Univariate polynomials and the contractibility of certain sets
aa r X i v : . [ m a t h . C A ] J a n UNIVARIATE POLYNOMIALS AND THE CONTRACTIBILITYOF CERTAIN SETS
VLADIMIR PETROV KOSTOV
Abstract.
We consider the set Π ∗ d of monic polynomials Q d = x d + P d − j =0 a j x j , x ∈ R , a j ∈ R ∗ , having d distinct real roots, and its subsets defined by fixingthe signs of the coefficients a j . We show that for every choice of these signs,the corresponding subset is non-empty and contractible. A similar result holdstrue in the cases of polynomials Q d of even degree d and having no real rootsor of odd degree and having exactly one real root. For even d and when Q d has exactly two real roots, the subset is contractible or empty. It is emptyexactly when the constant term and all odd coefficients are positive and thereis at least one even coefficient which is negative. Key words: real polynomial in one variable; hyperbolic polynomial; Descartes’rule of signs
AMS classification: Introduction
In the present paper we consider the general family of real monic univariatepolynomials Q d = x d + P d − j =0 a j x j . It is a classical fact that the subsets of R d ∼ = Oa . . . a d − of values of the coefficients a j for which the polynomial Q d has oneand the same number of distinct real roots are contractible open sets. These setsare the [ d/
2] + 1 open parts of R ,d := R d \ ∆ d , where ∆ d is the discriminant set corresponding to the family Q d . Remarks 1. (1) One defines the discriminant set by the two conditions:(a) The set ∆ d is defined by the equality Res( Q d , Q ′ d , x ) = 0, where Res( Q d , Q ′ d , x ) is the resultant of the polynomials Q d and Q ′ d , i. e. the determinant of thecorresponding Sylvester matrix.(b) One sets ∆ d = ∆ d \ ∆ d , where ∆ d is the set of values of the coefficients a j for which there is a multiple complex conjugate pair of roots of Q d and no multiplereal root.One observes that dim(∆ d ) =dim(∆ d ) = d − d ) = d −
2. Thus ∆ d isthe set of values of ( a , . . . , a d − ) for which the polynomial Q d has a multiple realroot.(2) The discriminant set is invariant under the one-parameter group of quasi-homogeneous dilatations a j u d − j a j , j = 0, . . . , d . Remark 1.
If one considers the subsets of R d for which the polynomial Q d hasone and the same numbers of positive and negative roots (all of them distinct) andno zero roots, then these sets will be the open parts of the set R ,d := R d \ (∆ d ∪{ a = 0 } ). To prove their connectedness one can consider the mapping “roots coefficients”. Given two sets of nonzero roots with the same numbers of negativeand positive roots (in both cases they are all simple) one can continuously deformthe first set into the second one while keeping the absence of zero roots, the numbersof positive and negative roots and their simplicity throughout the deformation. Theexistence of this deformation implies the existence of a continuous path in the set R ,d connecting the two polynomials Q d with the two sets of roots.In the present text we focus on polynomials without vanishing coefficients andwe consider the set R ,d := R d \ (∆ d ∪ { a = 0 } ∪ { a = 0 } ∪ · · · ∪ { a d − = 0 } ) . We discuss the question when its subsets corresponding to given numbers of positiveand negative roots of Q d and to given signs of its coefficients are contractible. Notation 1. (1) We denote by σ the d -tuple (sgn( a ) , . . . , sgn( a d − )), wheresgn( a j ) = + or − , by E d the set of elliptic polynomials Q d , i. e. polynomialswith no real roots, and by E d ( σ ) ⊂ E d the set consisting of elliptic polynomials Q d with signs of the coefficients defined by σ .(2) For d odd and for a given d -tuple σ , we denote by F d ( σ ) the set of monic realpolynomials Q d with signs of their coefficients defined by the d -tuple σ and havingexactly one real (and simple) root.(3) For d even, we denote by G d ( σ ) the set of polynomials Q d having signs of thecoefficients defined by the d -tuple σ and having exactly two simple real roots. Remark 2.
