. Consider some (closed) segment [b, c]inR. Let’s define the symbol µ={µb,µ 1,... ,µ r,µ c}corresponding to [b, c] such that all numbers µb, µc,µi(1 ≤i≤r) satisfy the following conditions: 1. 0 ≤r≤n. (Note that we consider also r= 0; in this case µ={µb,µ c}). 2. µb,µ c∈N∪{0};µ i∈N. (Nis the set of natural numbers). 3. r P i=1 µi+µb+µc≤n. 4. 1 ≤µ1≤···≤µ r. 386 L. Birbrair Let Mbe the set of all these symbols µ. Denote by Pn,µ the set of polynomials f∈Pnsuch that 1. fhas roots of multiplicities µband µcat the points band c correspondently. (Note that µb= 0 (or µc= 0) means that f(b)6= 0 (or f(c)6= 0)). 2. fhas exactly r(different) roots v16=v26=···6=v rof multiplicities µ1,µ 2,... ,µ r(correspondently) in the open interval (b, c). In other words Pn,µ is defined by the intersection of the zero divisor of fwith the segment [b, c]. Note that Pn,µ depends on the segment [b, c]. Theorem 2.1. The collection {Pn,µ}µ∈M is a semialgebraic stratification of Pn. Proof: It is clear that [ µ∈M Pn,µ =Pnbecause each polynomial belongs to some Pn,µ (for some µ). To prove that all Pn,µ are smooth submanifolds and semialgebraic sets we need the following. Lemma 2.1. Let µ={0, r z}| { 1,1,... ,1,0}(rcan be equal to 0). Then Pn,µ is an open semialgebraic set in Pn. Proof of Lemma 2.1: Let us prove that Pn,µ is semialgebraic. Each polynomial f∈Pn,µ can be presented in the following form: (1) f=(u−v 1 )·...·(u−v r)(u−`1)·...·(u−` s) ·(u 2+p 1u+q 1)·...·(u 2+p tu+q t) for some s, t ∈N∪{0}uniquely determined by fsuch that r+s+2t=n. For given pair (s, t) the set Pn,µ defines a semialgebraic set U(s, t) in the space Rnof coefficients (v1,... ,v r,` 1,... ,` s,p 1,q 1,... ,p t,q t)by the following inequalities:          b<v i<c, i=1,2,... ,r (it means: vi∈(b, c)) vi6=vjfor i6=j;i, j =1,2,... ,r ` i<bor `i>c, i=1,2,... ,s (it means: `i/∈[b, c]) p2 j−4qj<0,j=1,2,... ,t. Stratifications of polynomial spaces 387 Let Qs,t :Rn→Pnbe a map obtained by the opening brackets in (1). (Remined that Pnis identified to Rn). We get Pn,µ =[ s,t s+r+2t=n Qs,t(U(s, t)). Since Qs,t is an algebraic map we obtain (by Tarski-Seidenberg Theorem) that Pn,µ is a semialgebraic set. Let’s prove that Pn,µ is an open set. By the definition of Pn,µ the graph fconsidered in the subset (b, c)×Rintersects the “zero section” (b, c)×{0}transversally. Hence there exists a neighbourhood of fin Pnsatisfies the same property. It means that Pn,µ is an open subset. Lemma 2.1 is proved. Lemma 2.2. Pn,µ is a smooth submanifold of Pnand codim Pn,µ = µb+µc+ r P i=1 (µi−1). We prove this lemma in several steps. Claim. Let µsatisfies the following condition: r X i=1 µi+µb+µc=n. Then Pn,µ is a smooth submanifold of Pn. Proof: Consider a space Rrand a subset Vµ⊂Rrdefined in the following way: Vµ={(v1,... ,v r)∈R rsuch that (2)      vi∈(b, c),i=1,... ,r v i6=v j,if i6=j vi<v j ,if µi=µjand i<j }. Let Fµ:Vµ→Pnbe the following map (3) Fµ(v1,... ,v r)=(u−v 1 ) µ 1·(u−v 2 ) µ 2·...·(u−v r) µ r ·(u−b) µ b·(u−c) µ c. F µis a one-to-one map because each polynomial is uniquely defined by it’s roots. 