Open access peer-reviewed chapter

New Aspects of Descartes’ Rule of Signs

By Vladimir Petrov Kostov and Boris Shapiro

Submitted: July 30th 2018Reviewed: October 15th 2018Published: November 27th 2018

DOI: 10.5772/intechopen.82040

Downloaded: 261

Abstract

Below, we summarize some new developments in the area of distribution of roots and signs of real univariate polynomials pioneered by R. Descartes in the middle of the seventeenth century.

Keywords

  • real univariate polynomial
  • sign pattern
  • admissible pair
  • Descartes’ rule of signs
  • Rolle’s theorem

2010 Mathematics Subject Classication: Primary 26C10; Secondary 30C15

1. Introduction

The classical Descartes’ rule of signs claims that the number of positive roots of a real univariate polynomial is bounded by the number of sign changes in the sequence of its coefficients and it coincides with the latter number modulo 2. It was published in French (instead of the usual at that time Latin) as a small portion of Sur la construction de problèmes solides ou plus que solide which is the third book of Descartes’ fundamental treatise La Géométrie which, in its turn, is an appendix to his famous Discours de la méthode. It is in the latter chef d’oeuvre that Descartes developed his analytic approach to geometric problems leaving practically all proofs and details to an interested reader. This interested reader turned out to be Frans van Schooten, a professor of mathematics at Leiden who together with his students undertook a tedious work of making Descartes’ writings understandable, translating and publishing them in the proper language, that is, Latin. (For the electronic version of this book, see [13].) Mathematical achievements of Descartes form a small fraction of his overall scientific and philosophical legacy, and Descartes’ rule of signs is a small but important fraction of his mathematical heritage.

Descartes’ rule of signs has been studied and generalized by many authors over the years; one of the earliest can be found in [7], see also [4, 11]. (For some recent contributions, see [1, 2, 6, 10, 12, 14], to mention a few.)

In the present survey, we summarize a relatively new development in this area which, to the best of our knowledge, was initiated only in the 1990s (see [12]).

For simplicity, we consider below only real univariate polynomials with all nonvanishing coefficients. For a polynomial Pj=0dajxjwith fixed signs of its coefficients, Descartes’ rule of signs tells us what possible values the number of its real positive roots can have. For Pas above, we define the sequence of ±signs of length d+1which we call the sign pattern (SP for short) of P, namely, we say that a polynomial Pwith all nonvanishing coefficients defines the sign pattern σ(sd, sd1, , s0)if sj=sgnaj. Since the roots of the polynomials Pand Pare the same, we can, without loss of generality, assume that the first sign of a SP is always a +.

It is true that for a given SP with csign changes (and hence with p=dcsign preservations), there always exist polynomials Pdefining this sign pattern and having exactly pospositive roots, where pos=0,2,,cif cis even and pos=1,3,,cif cis odd (see, e.g., [1, 3]). (Observe that we do not impose any restriction on the number of negative roots of these polynomials.)

One can apply Descartes’ rule of signs to the polynomial 1dPxwhich has psign changes and csign preservations in the sequence of its coefficients and whose leading coefficient is positive. The roots of 1dPxare obtained from the roots of Pxby changing their sign. Applying the above result of [1] to 1dPx, one obtains the existence of polynomials Pwith exactly negnegative roots, where neg=0,2,,pif pis even and neg=1,3,,pif pis odd. (Here again we impose no requirement on the number of positive roots.)

A natural question apparently for the first time raised in [12] is whether one can freely combine these two results about the numbers of positive and negative roots. Namely, given a SP σwith csign changes and p=dcsign preservations, we define its admissible pair (AP for short) as posneg, where posc, negp, and the differences cposand pnegare even. For the SP σas above, we call cpthe Descartespair of σ. The main question under consideration in this paper is as follows.

Problem 1. Given a couple (SP, AP), does there exist a polynomial of degree dwith this SP and having exactly pospositive and exactly negnegative roots (and hence exactly dposneg/2complex conjugate pairs)?

If such a polynomial exists, then we say that it realizes a given couple (SP, AP). The present paper discusses the current status of knowledge in this realization problem.

Example 1. For d=4and for the sign pattern σ0++, the following pairs and only them are admissible: 22, 20, 02, and 00. The first of them is the Descartes’ pair of σ0.

It is clear that if a couple (SP, AP) is realizable, then it can be realized by a polynomial with all simple roots, because the property of having nonvanishing coefficients is preserved under small perturbations of the roots.