For an elliptic polynomial Q d , one has a >
0, because for a < F d ( σ )is opposite to sgn( a ). A polynomial from G d ( σ ) has two roots of same (resp. ofopposite) signs if a > a < Theorem 1. (1) For d even and for each d -tuple σ , the subset E d ( σ ) ⊂ E d isnon-empty and convex hence contractible.(2) For d odd and for each d -tuple σ , the set F d ( σ ) is non-empty and contractible.(3) For d even and for each d -tuple σ , the set G d ( σ ) is contractible or empty.This set is empty exactly when the constant term of Q d and all coefficients of odddegrees are positive and at least one coefficient of even degree (different from theleading coefficient and from the constant term) is negative. The theorem is proved in Section 4. The next result of this paper concerns hyper-bolic polynomials , i. e. polynomials Q d with d real roots counted with multiplicity. Notation 2.
We denote by Π d the hyperbolicity domain , i. e. the subset of R d forwhich the corresponding polynomial Q d is hyperbolic. The interior of Π d is the setof polynomials having d distinct real roots and its border ∂ Π d equals ∆ d ∩ Π d . Weset Π ∗ d := Π d \ (∆ d ∪ { a = 0 } ∪ { a = 0 } ∪ · · · ∪ { a d − = 0 } ) . Thus Π ∗ d is the set of monic degree d univariate polynomials with d distinct real rootsand with all coefficients non-vanishing. We denote by Π kd and Π ∗ kd the projectionsof the sets Π d and Π ∗ d in the space Oa d − k . . . a d − (hence Π dd = Π d and Π ∗ dd = Π ∗ d ),by ∂ Π kd the border of Π kd and by pos and neg the numbers of positive and negativeroots of a polynomial Q d having no vanishing coefficients. NIVARIATE POLYNOMIALS AND THE CONTRACTIBILITY OF CERTAIN SETS 3
Remarks 2. (1) For a hyperbolic polynomial with no vanishing coefficients, the d -tuple σ defines the numbers pos and neg . Indeed, by Descartes’ rule of signs a realunivariate polynomial Q d with c sign changes in its sequence of coefficients has ≤ c positive roots and the difference c − pos is even, see [13] and [10]. When applyingthis rule to the polynomial Q ( − x ) one finds that the number p of sign preservationsis ≥ neg and the difference p − neg is even. For a hyperbolic polynomial one has pos + neg = c + p = d , so in this case c = pos and p = neg .(2) By Rolle’s theorem the non-constant derivatives of a hyperbolic polynomial(resp. of a polynomial of the set Π ∗ d ) are also hyperbolic (resp. are hyperbolic withall roots non-zero and simple). Hence for two hyperbolic polynomials of the samedegree and with the same signs of their respective coefficients, their derivatives ofthe same orders have one and the same numbers of positive and negative roots.Our next result is the following theorem (proved in Section 5): Theorem 2.
For each d -tuple σ , there exists exactly one open component of the set Π ∗ d the polynomials Q d from which have exactly pos positive simple and neg negativesimple roots and have signs of the coefficients as defined by σ . This component iscontractible. In Section 2 we remind some results which are used in the proof of Theorem 2.In Section 3 we introduce some notation and we give examples concerning the setsΠ d and Π ∗ d for d = 1 and 2. These examples are used in the proof of Theorem 2. InSection 6 we make comments on Theorems 1 and 2 and we formulate open problems.2. Known results about the hyperbolicity domain
Before proving Theorems 1 and 2 we remind some results about the set Π d whichare due to V. I. Arnold, A. B. Givental and the author, see [3], [11] and [14] orChapter 2 of [17] and the references therein. Notation 3.