388 L. Birbrair Fµ(Vµ)=P n,µ because for each polynomial f∈Pn,µ we have the presentation (3). Now we have to prove that Fµis an immersion. We do it using the induction by n. If n= 1 we have: P1,{0,1}and P1,{1,0}are one points each, P1,{0,1,0}is an open interval (V{0,1,0}={v∈(b, c)},F{0,1,0}(v)=u−v). Now suppose that Fµis an immersion for each n<n 0 . Let’s prove it for n0. Suppose that r6= 0. For each polynomial f∈Pn0,µ we have the following presentation: (4) f=(u−v r ) µ r·h, where vrisarootoffbelonging to (b, c) and h∈Pn0−µr,ν,ν= {µb,µ 1,... ,µ r−1,µ c}. Let Gµr:(b, c)×Pn0−µr→Pn0be the following map: (5) Gµr(vr,g)=(u−v r ) µ r·g where g∈Pn0−µr. We will prove that Gµris a local immersion in a neighbourhood of a point (vr,g) such that vrisnotarootofg. Thus we can consider the map Fµas the following: Fµ(v1,... ,v r)=G µ r(v r ,F ν(v 1,... ,v r−1))¯¯Vµ and obtain that Fµis an immersion (Fνis an immersion by the induction hypothesis). Let Uvrand Ugbe neighbourhoods of vrand gsuch that for each v∈Uvrand for each ˜g∈Ugvisnotarootof˜g. In local coordinates (putting ˜u=u−vr) we obtain (6) ˜g(˜u)=˜u n 0 −µ r+˜g n 0 −µ r−1˜u n 0 −µ r−1+···+˜g 1˜u+˜g 0 , where ˜g06= 0 because vrisnotarootof˜g. Putting this presentation (6) to (5) we obtain Gµr(v, ˜g)=˜u n 0+q n 0 −1˜u n 0 −1+···+q 1˜u+q 0, Stratifications of polynomial spaces 389 where (7)                        qn0−1=˜g n 0 −µ r −1+µ r·w q n 0 −2=˜g n 0 −µ r −2+µ r·w·˜g n 0 −µ r −1+o(w) ...................................... q µ r=˜g 0+µ r·w·˜g 1+o(w) q µ r −1=µ r·w·˜g 0+o(w) q i=o(w) for i=0,... ,µ r−2. (We use the formula: (˜u+w)µr=˜u µ r+µ r ˜u µ r −1 w+o(w); w=vr−v.) From (7) we can see that the Jacobian matrix of Gµrhas an above triangular structure. Since ˜g06= 0 it has a maximal rank. Thus the map Gµris a local immersion in a neighbourhood of (vr,g). If r= 0 symbol µhas the following form: {µb,µ c}and µb+µc=n0. Hence Pn0,µ contains only one point f=(u−b) µ b·(u−c) µ c. So, we proved that Pn,µ is an immersed submanifold. To prove that Pn,µ is a submanifold we’ll show that Fµ(∂Vµ)∩Pn,µ =∅. (Note, that Vµis bounded.) Fµcan be extended to all Rrby the formula (3). The system (2) describes Vµis an open semialgebraic set in Rr. By the Second Main Structural Theorem (see [BR, p. 68]) each point v∈∂Vµsatisfies the following condition: "r _ i=1 ((vi=b)∨(vi=c))#∨ _ i6=j (vi=vj) . But by the definition of Pn,µ we get that Fµ(v)/∈Pn,µ because the polynomial Fµ(v) has the different multiplicity symbol. So, Pn,µ is a smooth submanifoled of Pn. The claim is proved. Remark 2.1. Since Vµis a semialgebraic set and Fµis a semialgebraic map we obtain (by Tarski-Seidenberg Theorem) that Pn,µ is also a semialgebraic set. Remark 2.2. Since Fµis an immersion then dim Pn,µ = dim Vµ=r=n−Ãr X i=1 (µi−1)+µb+µc!. So, we obtained a formula for codimension of Pn,µ for this special case. 390 L. Birbrair Proof of the lemma: Suppose that r P i=1 µi+µb+µc=m<n. For each polynomial f∈Pm,µ we have: (8) f=g·p where g∈Pm,µ and p∈Pn−m,{0,0}. Let e Fµ:Pm,µ ×Pn−m,{0,0}→Pnbe the map defined by the formula (8). The map is one-to-one because a polynomial is uniquely defined by it’s roots and sets of roots of gand pdo not intersect. Let us prove