In this short survey, we present what is currently known about Problem 1. After the pioneering observations of Grabiner [12] which started this line of research, important contributions to Problem 1 have been made by Albouy and Fu [1] who, in particular, described all non-realizable combinations of the numbers of positive and negative roots and respective sign patterns up to degree 6. Our results on this topic which we summarize below can be found in [5, 8, 9] and [15, 16, 17, 18, 19]. On the other hand, we find it surprising that such a natural classical question has not deserved any attention in the past, and we hope that this survey will help to change the situation. The current status of Problem 1 is not very satisfactory in spite of the complete results in degrees up to 8as well as several series of non-realizable cases in all degrees. There is still no general conjecture describing all non-realizable cases. It might happen that the answer to Problem 1 in sufficiently high degrees is very complicated.

On the other hand, besides Problem 1 as it is stated, there is a significant number of related basic questions which can be posed in connection to the latter Problem and are still waiting for their researchers. (Very few of them are listed in Section 5.)

One should also add that there is a number of completely different directions in which mathematicians are trying to extend Descartes’ rule of signs. They include, for example, rule of signs for other univariate analytic functions including exponential functions, trigonometric functions and orthogonal polynomials, multivariate Descartes’ rule of signs, tropical rule of signs, rule of signs in the complex domain, etc. (see, e.g., [6, 10, 14]) and references therein. But we think that Problem 1 is the closest one to the original investigations by Descartes himself.

The structure of this chapter is as follows. In Section 2, we provide the information about the solution of Problem 1 in degrees up to 11. In Section 3, we present several infinite series of non-realizable couples (SP, AP). Finally, in Section 4 we discuss two generalizations of Problem 1 and their partial solutions.

2. Solution of the realization problem 1 in small degrees

2.1 Natural Z2×Z2-action and degrees d=1, 2, and 3

Let us start with the following useful observation.

To shorten the list of cases (SP, AP) under consideration, we can use the following Z2×Z2-action whose first generator acts by.

Px1dPx,E1

and the second one acts by

PxPRxxdP1/x/P0.E2

Obviously, the first generator exchanges the components of the AP. Concerning the second generator, to obtain the SP defined by the polynomial PR, one has to read the SP defined by Pxbackward. The roots of PRare the reciprocals of these of Pwhich implies that both polynomials have the same numbers of positive and negative roots. Therefore, the SPs which they define have the same AP.

Remark 1. A priori the length of an orbit of any Z2×Z2-action could be 1, 2, or 4, but for the above action, orbits of length 1do not exist since the second components of the SPs defined by the polynomials Pxand 1dPxare always different. When an orbit of length 2occurs and dis even, then both SPs are symmetric w.r.t. their middle points (hence their last component equal +). Similarly, when dis odd, then one of the two SPs is symmetric w.r.t. its middle (with the last component equal to +), and the other one is antisymmetric. Thus, its last components equal .

It is obvious that all pairs or quadruples (SP, AP) constituting a given orbit are simultaneously (non-)realizable.

As a warm-up exercise, let us consider degrees d=1,2and 3. In these cases, the answer to Problem 1 is positive. We give the list of SPs, with the respective values cand pof their APs and examples of polynomials realizing the couples (SP, AP). In order to shorten the list, we consider only SPs beginning with two +signs; the cases when these signs are +are realized by the respective polynomials 1dPx. All quadratic factors in the table below have no real roots.

dSPcpAPP1++0101x+12+++0202x2+3x+2=x+1x+200x2+x+1++1111x2+x2=x1x+23++++0303x3+6x2+11x+6=x+1x+2x+301x3+3x2+4x+2=x+1x2+2x+2+++1212x3+2x2+x6=x1x+2x+310x3+5x2+4x10=x1x2+6x+10+++2121x3+x224x+36=x+6x2x301x3+2x219x+30=x+6x24x+5++1212x3+x24x4=x2x+1x+210x3+2x23x10=x2x2+4x+5

Example 2. For d=4, an example of an orbit of length 2is given by the couples

++22and++++22.

Here, both SPs are symmetric w.r.t. its middle.

For d=5, such an example is given by the couples

++23and+++32.

The first of the SPs is symmetric, and the second one is antisymmetric w.r.t. their middles.