We denote by K d the simplicial angle { x ≥ x ≥ · · · ≥ x d } ⊂ R d and by ˜ V the Vi`ete mapping˜ V : ( x , . . . , x d ) ( ϕ , . . . , ϕ d ) , ϕ j = X ≤ i
1) is the set { x = x > x = x >x } ⊂ R . The same notation is used for strata of Π d which is justified by parts(3) and (4) of Theorem 3. Theorem 3. (1) For k ≥ , every non-empty fibre ˜ f k of the projection π k : Π kd → Π k − d is either a segment or a point.(2) The fibre ˜ f k is a segment (resp. a point) exactly if the fibre is over a pointof the interior of Π k − d (resp. over ∂ Π k − d ).(3) The mapping ˜ V : K d → Π d is a homeomorphism.(4) The restriction of the mapping ˜ V to (the closure of ) any stratum of K definesa homeomorphism of the (closure of the) stratum onto its image which is (the closureof ) a stratum of Π d .(5) A stratum S of Π d defined by a multiplicity vector with ℓ components is asmooth ℓ -dimensional real submanifold in R d . It is the graph of a smooth ( d − ℓ ) -dimensional vector-function defined on the projection of the stratum in Oa d − ℓ . . . a d − . VLADIMIR PETROV KOSTOV
The field of tangent spaces to S continuously extends to the strata from the closureof S . The extension is everywhere transversal to the space Oa . . . a d − ℓ − . That is,the sum of the two vector spaces Oa . . . a d − ℓ − and (the extension of ) the field oftangent spaces to S is the space Oa . . . a d − .(6) For k ≥ , the set Π kd is the set of points on and between the graphs H k + and H k − of two locally Lipschitz functions defined on Π k − d whose values coincide on andonly on ∂ Π k − d . (We assume that the a d − k -coordinates of the points of H k + are notsmaller than the ones of the points of H k − .)(7) The graph H k + (resp. H k − ) consists of the closures of the strata whose mul-tiplicity vectors are of the form ( r, , s, , . . . ) (resp. (1 , r, , s, . . . ) ) and which haveexactly k − components. (In [17] it is written “ k components” which is wrong.)(8) For ≤ k ≤ ℓ , the projection S k of every ℓ -dimensional stratum S of Π d inthe space Oa d − k . . . a d − is the set of points on and between the graphs H k + ( S ) and H k − ( S ) of two locally Lipschitz functions defined on S k − whose values coincide onand only on ∂S k − . Remarks 3. (1) For k = 2, each fibre ˜ f is a half-line, see Example 2.(2) Consider two strata S and S of Π d defined by their multiplicity vectors µ ( S ) and µ ( S ). The stratum S belongs to the topological and algebraic closureof the stratum S if and only if the vector µ ( S ) is obtained from the vector µ ( S )by finitely-many replacings of two consecutive components by their sum. Remark 3.
For m ≥
2, consider the fibres of the projection π m ∗ : Π d → Π md , π m ∗ := π m +1 ◦ · · · ◦ π d . In particular, ˜ f d = f ⋄ d − . Suppose that such a fibre f ⋄ m is over a point A :=( a d − m , . . . , a d − ) ∈ Π md . When non-empty, the fibre f ⋄ m is either a point (when A ∈ ∂ Π md ) or a set homeomorphic to a ( d − m )-dimensional cell and its boundary(when A ∈ Π md \ ∂ Π md ). This follows from part (6) of Theorem 3. The bound-ary of the cell can be represented as consistsing of two 0-dimensional cells (theseare the graphs of the functions H m +1 ± | A ), of two 1-dimensional cells (the graphs of H m +2 ± | ( π m +1 ) − ( A ) ), of two 2-dimensional cells (the graphs of H m +3 ± | ( π m +1 ◦ π m +2 ) − ( A ) ), . . . , of two ( d − m − H d ± | (( π m +1 ◦ π m +2 ◦···◦ π d − ) − ( A ) ).In Chapter 2 of [17] one can find also results concerning the hyperbolicity domainwhich are exposed in the thesis [21] of I. M´eguerditchian.3. Notation and examples
Notation 4.
Given a d -tuple σ = ( β , . . . , β d − ), where β j = + or − , we denoteby R ( σ ) the subset of R d ∼ = Oa · · · a d − defined by the conditions sgn( a j ) = β j , j = 0, . . . , d −
1, and we set Π ∗ d,σ := Π ∗ d ∩ R ( σ ). For a set T ⊂ Oa · · · a d − , wedenote by T k its projection in the space Oa d − k · · · a d − . Example 1.
For k = 1 and for a j = 0, j = 0, . . . , d −
2, there exists a hyperbolicpolynomial of the form ( x + a d − ) x d − with any a d − ∈ R , so Π d = R . If one choosesany hyperbolic degree d polynomial Q ∗ d with distinct roots, the shift x x + g results in a d − a d − + dg , so there exist such polynomials Q ∗ d with any values of a d − . In addition, one can perturb the coefficients a , . . . , a d − to make them allnon-zero by keeping the roots real and distinct. Thus Π ∗ d = R ∗ = R \ { a d − = 0 } , NIVARIATE POLYNOMIALS AND THE CONTRACTIBILITY OF CERTAIN SETS 5
Figure 1.