that e Fµis an immersion. The proof uses the same arguments as the proof of the claim. Consider the same set Vµdescribed by inequalities (2) and construct a map Gµ:Vµ×Pn−m,{0,0}→Pnsuch that (9) Gµ(v,p)=F µ (v)·p, where Fµis a map constructed in the proof of the claim (the formula (3)). Since Fµ:Vµ→Pm,µ is a diffeomorphism it is enough to prove that Gµ is an immersion. Let us prove it by induction by m.Ifm= 0 then Gµis just the identity map on Pn−m,{0,0}and Pn−m,{0,0}is an open set (by Lemma 2.1). Suppose that we proved the statement for m<m 0 . Let’s prove it for m0. Consider the polynomial f∈Pn,µ. Suppose that r6= 0. Take the root vr. We have the following presentation (10) f=(u−v r ) µ r·h, where h∈Pn−µr,ν,ν={µb,µ 1,... ,µ r−1,µ c}. The continuation of the proof is the same as in the claim. Let r= 0. It means that µ={µb,µ c}. In this case Vµ=∅.Thus it is enough to prove that the maps Gµb:Pn−µb,{0,0}→Pnand Gµc: Pn−m,{0,0}→Pn−µbdefined as follows Gµb(h)=(u−b) µ b·h, Gµc(g)=(u−c) µ c·g are immersions. We will show it for Gµb.ForG µ cthe proof is the same. Let h∈Pn−µb,{0,0}. Putting ˜u=u−bwe obtain h(˜u)=˜u n−µ b+h n−µ b −1˜u n−µ b −1+···+h 0 Stratifications of polynomial spaces 391 and Gµb(h)(˜u)=˜u n+h n−µ b −1˜u n−1+···+h 0˜u µ b. So, Pn,µ =Gµb◦Gµc(Pn−m,{0,0}) is an immersed submanifold. Let us prove that Pn,µ is a submanifold. The map Gµ, defined by formula (9), can be extended to the set Rr×Pn−m. Let (v,p)∈∂(Vµ× Pn−m,{0,0}). It means that either v∈∂Vµor p∈∂Pn−m,{0,0}. The case v∈∂Vµwas considered in the proof of the claim. Let p∈∂Pn−m,{0,0}. It means that Phasarootin[b, c]. Hence, f=Gµ(v,p) does not belong to Pn,µ. Since Pn−m,{0,0}is unbounded it is also necessary to prove that if a sequence (vk,p k) tends to infinity for k→∞then so for Gµ(vk,p k). But it is a partial case of Proposition 1.5.5 [BR]. Since Gµis a homeomorphism to the image dim Pn,µ =r+n−m=n−Ãr X i=1 (µi−1)+µb+µc!. It proves the codimension formula. Lemma 2.2 is proved. Remark 2.3. We also proved that Pn,µ is a semialgebraic set. Let’s define a codimension of µas a codimension of the corresponding stratum Pn,µ if Pn,µ 6=∅:c(µ)=codimP n,µ. This definition is correct (does not depend on n), because by Lemma 2.2 we have: c(µ)= r X i=1 (µi−1)+µb+µc. Now we are going to define some semiorder relation Ron the set of multiplicity symbols M. We do it in the following way. 1. If c(µ1)=c(µ 2 ) then (µ1,µ 2) and (µ2,µ 1) do not belong to R. 2. Let c(µ2)=c(µ 1 ) + 1. Then (µ1,µ 2)∈Rif these symbols satisfy one of the following conditions: a) r2=r1+1,µ 1 b=µ 2 b,µ 1 c=µ 2 cand there exists µ2 jsuch that µ2 i=µ1 ifor all i6=j(and hence µ2 j= 2). b) r2=r1−1, µ1 b=µ2 b,µ1 c=µ2 cand there exist µ1 s,µ1 jand µ2 k such that µ2 i=µ1 ifor all i6=s, j, k (and hence µ1 s+µ1 j=µ2 k). c) r2=r1−1, µ1 b=µ2 band there exists µ1 jsuch that µ2 i=µ1 i for all i6=j(and hence µ1 c+µ1 j=µ2 c). 