Finally, for d=3, the following four couples (SP, AP)

+++12;+++21;+12;+++21.

constitute one orbit for d=3. In this example all admissible pairs are Descartes’ pairs.

2.2 Degrees d4

It turns out that for d4, it is no longer true that all couples (SP, AP) are realizable by polynomials of degree d. Namely, the following result can be found in [12]:

Theorem 1. The only couples (SP, AP) which are non-realizable by univariate polynomials of degree 4are

++02and++++20.

It is clear that these two cases constitute one orbit of the Z2×Z2-action of length 2(the SPs are the same when read the usual way and backward).

Proof. The argument showing non-realizability in Theorem 1 is easy. Namely, if a polynomial

Px4+a3x3+a2x2+a1x+a0

realizes the second of these couples and has two positive roots α<βand no negative roots, then for any uαβ, the values of the monomials x4, a2x2, and a0are the same at uand u, while the monomials a3x3and a1xare positive at uand negative at u. Hence, Pu<Pu<0. As P0>0and limxPx=+, the polynomial Phas two negative roots as well—a contradiction.

For d=4, realizability of all other couples (SP, AP) can be proven by producing explicit examples.

Remark 2. In [19] a geometric illustration of the non-realizability of the two cases mentioned in Theorem 1 is proposed. Namely, one considers the family of polynomials Qx4+x3+ax2+bx+cand the discriminant set

ΔabcR3ResQQ=0,

where Res QQis the resultant of the polynomials Qand Q. The hypersurface Δ=0partitions R3into three open domains, in which the polynomial Qhas 0, 1, or 2complex conjugate pairs of roots, respectively. These domains intersect the 8open orthants of R3defined by the coordinate system abc, and in each of these intersections, the polynomial Qhas one and the same number of positive, negative, and complex roots, as well as the same signs of its coefficients. The non-realizability of the couple ++++20can be interpreted as the fact that the corresponding intersection is empty. Pictures of discriminant sets allow to construct easily the numerical examples mentioned in the proof of Theorem 1.

It remains to be noticed that for α>0and β>0, the polynomials Pxand βPαxhave one and the same numbers of positive, negative, and complex roots. Therefore, it suffices to consider the family of polynomials Qin order to cover all SPs beginning with ++. The ones beginning with +will be covered by the family Qx.

For degrees d=5and 6, the following result can be found in [1].

Theorem 2. (1) The only two couples (SP, AP) which are non-realizable by univariate polynomials of degree 5are:

++03and+++30.

(2) For degree d=6, up to the above Z2×Z2-action, the only non-realizable couples (SP, AP) are:

++02;++04;+++02;+++04.

The two cases of Part (1) of Theorem 2 also form an orbit of the Z2×Z2-action of length 2. Each of the first two cases of Part (2) defines an orbit of length 2, while each of the last two cases defines an orbit of length 4.

For d=7, the following theorem is contained in [8].

Theorem 3. For univariate polynomials of degree 7, among their 1472 possible couples (SP, AP) (up to the Z2×Z2-action), exactly the following 6are non-realizable:

+++05;++++05;+++03;++++05;++03;++05.

The lengths of the respective orbits in these 6cases are 4, 2, 4, 4, 2, and 2.

The case d=8has been partially solved in [8] and completely in [16]:

Theorem 4. For degree d=8, among the 3648 possible couples (SP, AP) (up to the Z2×Z2-action), exactly the following 19are non-realizable:

++++06;+++06;++++06;+++++06;++++02;++++02;+++02;+++04;(+++,02;+++04;++02;(++,04;++06;+++++06;+++04;+++04;++++04;++++04;++++04.

The lengths of the respective orbits are 2, 4, 4, 4, 2, 4, 4, 4, 2, 2, 2, 2, 2, 4, 4, 4, 4, 4, and 4.

Remark 3. As we see above, for d=4, 5, 6, 7, and 8, up to the Z2×Z2-action, the numbers of non-realizable cases are 1, 1, 4, 6, and 19, respectively. The fact that these numbers increase more when d=5and d=7than when d=4and d=6could be related to the fact that the maximal possible number of complex conjugate pairs of roots of a real univariate degree dpolynomial is d/2. This number increases w.r.t. d1/2when dis even and does not increase when dis odd.

Observe that for d8, all examples of couples (SP, AP) which are non-realizable are with APs of the form ν0or 0νand νN. Initially, we thought that this is always the case. However, recently it was proven that, for higher degrees, this fact is no longer true (see [17]):

Theorem 5. For d=11, the following couple (SP, AP)

++++++18

is non-realizable. The Descartespair in this case equals 38.