The discriminant set of the family of polynomials x + x + bx + c and the sets Π ∗ ,σ ∩ { a = 1 } .Π ∗ d ∩ { a d − > } = { R ∗ + : a d − > } , Π ∗ d ∩ { a d − < } = { R ∗− : a d − < } . Example 2.
One can extend part (7) of Theorem 3 to k = 2 and say that theborder of the set Π d is the set H while H − is empty. The set H is the projectionin R ∼ = Oa d − a d − of the stratum of Π d consisting of polynomials having a d -fold real root: ( x + λ ) d . Its multiplicity vector equals ( d ). Hence a d − = dλ , a d − = d ( d − λ /
2, so H : a d − = ( d − a d − / d . One can also extend part (6)of Theorem 3 to k = 2 and observe thatΠ ∗ d = { a d − = 0 = a d − , a d − < ( d − a d − / d } , Π ∗ d ∩ { a d − > , a d − > } = { a d − > , < a d − < ( d − a d − / d } =: Π ∗ d ++ , Π ∗ d ∩ { a d − < , a d − > } = { a d − < , < a d − < ( d − a d − / d } =: Π ∗ d − + , Π ∗ d ∩ { a d − > , a d − < } = { a d − > , a d − < } =: Π ∗ d + − andΠ ∗ d ∩ { a d − < , a d − < } = { a d − < , a d − < } =: Π ∗ d −− . To obtain similar formulas for Π d instead of Π ∗ d one has to lift the condition a d − = 0 = a d − and to replace everywhere the inequality a d − < ( d − a d − / d by a d − ≤ ( d − a d − / d . Example 3.
For k = 3 (hence σ = ( β , β , β )), we set a := a , a := b , a := c , andwe consider the polynomial Q := x + ax + bx + c . Taking into account the group ofquasi-homogeneous dilatations which preserves the discriminant set (see part (2) of VLADIMIR PETROV KOSTOV
Remarks 1) one concludes that each set Π ∗ ,σ is diffeomorphic to the correspondingdirect product (Π ∗ ,σ ∩ { a = 1 } ) × (0 , ∞ ) if β = + or (Π ∗ ,σ ∩ { a = − } ) × ( −∞ ,
0) if β = − . Set σ ′ := ( − β , β , − β ). Using the same group of dilatations with u = − ∗ ,σ ′ ∩{ a = − } is diffeomorphic to the set Π ∗ ,σ ∩{ a = 1 } .Therefore in order to prove that all sets Π ∗ ,σ are contractible it suffices to showthis for the sets Π ∗ ,σ ∩ { a = 1 } with β = +. The latter sets are shown in Fig. 1.The figure represents the discriminant set of the polynomial Q • := x + x + bx + c ,i. e. the set Res( Q • , Q • ′ , x ) = 4 b − b − bc + 27 c + 4 c = 0 . This is a curve in R := Obc having a cusp at ( b, c ) = (1 / , /
27) which correspondsto the polynomial ( x + 1 / . The four sets Π ∗ ,σ ∩ { a = 1 } are the intersectionsof the interior of the curve with the open coordinate quadrants. The intersectionswith { b > , c > } and { b > , c < } are bounded curvilinear triangles.4. Proof of Theorem 1
Part (1). Each set E d ( σ ) is non-empty. Indeed, given a polynomial Q d with a > C > Q d + C is elliptic.If the polynomials Q d, and Q d, belong to the set E d ( σ ), then for t ∈ [0 , Q ♯d := tQ d, + (1 − t ) Q d, also belongs to it. Indeed, the signs of therespective coefficients are the same and if Q d, ( x ) > Q d, ( x ) >
0, then Q ♯d ( x ) >
0. Thus the set E d ( σ ) is convex hence contractible.Part (2). Each set F d ( σ ) is non-empty. Indeed, for C > Q d + sgn( a ) C has a single real root which is simple and the sign of thisroot is opposite to the sign of Q d (0). For a given polynomial Q d ∈ F d ( σ ), denotethis root by ξ . Then for t ∈ (0 , Q ♭d := tQ d + (1 − t )( x d − ξ d ) ∈ F d ( σ ) . Indeed,1) the signs of the respective coefficients of Q d and Q ♭d are the same and2) both polynomials Q d and x d − ξ d are positive for x > ξ and negative for x < ξ ,and one has sgn( a ) = − sgn( ξ d ) = − sgn( ξ ).On the other hand, consider the polynomial x d − ξ d , ξ = 0, and the sum T := x d − ξ d + d − X j =0 a j x j , where d − X j =0 a j ∈ (0 , ε ) , ε > , and the signs of the quantities a j are defined by the d -tuple σ . For each ε > T of such polynomials T is open and contractible. For ε > T ⊂ F d ( σ ).For each closed subset Q of the set F d ( σ ), there exists ε ∗ > Q d ∈ Q and for t ∈ (0 , ε ∗ ], one has Q ♭d ∈ T ⊂ F d ( σ ).Hence the set Q can be contracted into a subset of the contractible set T . Hence F d ( σ ) is contractible.Part (3). We consider two cases: Case A). The two real roots of Q d have the same sign (i. e. a > ). NIVARIATE POLYNOMIALS AND THE CONTRACTIBILITY OF CERTAIN SETS 7
We assume that they are positive (otherwise one can consider the polynomial Q d ( − x ) with the d -tuple resulting from σ via x
7→ − x ). Denote them by 0 < ξ < η .Suppose first that there exists a coefficient of Q d of a monomial of odd degree m which is negative. There exists a unique polynomial of the form R := x d − Ax m + B , A > B >
0, such that R ( ξ ) = R ( η ) = 0. Indeed, the conditions ξ d − Aξ m + B = η d − Aη m + B = 0imply A = ( ξ d − η d ) / ( ξ m − η m ) > B = ξ m η m ( ξ d − m − η d − m ) / ( ξ m − η m ) > R thus defined has no other real rootsthan ξ and η . Hence for t ∈ (0 , tQ d + (1 − t ) R belongs to theset G d ( σ ). Indeed, both polynomials Q d and R are positive on ( −∞ , ξ ) ∪ ( η, ∞ )and negative on ( ξ, η ); the signs of the non-zero coefficients of R are the same asthe signs of the respective coefficients of Q d . The contractibility of the set G d ( σ ) isproved then as the one of the set F d ( σ ). The set G d ( σ ) is not empty, because onecan perturb the polynomial R to make all its coefficients non-zero and with signsdefined by σ ; the perturbed polynomial will have exactly two simple positive rootsand ( d − / Q d can have two real positive roots only if there is at least one coefficientof even degree which is negative. However in this case the set G d ( σ ) is empty, seeProposition 4 in [7]. Case B). The two real roots of Q d have opposite signs (hence a < ). Denote these roots by − η < < ξ . One can suppose that η = ξ , otherwise onecan change the roots slightly without changing the signs of the coefficients of Q d and without introducing other real roots. We suppose that ξ > η , otherwise onecan consider Q d ( − x ) instead of Q d ( x ).Suppose that there is a negative coefficient of Q d of odd degree m . We constructa polynomial S := x d − Ax m − B , A > B >
0, such that S ( ξ ) = S ( − η ) = 0. Thelatter two equalities imply A = ( ξ d − η d ) / ( ξ m + η m ) > B = ξ m η m ( ξ d − m + η d − m ) / ( ξ m + η m ) > . By Descartes’ rule of signs the polynomial S has no real roots other than ξ and − η .Hence for t ∈ (0 , tQ d + (1 − t ) S belongs to the set G d ( σ ) and thecontractibility of the set G d ( σ ) is proved as the one of the set F d ( σ ).Suppose that all odd coefficients of Q d are positive. Then for each a >
0, onehas Q d ( a ) > Q d ( − a ) and hence ξ < η (because Q d (0) < Q d ( x ) to Q d ( − x ) to have ξ > η and all odd coefficients of Q d are now negative. Soone can again construct the polynomial S and prove the contractibility of the set G d ( σ ). 5. Proof of Theorem 2
Denote by M a component of the set Π ∗ d defined after a d -tuple σ and by M k its projection in the space Oa d − k · · · a d − . It is shown in [19] (see Proposition 1therein) that M is non-empty. In this section we prove by induction on d thefollowing statement: VLADIMIR PETROV KOSTOV
Proposition 1.