398 L. Birbrair Then applying the same arguments as in the Step 2 we obtain that FT,² is a diffeomorphism to the image in a neighbourhood of f. Step 4: Lemma 2.3. Let the pair (T,f)satisfies the condition of the theorem. Then there exist δ>0, a finite set of points {wk}and a special interval exchange transformation S:[s, t)→[s, t)such that 1) There exist ²>0and a neighbourhood Ufsuch that FS,²¯¯Uf= FT,²¯¯Uf. 2) [wk−δ, wk+δ]∩[wj−δ, wj+δ]=∅for k6=j. 3) Let Ω=[ k [w k−δ, wk+δ). Then S¯¯Ωis a permutation of intervals {[wk−δ, wk+δ)}and S¯¯[s,t)−Ω=id. Proof: Set {wk}=(z(f)∩[s, t)) ∪T(z(f)∩[s, t)). Take δ>0 such that it satisfies the property 2) and for every v∈ z(f)∩[s, t)wehave[v−δ, v +δ]⊂Int ∆i(∆iis the corresponding to v subinterval (see the condition of the theorem)). Set Ω0=[ v∈z(f)∩[s,t) [v−δ, v +δ) and S¯¯Ω0=T¯¯Ω0. Let us extend the map Sto Ω −Ω0such that S¯¯Ωwill be a permutation of intervals. And finally, let us extend Sto [s, t)−Ω as the identity map. A neighbourhood Ufof fand ²>0 we can find in the same way as in the Steps 2 and 3. Step 5: Now we can restrict our consideration to permutations of intervals. For each permutation Sof intervals we have: S=S1◦S2◦ ···◦Sm, where {Si}m i=1 are permutations of intervals corresponding to the standard generators of the permutation group Sp(p=#{ω i }). Each generator is an interchanging of a pair of intervals, which were considered in the Steps 2 and 3. It completes the proof of the theorem. Stratifications of polynomial spaces 399 2.4. Differential triviality of the stratifications defined by the multiplicity symbol. Let Abe a stratified set and {Ki}be a stratification. We say that this stratification is differentially trivial if for each stratum Kiand for each two points x1,x 2∈K ithere exist neighbourhoods of these points Ux1⊂Aand Ux2⊂Aand a diffeomorphism F:Ux1→Ux2such that F(Ux1∩Kj)=U x 2∩K jfor every jsuch that Ki⊂C`(Kj). Theorem 2.4. The stratification {Pn,µ}µ∈M is differentially trivial. Proof: Let f1,f 2∈P n,µ. Let m= r P i=1 µi+µb+µc. Then we have: f1=˜ f1·g1,˜ f2·g2, where ˜ f1,˜ f2∈Pm,µ and g1,g 2∈P n−m,{0,0}. Lemma 2.4. There exist neighbourhoods U˜ f1of ˜ f1and U˜ f2of ˜ f2in Pmand a diffeomorphism H:U˜ f1→U˜ f2such that for each ν≺µ H(Pm,ν ∩U˜ f1)=P m,ν ∩U˜ f2and H(Pm,µ ∩U˜ f1)=P m,µ ∩U˜ f2. Proof of the lemma: Let µ={µb,µ 1,... ,µ r,µ c},v i∈(b, c) and ωi∈ (b, c)(i=1,... ,r) be roots of ˜ f1and ˜ f2(correspondently) corresponding to µi. Take δ>0 such that for all i=1,... ,r we have: 1) [vi−δ, vi+δ]∩[vj−δ, vj+δ]=∅if i6=j. 2) [vi−δ, vi+δ]∩[b, b +δ]=∅. 3) [vi−δ, vi+δ]∩[c−δ, c]=∅. Let for this δthe same properties hold for ωiand 4) [vi−δ, vi+δ]∩[wj−δ, wj+δ]=∅if wj6=vi. Denote by Ω1= r [ i=1 [vi−δ, vi+δ), Ω2= r [ i=1 [wi−δ, wi+δ) and Ω = Ω1∪Ω2. Let us define a permutation Sof intervals on [b, c]inthe following way. Step 1: Define Son Ω1:S(u)=u−v i+w ifor u∈[vi−δ, vi+δ). Step 2: Define Son Ω2−Ω1such that S¯¯Ωbe a permutation of intervals. Step 3: Define S(u)=ufor u∈[b, c]−Ω. It is clear that FS,²(˜ f1)= ˜ f 2for every ²>0. By Proposition 2.5 we have FS,²(Pm,ν)=P m,ν for every multiplicity symbol ν. By Theorem 2.3 there exist neighbourhoods U˜ f1and U˜ f2such that H=FS,² is a diffeomorphism of U˜ f1onto U˜ f2. The lemma is proved. 400 L. Birbrair Remark 2.5. The neighbourhoods U˜ f1and U˜ f2can be choosen such that for each h∈U˜ f1∪U˜ f2for sufficiently small δ0>0, z(h)⊂(−², ²)× (b−δ0,c+δ0). Now take ²,δ0and neighbourhoods Ug1and Ug2such that: 1. For each g∈Ug1∪Ug2z(g)∩((−², ²)×(b−δ0,c+δ0)) = ∅. 