There is a strong evidence that for d=9, the similar couple (SP, AP)

+++++16

is also non-realizable. (Its Descartes’ pair equals 36.) If this were true, then 9would be the smallest degree with an example of a non-realizable couple (SP, AP) for which both components of the AP are nonzero. When studying the cases d=8and d=11(see [16] and [17]), discriminant sets have been considered (see Remark 2).

Summarizing the above, we have to admit that the information in low degrees available at the moment does not allow us to formulate a consistent conjecture describing all non-realizable couples in an arbitrary degree which we could consider as sufficiently well motivated.

3. Series of examples of (non-)realizable couples (SP, AP)

In this section we present a series of couples (non-)realizable for infinitely many degrees. We decided to include those proofs of the statements formulated below which are short and instructive.

3.1 Some examples of realizability and a concatenation lemma

Our first examples of realizability deal with polynomials with the minimal possible number of real roots:

Proposition 1. For deven, any SP whose last component is a +(resp. is a ) is realizable with the AP 00(resp. 11). For dodd, any SP whose last component is a +(resp. is a ) is realizable with the AP 01(resp. 10).

Proof. Indeed, for any given SP, it suffices to choose any polynomial defining this SP and to increase (resp. decrease) its constant term sufficiently much if the latter is positive (resp. negative). The resulting polynomial will have the required number of real roots.□

Our next example deals with hyperbolic polynomials, that is, real polynomials with all real roots. Several topics concerning hyperbolic polynomials are developed in [18].

Proposition 2. Any SP is realizable with its Descartespair.

Proposition 2 will follow from the following concatenation lemma whose proof can be found in [8].

Lemma 1. Suppose that monic polynomials P1and P2, of degrees d1and d2resp., realize the SPs +σ̂1and +σ̂2, where σ̂jare the SPs defined by Pjin which the first +is deleted. Then:

  1. If the last position of σ̂1is a +, then for any ε>0small enough, the polynomial εd2P1xP2x/εrealizes the SP +σ̂1σ̂2and the AP pos1+pos2neg1+neg2.

  2. If the last position of σ̂1is a , then for any ε>0small enough, the polynomial εd2P1xP2x/εrealizes the SP +σ̂1σ̂2and the AP pos1+pos2neg1+neg2.(Here σ̂2is the SP obtained from σ̂2by changing each +by a and vice versa.)

The concatenation lemma allows to deduce the realizability of couples (SP, AP) with higher values of dfrom that of couples with smaller din which cases explicit constructions are usually easier to obtain. On the other hand, non-realizability of special cases cannot be concluded using this lemma.

Example 3. Denote by τthe last entry of the SP σ̂1. We consider the cases

P2x=x1,x+1,x2+2x+2,x22x+2withpos2neg2=10,01,00,00resp.

When τ=+,then one has, respectively,

σ̂2=,+,++,+,

and the SP of εd2P1xP2x/εequals

+σ̂1,+σ̂1+,+σ̂1++,+σ̂1+.

When τ=,then one has, respectively,

σ̂2=+,,,+,

and the SP of εd2P1xP2x/εequals

+σ̂1+,+σ̂1,+σ̂1,+σ̂1+.

Proof of Proposition 2. We will use induction on the degree dof the polynomial. For d=1, the SP +(resp. ++) is realizable with the AP 10(resp. 01) by the polynomial x1(resp. x+1).

For d=2, we apply Lemma 1. Set P1x+1and P2x1. Then, for ε>0small enough, the polynomials

εP1xP2x/ε=x+1xε=x2+1εxεandεP2xP1x/ε=x1x+ε=x2+1+εxε

define the SPs ++and +, respectively, and realize them with the AP 11. In the same way, one can concatenate P1(resp. P2) with itself to realize the SP +++with the AP 02(resp. the SP ++with the AP 20). These are all possible cases of monic hyperbolic degree 2polynomials with nonvanishing coefficients.

For d2, in order to realize a SP σwith its Descartes’ pair cp, we represent σin the form σuv, where uand vare the last two components of σand σis the SP obtained from σby deleting uand v. Then, we choose P1to be a monic polynomial realizing the SP σu:

  1. With the AP c1p, and we set P2x1, if u=v.

  2. With the AP cp1, and we set P2x+1, if u=v. □

Our next result discusses (non-)realizability for polynomials with only two sign changes (see [8, 9]).