For k ≥ , the set M k is the set of points between the graphs L k ± of two continuous functions defined on M k − . Remark 4.
The functions L k ± can be extended to continuous functions defined on M k − , whose values might coincide (but this does not necessarily happen) only on ∂M k − , see Remark 5.Contractibility of M follows directly from the proposition. Indeed, one cansuccessively contract M into its projections M d − , M d − , . . . , M . The latter isone of the sets Π ∗ d ±± defined in Example 2 which are contractible. Proof of Proposition 1.
Throughout the proof we denote by H k ± not only the graphsmentioned in Theorem 3, but also the corresponding functions.A) We prove Proposition 1 by induction on d . The induction base are the cases d = 2 and d = 3. For d = 2, the proof of the proposition results directly from thecontractibility of the sets Π ∗ d ±± , see Example 2. For d = 3, the proposition followsfrom Example 3.Suppose that Proposition 1 holds true for d = d ≥
3. Set d := d + 1. Theintersection Π d ∩ { a = 0 } can be identified with Π d − (via a shift by 1 of theindices of the coefficients a j ). In order not to change these indices, we assume thatthe polynomials of Π d − are of the form Q d − := x d − + a d − x d − + · · · + a x + a .B) Starting with a component U of the set Π ∗ d − (hence U = U d − ), we constructin several steps the components U + and U − of the set Π ∗ d sharing with U the signsof the coefficients a d − , . . . , a . One has a > U + and a < U − .At the first step we construct the sets U ± , as follows. We remind that theprojections π k and their fibres ˜ f k were defined in part (1) of Theorem 3. Eachfibre ˜ f d of the projection π d which is over a point of U is a segment, see part(1) of Theorem 3. If Q d − ∈ U , then for ε > xQ d − ± ε are hyperbolic. Indeed, all roots of Q d − are real and simple. The set U + , (resp. U − , ) is the union of the interior points of these fibres ˜ f d which arewith positive (resp. with negative) a -coordinates. Thus the set U + , (resp. U − , )consists of the points of the fibres ˜ f d (that are over U ) which are between the graphsof the restrictions to U of the functions H d + and 0 (resp. 0 and H d − , see part (6) ofTheorem 3). Hence the sets U ± , are open, non-empty and contractible.C) Recall that the set U consists of all the points between the graphs L d − ± oftwo continuous functions defined on U d − . Depending on the sign of a in U , partof one of these graphs (or the whole of it) could belong to the hyperplane a = 0.Consider a fibre ˜ f d over a point of one of the graphs L d − ± and not belongingto the hyperplane a = 0. A priori the two endpoints of the fibre cannot have a -coordinates with opposite signs, because then the intersection of the fibre with thecoordinate hyperplane a = 0 would be an interior point for Π d hence not a point of ∂U , see part (2) of Theorem 3. Both endpoints cannot have non-zero coordinates ofthe same sign, because again this would not be a point of ∂U . Hence the followingthree possibilities remain:a) both endpoints have zero a -coordinates;b) one endpoint has a zero and the other endpoint has a positive a -coordinate; NIVARIATE POLYNOMIALS AND THE CONTRACTIBILITY OF CERTAIN SETS 9 c) one endpoint has a zero and the other endpoint has a negative a -coordinate.D) Consider the points of the graph L d − which do not belong to the hyperplane a = 0 (for L d − − the reasoning is similar). If for B ∈ ( L d − \ { a = 0 } ), possibilitya) takes place, then there is nothing to do. Suppose that possibility b) takes place.Denote by a j,B the coordinates of the point B (hence a ,B = 0). For each suchpoint B , fix the coordinates a j = a j,B for j = 1 and vary a . Then for some a = a ,C > a ,B , one has either a ,C = 0 or the point C belongs to the graph H d − . In both these situations we add to the set U , + the points of the interior ofall fibres ˜ f d over the interval [ a ,B , a ,C ) (with a j = a j,B for j = 1), over all points B ∈ L d − \ { a = 0 } .If possibility c) takes place, then we fix again a j,B for j = 