2. The map L(g)=g−g 1+g 2maps Ug1onto Ug2. Let Uf1=U˜ f1·Ug1and Uf2=U˜ f2·Ug2. Define a map ψ:Uf1→Uf2 in the following way. For each p∈Uf1we have a unique presentation p=˜p·gsuch that ˜p∈U˜ f1,g∈Ug1(because the resultant of ˜ f1and g1 is not equal to zero). Set ψ(p)=H(˜p)·L(g). Since Hand Lare local diffeomorphisms and the resultant of ˜ f2and g2is nondegenerate ψis a diffeomorphism. By lemma it has required properties. Theorem 2.4 is proved. Since the stratification is differentially trivial and semialgebraic it is Whitney regular (see, for example, [GWPL]). Hence we have Theorem 2.5. The stratification {Pn,µ}µ∈M defined in section 1 is Whitney regular (satisfies the axioms “a” and “b” of Whitney). Theorem 2.6. The stratifications defined in section 2.2 are Whitney regular. The proof is the same as the proof of Theorem 2.4. 3. Stratifications of complex monic polynomials 3.1. Multiplicity symbol. Let Pn(C) be a space of all monic polynomials of degree nwith complex coefficients. Let N⊂Cbe a semialgebraic closed subset. We have N=Int(N)∪Smooth(∂N)∪Sing(∂N), where Smooth(∂N) is a set of Cρ-smooth points of ∂N (we can suppose the order of differentiability 0≤ρ≤∞), Sing(∂N)def =∂N −Smooth(∂N). Since Nis semialgebraic we have # Sing(∂N)<∞for any ρ. The set Smooth(∂N) can be obtained as a union of two connected components S1and S2defined in the following way: S1={x∈Smooth(∂N) and there exists ²>0 such that B(x, ²)∩Int(N)6=∅},S2= Smooth(∂N)−S1(B(x, ²) means a ball with the center xand radius ²). Now let us define a multiplicity symbol µ(N)={µ1,...,µ r 1,η 1,...,η r 2, ζ 1,...,ζ r 3,θ 1,...,θ r 4}as a collection of natural numbers corresponding to the set Nsatisfy the following properties: Stratifications of polynomial spaces 401 1. r1≥0, r2≥0, r3≥0, r4= # Sing(N). 2. If Int(N)=∅then r1=r2=0. 3. If S1=∅then r2=0. 4. If S2=∅then r3=0. 5. r1 P i=1 µi+ r2 P j=1 ηj+ r3 P s=1 ζs+ r4 P k=1 θk≤n. 6. 0 <µ 1≤µ 2≤···≤µ r 1, 0<η 1≤η 2≤···≤η r 2, 0<ζ 1≤ζ 2≤···≤ζ r 3. Denote by Mn(N) the set of all multiplicity symbols for Nand n fixed. Let Pn,µ(N)(C) be the set of polynomials f∈Pn(C) such that 1. fhas exactly r1roots v16=v26=···6=v r 1on Int(N) with multiplicities µ1,...,µ r 1(correspondently). 2. fhas exactly r2roots ˜v16=˜v 26=···6=˜v r 2on S1with multiplicities η1,...,η r 2. 3. fhas exactly r3roots v0 16=v0 26=···6=v0 r 3on S2with multiplicities ζ1,...,ζ r 3. 4. fhas a root at the point yk∈Sing(N) with a multiplicity θk. (In the case f(yk)6= 0 set θk= 0). The main goal of this part is the following result. Theorem 3.1. The collection {Pn,µ(N)(C)}µ(N)∈Mn(N)isaCρ-differentially trivial (and thus Whitney regular) stratification of Pn(C). The proof contains several steps. Definition 3.1. A zero-symbol 0(N) is a multiplicity symbol such that the first three parts µ,ηand ζare empty (r1=r2=r3= 0) and all θ-s are equal to zero. Lemma 3.1. Pn,0(N)is an open semialgebraic subset of Pn(C). Proof: Each polynomial f∈Pn(C) can be presented in the following form: (11) f=(u−v 1 )(u−v2)·...