Proposition 3. Consider a sign pattern σ¯with 2sign changes, consisting of mconsecutive pluses followed by nconsecutive minuses and then by qconsecutive pluses, where m+n+q=d+1.Then:

  1. For the pair 0d2,this sign pattern is not realizable if

    κdm1mdq1q4;E3

  • The sign pattern σ¯is realizable with any pair of the form 2v, except in the case when dand mare even, n=1(hence qis even), and v=0.

  • Certain results about realizability are formulated in terms of the ratios between the quantities pos, neg, and d. The following proposition is proven in [8].

    Proposition 4. For a given couple (SP, AP), if minposneg>d4/3, then this couple is realizable.

    3.2 The even and the odd series

    Suppose that the degree dis even. Then, the following result holds (see Proposition 4 in [8]):

    Proposition 5. Consider the SPs satisfying the following three conditions:

    1. Their last entry (i.e., the sign of the constant term) is a +.

    2. The signs of all odd monomials are +.

    3. Among the remaining signs of even monomials, there are exactly 1signs (at arbitrary positions).

    Then, for any such SP, the APs 20,40,,20, and only they, are non-realizable.

    Suppose now that the degree d5is odd. For 1kd3/2, denote by σkthe SP beginning with two pluses followed by kpairs +and then by d2k1minuses. Its Descartes’ pair of σkequals 2k+1d2k1. The following proposition is proven in [19].

    Theorem 6. (1) The SP σkis not realizable with any of the pairs 30,50,,2k+10; (2) The SP σkis realizable with the pair 10; (3) The SP σkis realizable with any of the APs 2+12r, =0,1,,k, and r=1,2,,d2k1/2.

    One can observe that Cases (1), (2), and (3) exhaust all possible APs posneg.

    4. Similar realization problems

    In this section, we consider realization problems similar or motivated by Problem 1. A priori it is hard to tell which of these or similar problems might have a reasonable answer.

    4.1 D-Sequences

    Consider a real polynomial Pof degree dand its derivative. By Rolle’s theorem, if Phas exactly rreal roots (counted with multiplicity), then the derivative Phas r1+2real roots (counted with multiplicity), where N0. It is possible that Phas more real roots than P. For example, for d=2and P=x2+1, one gets P'=2xwhich has a real root at 0, while Phas no real roots at all. For d=3, the polynomial P=x3+3x28x+10=x+5(x12+1)has one negative root and one complex conjugate pair, while its derivative P=3x2+6x8has one positive and one negative root.

    Now, for j=0, , and d1, denote by rjand cjthe numbers of real roots and complex conjugate pairs of roots of the polynomial Pj(both counted with multiplicity). These numbers satisfy the conditions

    rjrj+1+1,rj+2cj=dj.E4

    Definition 1. A sequence (r02c0,r12c1,,rd12cd1)satisfying conditions (4) will be called a D-sequence of length d. We say that a given D-sequence of length dis realizable if there exists a real polynomial Pof degree dwith this D-sequence, where for j=0,,d1, all roots of Pjare distinct.

    Example 4. One has rd1=1and cd1=0. Clearly, one has either rd2=2, cd2=0or rd2=0, cd2=1. For small values of d, one has the following D-sequences and respective polynomials realizing them:

    d=110xd=22010x210210x2+1d=3302010x3x120210x3+x122010x3+10x2+26x.

    The following question where a positive answer to which can be found in [15] seems very natural.

    Problem 2. Is it true that for any dN, any D-sequence is realizable?

    4.2 Sequences of admissible pairs

    Now, we are going to formulate a problem which is a refinement of both Problems 1 and 2.