1 and vary a . We addto the set U , − the points of the interior of all fibres ˜ f d over the interval [ a ,B , a ,C )(with a j = a j,B for j = 1), over all points B ∈ L d − \ { a = 0 } .E) We perform a similar reasoning and construction with L d − − (in which therole of H d − is played by H d − − ). In this case one has a ,C < a ,B and the interval[ a ,B , a ,C ) is to be replaced by the interval ( a ,C , a ,B ].F) Thus we have enlarged the sets U , ± ; the new sets are denoted by U , ± . Thesets U , ± and U , ± satisfy the conclusion of Proposition 1. We denote the graphs L k ± defined for the sets U , ± and U , ± by L k , ± and L k , ± . The construction of thesegraphs implies that they are graphs of continuous functions (because such are thegraphs H k ± ). The set U , + ∪ U , − (resp. U , + ∪ U , − ) contains all points of the set( π d ) − ( U ) ∩ Π ∗ d,σ (resp. ( π d − ◦ π d ) − ( U d − ) ∩ Π ∗ d,σ ).G) We remind that ˜ f d = f ⋄ d − , see Remark 3. Suppose that the sets U s, ± ,2 ≤ s ≤ d −
3, are constructed such that they satisfy the conclusion of Proposition 1(the graphs L k ± are denoted by L ks, ± ) and that the set U s, + ∪ U s, − contains all pointsof the set ( π d − s +1 ◦ · · · ◦ π d ) − ( U d − s ) ∩ Π ∗ d,σ . Consider a point D ∈ L d − s + whichdoes not belong to the hyperplane a s = 0. For the fibre f ⋄ d − s of the projection π d − s +1 ◦ · · · ◦ π d which is over D (see Remark 3) one of the three possibilities takesplace:a’) the minimal and the maximal possible value of the a s -coordinate of the pointsof the fibre are zero;b’) the minimal possible value is 0 and the maximal possible value is positive;c’) the minimal possible value is negative and the maximal possible value is 0.It is not possible to have both the maximal and minimal possible value of a s non-zero, because in this case the point D does not belong to the set ∂U d − s . Thisis proved by analogy with C). With regard to Remark 3, when the fibre f ⋄ d − s is nota point, then the maximal (resp. the minimal) value of a s is attained at one of the0-dimensional cells (resp. at the other 0-dimensional cell) and only there. This canbe deduced from part (2) of Theorem 3.H) When possibility a’) takes place, then there is nothing to do. Suppose thatpossibility b’) takes place. Denote by a j,D the coordinates of the point D (hence a ,D = · · · = a s − ,D = 0). Fix a j,D for j = s and vary a s . Then for some a s = a s,E > a s,D , one has either a s,E = 0 or the point E belongs to the graph H d − s + . In this case we add to the set U s, + the points of the interior of all fibres f ⋄ d − s over theinterval [ a s,D , a s,E ) (with a j = a j,D for j = s ), over all points D ∈ L d − s + \ { a s = 0 } .If possibility c’) takes place, then we fix again a j,D for j = s and vary a s . Weadd to the set U s, − the points of the interior of all fibres f ⋄ d − s over the interval[ a s,D , a s,E ) (with a j = a j,D for j = s ), over all points D ∈ L d − s + \ { a s = 0 } .We consider in a similar way the graph L d − s − in which case the role of H d − s + isplayed by H d − s − , one has a s,E < a s,D and the interval [ a s,D , a s,E ) is to be replacedby the interval ( a s,E , a s,D ].I) We have thus constructed the sets U s +1 , ± which satisfy the conclusion ofProposition 1; the set U s +1 , + ∪ U s +1 , − contains all points of the set ( π d − s ◦ · · · ◦ π d ) − ( U d − s − ) ∩ Π ∗ d,σ . It should be noticed that as the fibres f ⋄ d − s contain cells ofdimension from 0 to s , all graphs L ks, ± would have to be changed when passing from L ks, ± to L ks +1 , ± . The new graphs are graphs of continuous functions; this followsfrom the construction and from the fact that such are the graphs H k ± .J) One can construct the sets U d − , ± in a similar way. The only difference is thefact that there is a graph H , but not a graph H − , see Example 2. We set U ± := U d − , ± . The set U + ∪ U − contains all points from the set ( π ◦· · ·◦ π d ) − ( U ) ∩ Π ∗ d,σ .The sets U ± satisfy the conclusion of Proposition 1. Hence they are contractible. (cid:3) Remark 5.