·(u−v n), v 1,v 2,...,v n∈Care the roots of f. 402 L. Birbrair Let v=(v 1 ,...,v n)∈C nbe a “root vector”. Consider a subset V0(N)⊂Cndefined as follows V0(N)={v∈Cn;v=(v 1 ,...,v n),v i/∈Nfor i=1,...,n}. Let Q:Cn→Pn(C) be a map obtained by opening brackets in (11). We have Pn,0(N)(C)=Q(V 0(N)). Thus Pn,0(N)(C) is semialgebraic (by Tarski-Seidenberg Theorem). Since Nis a closed subset of CPn,0(N)is open (by continuaty of roots [BR]). Lemma 3.2. For each µ(N)∈M n (N)the set Pn,µ(N)(C)is a semialgebraic subset and an immersed submanifold of Pn(C). To prove the lemma we need one proposition which is in fact the statement of the lemma in some particular case. Proposition 3.1. Let µ(N)={(µ i ),(η j ),(ζ s ),(θ k )}∈M n (N)and r1 P i=1 µi+ r2 P j=1 ηj+ r3 P s=1 ζs+ r4 P k=1 θk=n. Then Pn,µ(N)(C)is a semialgebraic subset and an immersed submanifold of Pn(C). Proof: Let Vµ(N)={v=(v 1 ,...,v r 1,˜v 1,...,˜v r 2,...,v0 1,...,v0 r 3}be a subset of Cr(r=r1+r2+r3) obtained as a set of solutions of the following system of equations and inequalities          vi∈Int(N), ˜vj∈S1, v0 s∈S2, vi16=vi2,˜vj16=˜v j 2 ,v0 s 1=v0 s 2for i16=i2,j 26=j 2,s 16=s 2. By the definition Vµ(N)is a Cρsemialgebraic submanifold of Cr. Let Fµ(N):Cr→Pn(C) be a map defined in the following way Fµ(N)(v1,...,v r 1,˜v 1,...,˜v r 2,v0 1,...,v0 r 3) =(u−v 1 ) µ 1·...·(u−v r 1) µ r 1(u−˜v 1) η 1·...·(u−˜v 2) η r 2(u−v0 1) ζ 1 ·...·(u−v0 r 3) ζ r 3(u−y 1) θ 1·...·(u−y r 4) θ r 4 where Sing(∂N)={y 1 ,...,y r 4}. Observe that if θk= 0 we have (u− yk)θk=1. Clearly the map Fµ(N)is an algebraic map. The proof of the fact that Fµ(N)is a local immersion is the same as in Lemma 2.2. Stratifications of polynomial spaces 403 Proof of Lemma 3.2: Now consider a general case: r1 X i=1 µi+ r2 X j=1 ηj+ r3 X s=1 ζs+ r4 X k=1 θk≤n. Each polynomial f∈Pn,µ(N)(C) can be presented in the following form: f=g·h, where g∈Pm,µ(N)(C) such that m= r1 P i=1 µi+ r2 P j=1 ηj+ r2 P s=1 ζs+ r4 P k=1 θkand h∈Pn−m,0(N)(C). We define a map Gµ(N):Vµ(N)×Pn−m,0(N)(C)→Pn,µ(N)(C)inthe following way: Gµ(N)(v,h)=F µ(N) (v)·h. By the same argument as in Lemma 2.2 we obtain that Gµ(N)is a local immersion. By Tarski-Seidenberg Theorem Pn,µ(N)(C) is a semialgebraic set. Remark 3.1. In contrast to the real case the maps Fµ(N)and Gµ(N) are not necessary one-to-one. One can show that they are covering maps. Remark 3.2. By the immediate calculations we obtain Codim Pn,µ(N)=2 r 1 X i=1 (µi−1) + r2 X j=1 (2ηj−1) + r3 X s=1 (2ζs−1)+2 r 4 X k=1 θk. In the next sections we are going to prove the differential triviality of the partition {Pn,µ(N)(C)}. 3.2. Disk-exchange transformations. Let N⊂Cbe a closed semialgebraic subset. Let {x1,...,x p}be a finite subset of Int(N). Consider ²>0 such that 1. B(xi,²)⊂Int(N) for all i=1,...,p. 2. B(xi1,²)∩B(x i 2,²)=∅for i16=i2. Let {˜x1,...,˜x t 1},{x 0 1,...,x 0 i 2}be finite subsets of S1and S2correspondently and δ1,δ2be numbers such that 3. Each pair (B(˜xj,δ 1,B(˜xj,δ 1)∩∂N)isC ρ -diffeomorphic to the pair (B(0,1),B(0,1) ∩R). 404 L. Birbrair Each pair (B(x0 s,δ 2),B(x 0 s,δ 2)∩∂N)isC ρ -diffeomorphic to the pair (B(0,1),B(0,1) ∩R). 