    Recall that for a real polynomial Pof degree d, the signs of its coefficients ajdefine the sign patterns σ0,σ1,,σd1corresponding to Pand to all its derivatives of order d1since the SP σjis obtained from σj1by deleting the last component. We denote by ckpkand posknegkthe Descartes’ and admissible pairs for the SPs σk,k=0,,d1. The following restrictions follow from Rolle’s theorem:

    posk+1posk1,negk+1negk1andposk+1+negk+1posk+negk1.E5

    It is always true that

    posk+1+negk+1+3posknegk2N.E6

    Definition 2. Given a sign pattern σ0of length d+1, suppose that for k=0,,d1, the pair posknegksatisfies the conditions

    poskck,ckposk2Z,negkpk,pknegk2Z,andsgnak=1posk.E7

    as well as the inequalities (5)–(6). Then, we say that

    pos0neg0posd1negd1E8

    is a sequence of admissible pairs (SAPs). In other words, it is a sequence of pairs admissible for the sign pattern σ0in the sense of these conditions. We say that a given couple (SP, SAP) is realizable if there exists a polynomial Pwhose coefficients have signs given by the SP σ0, and such that for k=0,,d1, the polynomial Pkhas exactly poskpositive and negknegative roots, all of them being simple. Complex roots are also supposed to be distinct.

    Remark 4. If one only knows the SAP 8, the SP σ0can be restituted by the formula

    σ0=+1posd11posd21pos0.

    Nevertheless, in order to make comparisons with Problem 1 more easily, we consider couples (SP, SAP) instead of just SAPs. But for a given SP, there are, in general, several possible SAPs which is illustrated by the following example.

    Example 5. Consider the SP of length d+1with all pluses. For d=2and 3, there are, respectively, two and three possible SAPs:

    0201,0001,ford=2and030201,010201,010001ford=3.

    For d=4,5,6,7,8,9,10, the numbers Adof SAPs compatible with the SP of length d+1having all pluses are

    7,12,30,55,143,273,and728,

    respectively. One can show that Ad2Ad1, if d2is even, and Ad3Ad1/2, if d3is odd (see [5]).

    Example 6. There are two couples (SP, SAP) corresponding to the couple (SP, AP) C(++++, 02); we also say that the couple Ccan be extended into these couples (SP, SAP). These are

    (++++,02,21,11,01)and(++++,02,01,11,01).

    Indeed, by Rolle’s theorem, the derivative of a polynomial realizing the couple Chas at least one negative root. By conditions (7), this derivative (whose degree equals 3) has an even number of positive roots. This yields just two possibilities for pos1neg1, namely, 21and 01. The second derivative is a quadratic polynomial with positive leading coefficient and negative constant term. Hence, it has a positive and a negative root. The realizability of the above two couples (SP, SAP) is proven in [5].

    Our final realization problem is as follows:

    Problem 3. For a given degree d, which couples (SP, SAP) are realizable?

    Remarks 1. (1) This problem is a refinement of Problem 1, because one considers the APs of the derivatives of all orders and not just the one of the polynomial itself (see Remark 4). Therefore, if a given couple (SP, AP) is non-realizable, then all couples (SP, SAP) corresponding to it in the sense of Example 6 are automatically non-realizable.

    (2) Obviously, Problem 3 is a refinement of Problem 2—in the latter case, one does not take into account the signs of the real roots of the polynomial and its derivatives.

    (3) When we deal with couples (SP, SAP), we can use the Z2-action defined by (1). Therefore, it suffices to consider the cases of SPs beginning with ++. The generator (2.2) of the Z2×Z2-action cannot be used, because when the derivatives of a polynomial are involved, the polynomial loses its last coefficients. Due to this circumstance, the two ends of the SP cannot be treated equally.

    The following proposition is proven in [5]:

    Proposition 6. For any given SP of length d+1and d1, there exists a unique SAP such that pos0+neg0=d. This SAP is realizable. For the given SP, this pair pos0neg0is its Descartespair.

    Example 7. For even d, consider the SP with all pluses. Any hyperbolic polynomial with all negative and distinct roots realizes this SP with SAP

    0d0d101.

    One can choose such a polynomial Pwith all d1distinct critical values. Hence, in the family of polynomials P+tand t>0, one encounters polynomials realizing this SP with any of the SAPs

    0d20d10d201,=0,1,d/2.

    In the same way, for odd d, the SP +++is realizable with the SAP

    1d10d10d201

    by some hyperbolic polynomial Rwith all distinct roots and critical values. In the family of polynomials Rsand s>0, one encounters polynomials realizing this SP with any of the SAPs

    1d120d10d201,=0,1,d1/2.

    For d5, the following exhaustive answer to Problem 3 is given in [5]:

    1. A. For d=1, 2, and 3, all couples (SP, SAP) are realizable.

    2. B. For d=4, the couple (SP, SAP)

    ++++20211101,

    and only it (up to the Z2-action), is non-realizable. Its non-realizability follows from one of the couples (SP, AP) C++++20(see Theorem 1).