The functions L k ± encountered throughout the proof of the propositioncan be extended by continuity on the closures of the sets on which they are defined,because this is the case of the functions H k ± . Moreover, fibres ˜ f k which are pointsappear only in case they are over points of the graphs H k − ± . Hence this describesthe only possibility for the values of the functions with graphs L k ± to coincide.6. Comments and open problems
One could try to generalize Theorem 2 by considering instead of the set Π ∗ d theset R ,d , i. e. by dropping the requirement the polynomial Q d to be hyperbolic. Soan open problem can be formulated like this: Open problem 1.
For a given degree d , consider the triples ( σ, pos, neg ) com-patible with Descartes’ rule of signs. Is it true that for each such triple, the corre-sponding subset of the set R ,d is either contractible or empty? The difference between this open problem and Theorem 2 is the necessity tocheck whether the subset is empty or not (see part (3) of Theorem 1). For instance,if d = 4, then for neither of the triples((+ , − , + , +) , ,
0) and (( − , − , − , +) , , x + a x + a x + a x + a with signs of the coefficients a j as defined by σ and with 2positive and 0 negative or with 0 positive and 2 negative roots respectively, see [12](all roots are assumed to be simple).The question of realizability of triples ( σ, pos, neg ) has been asked in [2]. Theexhaustive answer to this question is known for d ≤
8. For d = 4, it is due toD. Grabiner ([12]), for d = 5 and 6, to A. Albouy and Y. Fu ([1]), for d = 7 andpartially for d = 8, to J. Forsg˚ard, V. P. Kostov and B. Shapiro ([7] and [8]) and NIVARIATE POLYNOMIALS AND THE CONTRACTIBILITY OF CERTAIN SETS 11 for d = 8 the result was completed in [15]. Other results in this direction can befound in [4], [6] and [16]. Remarks 4. (1) It is not easy to imagine how one could prove that all componentsof R ,d are either contractible or empty without giving an exhaustive answer to thequestion which triples ( σ, pos, neg ) are realizable and which are not. Unfortunately,at present, giving such an answer for any degree d is out of reach.(2) If one can prove not contractibility of the non-empty components, but onlythat they are (simply) connected, would also be of interest.For a degree d univariate real monic polynomial Q d without vanishing coef-ficients, one can define the couples ( pos ℓ , neg ℓ ) of the numbers of positive andnegative roots of Q ( ℓ ) d , ℓ = 0, 1, . . . , d −
1. One can observe that the d couples( pos ℓ , neg ℓ ) define the signs of the coefficients of Q d and that their choice must becompatible not only with Descartes’ rule of signs, but also with Rolle’s theorem.We call such d -tuples of couples compatible for short. We assume that for ℓ = 0, 1, . . . , d −
1, all real roots of Q ( ℓ ) d are simple and non-zero.To have a geometric idea of the situation we define the discriminant sets ˜∆ j , j = 1, . . . , d as the sets ∆ j defined in the spaces Oa d − j . . . a d − for the polynomials Q ( d − j ) d . In particular, ˜∆ d = ∆ d . For j = 1, . . . , d −
1, we set ∆ j := ˜∆ j × Oa . . . a d − j − . We define the set R ,d as R ,d := R d \ (cid:0) ( ∪ dj =1 ∆ j ) ∪ ( ∪ d − j =0 { a j = 0 } ) (cid:1) . For d ≤
5, the question when a subset of R ,d defined by a given compatible d -tupleof couples ( pos ℓ , neg ℓ ) is empty is considered in [5]. Open problem 2.
Given the d compatible couples ( pos ℓ , neg ℓ ) , is it true thatthe subset of R ,d defined by them is either connected (eventually contractible) orempty? In other words, is it true that each d -tuple of such couples defines eitherexactly one or none of the components of the set R ,d ? Some problems connected with comparing the moduli of the positive and negativeroots of hyperbolic polynomials are treated in [18], [20] and [19]. Other problemsconcerning hyperbolic polynomials are to be found in [17]. A tropical analog ofDescartes’ rule of signs is discussed in [9].
Acknowledgement.
B. Z. Shapiro from the University of Stockholm attractedthe author’s attention to Open Problem 1 and suggested the proof of part (1) ofTheorem 1.
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Universit´e Cˆote d’Azur, CNRS, LJAD, France
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