4. B(˜xj,δ 1)∩B(x 0 s,δ 2)=∅for all j=1,...,t 1,s=1,...,t 2. 5. B(˜xj1,δ 1)∩B(˜xj2,δ 1)=∅for j16=j2. B(x0 s1,δ 2)∩B(x 0 s 2,δ 2)=∅for s16=s2. 6. B(xi,²)∩B(˜xj,δ 1)=∅ B(x i ,²)∩B(x 0 s,δ 2)=∅for all i=1,...,p,j=1,...,t 1,s= 1,...,t 2. Let B=Ãp [ i=1 B(xi,²) !∪  t 1 [ j=1 B(˜xj,δ 1)  ∪Ãt 2 [ s=1 B(x0 s,δ 2) !. Definition 3.2. A map T:C→Cis called a disk-exchange transformation associated to Nif it satisfies the following conditions: 1. Tis a bijection. 2. T¯¯C−B= id. 3. For each i1there exists i2such that T¯¯B(xi2,²)is a Cρ-diffeomorphism of B(xi1,²)ontoB(x i 2 ,²) such that T(xi1)=x i 2. 4. For each j1(or s1) there exists j2(or s2, correspondently), such that T¯¯B(˜xj1,δ1)(or T¯¯B(x0 s1,δ2))isaC ρ -diffeomorphism onto B(˜xj2δ1) (or B(x0 s2,δ 2)) such that T(˜xj1)=˜x j 2(or T(x0 s1)=x 0 s 2) and T(B(˜xj1,δ 1)∩N)=B(˜xj2,δ 1)∩N (or T(B(x0 s1,δ 2)∩N)=B(x 0 s 2,δ 2)∩Ncorrespondently). The points xi,˜x j ,x 0 swe call centers of the disk-exchange transformation. Let us define a map T∗:Pn(C)→Pn(C) associated to a given diskexchange transformation T. Let f∈Pn(C) be presented in the following standard form: f=(u−v 1 ) α 1·...·(u−v k) α k. Set T∗(f)=(u−T(v 1 ))α1·...·(u−T(v k))αk. Stratifications of polynomial spaces 405 Proposition 3.2. Let f∈Pn,µ(N)(C). Then for each disk-exchange transformation Tassociated to Nwe have T∗(f)∈Pn,µ(N)(C). This proposition follows from the fact that N,∂N and C−Nare invariant under the map T. Remark 3.3. The disk-exchange transformations associated to Ndo not form a group for arbitrary chosen N. Now we prove the theorem analogous to the Theorem 2.3. Theorem 3.2. Let µ(N)∈M m (N)and r1 X i=1 µi+ r2 X j=1 ηj+ r3 X s=1 ζs+ r4 X k=1 θk=m. Let f1,f 2∈P m,µ(N)(C). Then 1) There exists a disk-exchange transformation T:C→Csuch that T∗(f1)=f 2 . 2) There exist neighbourhoods Uf1and Uf2in Pm(C)such that T∗: Uf1→Uf2is a Cρ-diffeomorphism. Proof: 1) Let Z1be a zero set of f1and Z2be a zero set of f2.We have Zq=Z1 1∪Z2 q∪Z3 q∪Z4 q(q= 1 or 2), where Z1 q=Zq∩Int(N),Z 2 q =Z q ∩S 1 ,Z 3 q =Z q ∩S 2 ,Z 4 q =Z q ∩Sing(∂N). Let Z1=Z1 1∪Z1 2,Z 2 =Z 2 1 ∪Z 2 2 ,Z 3 =Z 3 1 ∪Z 3 2 ,Z 4 =Z 4 1 ∪Z 4 2 . (Observe that by the definition of the multiplicity symbol we have Z4= Z4 1=Z4 2). Let Z=Z1∪Z2. Let P:Z→Zbe a permutation of points satisfies the following conditions: 1. P¯¯Z4= id. 2. P(Z` 1)=Z ` 2(`=1,2,3,4). 3. If vi∈Int(N)isarootoff 1with multiplicity µithen P(vi)isa root of f2with the same multiplicity µi. 4. If ˜vj∈S1isarootoff 1with multiplicity ηjthen P(˜vj)isaroot of f2with multiplicity ηj. 5. If v0 s∈S2isarootoff 1with multiplicity ζsthen P(v0 s)isaroot of f2with multiplicity ζs. 406 L. Birbrair Let Z1={x1,...,x p},Z2={˜x 1,...,˜x t 1},Z3={x 0 1,...,x 0 r 2}. Let ², δ1and δ2be numbers such that conditions 1-6 from the definition of the disk-exchange transformation are satisfied. We define a disk-exchange transformation T:C→Cwith centers xi,˜x jand x0 sby T(xi)=P(x i ), T(˜xj)=P(˜xj) and T(x0 s)=P(x 0 s ). Clearly T∗(f1)=f 2 . 