    One can observe that the couple Ccan be uniquely extended into a couple (SP, SAP). Indeed, the first derivative has a positive constant term hence an even number of positive roots. This number is positive by Rolle’s theorem. Hence, the AP of the first derivative is 21. In the same way, one obtains the APs 11and 01for the second and third derivatives, respectively.

    1. C. For d=5, the following couples (SP, SAP), and only they, are non-realizable:

    (+++++,21,20,21,11,01),(+++++,01,20,21,11,01),(++++,30,20,21,11,01),(++++,10,20,21,11,01),(+++,30,31,21,11,01).

    The non-realizability of the first four of them follows from that of the couple C. The last one is implied by part (1) of Theorem 2; it is true that the couple (SP, AP) +++30extends in a unique way into a couple (SP, SAP), and this is the fifth of the five such couples cited above.

    One of the methods used in the study of couples (SP, AP) or (SP, SAP) is the explicit construction of polynomials with multiple roots which define a given SP. Such constructions are not difficult to carry out because one has to use families of polynomials with fewer parameters. Once a polynomial with multiple roots is constructed, one has to justify the possibility to deform it continuously into a nearby polynomial with all distinct roots. Multiple roots can give rise to complex conjugate pairs of roots. An example of such a construction is the following lemma from [5].

    Lemma 2. Consider the polynomials Sx+13xa2and Tx+a2x13and a>0. Their coefficients of x4are positive if and only if, respectively, a<3/2and a>3/2. The coefficients of the polynomial Sdefine the SP

    +++++fora036/3,++++fora36/336,+++fora362/3and++++fora2/33/2.

    The coefficients of Tdefine the SP

    ++++fora3/23+6/3,+++fora3+6/33+6and++++fora>3+6.

    5. Outlook

    1. Our first open question deals with the limit of the ratio between the quantities Rdof all realizable and Adof all possible cases of couples (SP, AP) as d. In principle, one does not have to take into account the Z2×Z2-action in order not to face the problem of the two different possible lengths of orbits (2and 4).

      1. A priori, for d4, one has Rd/Ad01. It would be interesting to find out whether this ratio has a limit as dand, if “yes,” whether this limit is 0and 1or belongs to 01. In the latter case, it would be interesting to find the exact value.

      2. A less ambitious open problem is to find an interval αβ01to which this ratio belongs for any dN, d4, or at least for dsufficiently large.

    2. A related problem would be to find sufficient conditions for realizability based on the ratios between the quantities pos, neg, and d. On the one hand, when the ratios pos/dand neg/dare both large enough, one has realizability (see Proposition 4). On the other hand, in all examples of non-realizability known up to now, one of the quantities posand negis either 0or is very small compared to the other one. Thus, it would be interesting to understand the role of these ratios for the (non)-realizability of the couples (SP, AP).

    3. Our third open question is about the realizability of couples (SP, SAP). For d5, the non-realizability of all non-realizable couples (SP, SAP) results from the non-realizability of the corresponding couples (SP, AP). In principle, one could imagine a situation in which there exists a couple (SP, AP) extending into several couples (SP, SAP) some of which are realizable and the remaining are not. Whether, for d6, such couples (SP, AP) exist or not is unknown at present.

    4. Our final natural and important question deals with the topology of intersections of the set of real univariant polynomials with a given number of real roots with orthants in the coefficient space (which means fixing the signs of the coefficients). It is well known that the set of monic univariate polynomials of a given degree and with a given number of real roots is contractible. When we cut this set with the union of coordinate hyperplanes (coordinates being the coefficients of polynomials), then it splits into a number of connected components. In each such connected component, the number of positive and negative roots is fixed. But, in principle, it can happen that different connected components correspond to the same pair (pos, neg). Could this really happen? Are all such connected components contractible, or they can have some nontrivial topology?

    Acknowledgments

    The authors want to thank the Department of Mathematics of the Université Côte d’Azur and Stockholms Universitet for the hospitality, financial support, and nice working conditions during our several visits to each other.

    Notes

    • To René Descartes, a polymath in philosophy and science.

    © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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    Vladimir Petrov Kostov and Boris Shapiro (November 27th 2018). New Aspects of Descartes’ Rule of Signs, Polynomials - Theory and Application, Cheon Seoung Ryoo, IntechOpen, DOI: 10.5772/intechopen.82040. Available from:

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