2) Step 1: Let us consider some particular case. Let v1 i0,v2 i 0∈Int(N) be roots of f1and f2(correspondently) with multiplicity µi0. Suppose that other roots of f1and f2are the same; v1 i=v2 i(i6=i0), ˜v1 j=˜v 2 j , v 01 s=v 02 s(for all j, s). The disk-exchange transformation Tconstructed in the part 1) is just an interchanging of the two small disks with centers at v1 i0and v2 i0and radius ². We have f1=(u−v 1 i 0) µ i 0·h(12) f2=(u−v 2 i 0) µ i 0·h,(13) where h∈Pm−µi0(C). Let U1 0be a small neighbourhood of (u−v1 i0)µi0in Pµi0(C) and Uhbe a small neighbourhood of hin Pm−µi0(C) such that the map φ1:U1 0×Uh→ Pm(C) obtained by “opening brackets” in (12) is a diffeomorphism to the image. (φ1is a local diffeomorphism because the Jacobian Jφ1at the point ((u−v1 i0)µi0,h) is a resultant of these polynomials [J] and they have not common roots). Analogically let U2 0be a small neighbourhood of (u−v2 i0)µi0in Pµ0(C) such that φ2:U2 0×Uh→Pm(C) obtained by “opening brackets” in (13) is a diffeomorphism to the image. Let Uf1=φ1(U1 0×Uh) and Uf2=φ2(U2 0×Uh). Let e T:U1 0→U2 0be the map defined in the same way as T∗but in the space Pµ0(C). Observe that e Tis a diffeomorphism. Thus in the coordinate system defined by φ1and φ2the map T∗can be written in the following way: T∗=(e T,id). Since e Tis a diffeomorphism T∗is also diffeomorphism. Step 2: Let ˜v1 j0,˜v 2 j 0be roots of f1and f2(correspondently) with multiplicity ηj0. Suppose also (as in Step 1) that other roots of f1and f2are the same. The proof is absolutely the same as in Step 1. Step 3: The same arguments for f1and f2with roots v01 s0,v02 s0with multiplicity ζs0and the same other roots. Step 4: For every pair f1,f 2∈P m,µ(N)(C) a disk-exchange transformation Tsuch that T∗(1) = f2can be obtained as a composition of maps considered in Steps 1, 2, 3. The theorem is proved. Stratifications of polynomial spaces 407 3.3. Differential triviality. Theorem 3.3. The family {Pn,µ(N)(C)}µ(N)∈Mn(N)is Cρ-differentially trivial. Proof: Let f1,f 2∈P n,µ(N)(C)wehave: f 1=˜ f 1·g 1 ,(14) f2=˜ f2·g2,(15) where ˜ f1,˜ f2∈Pm,µ(N)(C) with m=Σµ i+Ση j+Σζ s+Σθ k ,g 1 ,g 2∈ P n−m,0(N)(C). Let Tbe a disk-exchange transformation connecting ˜ f1and ˜ f2constructed in Theorem 3.2. Let U˜ f1and U˜ f2be small neighbourhoods such that T∗:U˜ f1→U˜ f2isaCρ-diffeomorphism. Let Ug1and Ug2be two small open balls in Pn−m(C) belonging to Pn−m,0(N)(C). (These two balls exist because Pn−m,0(N)(C) is an open subset of Pn−m(C)by Lemma 3.1). Let H:Ug1→Ug2be a diffeomorphism (can be a translation map). Let ψ1:Ug1×U˜ f1→Pn(C) and ψ2:Ug2×U˜ f2→Pn(C) be maps defined by “opening brackets” in (14) and (15). ψ1and ψ2 are local diffeomorphisms by the same argument as in Theorem 3.2. Set Uf1=ψ1(Ug1×U˜ f1) and Uf2=ψ2(Ug2×U˜ f2). Thus the map G=ψ2◦(H,T∗)◦ψ−1 1:Uf1→Uf2 defines a required diffeomorphism. End of the proof of Theorem 3.1: So, we obtained that the family {Pn,µ(N)(C)}µ(N)∈Mn(N)satisfies the following conditions: 1. [ µ(N)∈Mn(n) Pn,µ(N)(C)=P n (C). 2. Pn,µ(N)(C) are immersed submanifolds and semialgebraic subsets of Pn(C). 3. Pn,µ1(N)(C)∩Pn,µ2(N)(C)=∅for µ1(N)6=µ2(N). 4. The family is Cρ-differentially trivial. Thus (by [GPWL]) we can conclude that {Pn,µ(N)(C)}µ(N)∈Mn(N)is a Whitney regular stratification of Pn(C). The theorem is proved.