This is a preprint of: “Periodic orbits of discrete and continuous dynamical systems via Poincaré- Miranda theorem”, Armengol Gasull, Vı́ctor Mañosa, Discrete Contin. Dyn. Syst. Ser. B, vol. 25(2), 651–670, 2020. DOI: [10.3934/dcdsb.2019259] Periodic orbits of discrete and continuous dynamical systems ∗ via Poincaré-Miranda theorem Armengol Gasull(1) and Vı́ctor Mañosa (2) (1) Departament de Matemàtiques Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain gasull@mat.uab.cat (2) Departament de Matemàtiques, Universitat Politècnica de Catalunya Colom 11, 08222 Terrassa, Spain victor.manosa@upc.edu September 17, 2018 Abstract We present a systematic methodology to determine and locate analytically isolated periodic points of discrete and continuous dynamical systems with algebraic nature. We apply this method to a wide range of examples, including a one-parameter family of counterexamples to the discrete Markus-Yamabe conjecture (La Salle conjecture); the study of the low periods of a Lotka-Volterra-type map; the existence of three limit cycles for a piece-wise linear planar vector field; a new counterexample of Kouchnirenko’s conjecture; and an alternative proof of the existence of a class of symmetric central configuration of the (1 + 4)-body problem. Mathematics Subject Classification 2010: 37C25, 39A23 (Primary); 13P15, 34D23, 70F15, 70K05 (Secondary). Keywords: Poincaré-Miranda theorem; Periodic orbits; Lotka-Volterra maps; Thue-Morse maps; Discrete Markus-Yamabe conjecture; Kouchnirenko’s conjecture; Limit cycles; Planar piecewise linear systems; Central configurations. ∗ Acknowledgements. The authors are supported by Ministry of Economy, Industry and Competitiveness–State Research Agency of the Spanish Government through grants MTM2016-77278-P (MINECO/AEI/FEDER, UE, first author) and DPI2016-77407-P (MINECO/AEI/FEDER, UE, second author). The first author is also supported by the grant 2017-SGR-1617 from AGAUR, Generalitat de Catalunya. The second author acknowledges the group’s research recognition 2017-SGR-388 from AGAUR, Generalitat de Catalunya. This work was completed at the Erwin Schrödinger International Institute for Mathematics and Physics, when the authors participated in the ESI Research in Teams project 2018. 1 1 Introduction and main results Periodic orbits are one of the main objects of study of the theory of dynamic systems. A priori there are many ways to prove the existence periodic orbits, for instance one can try to apply the plenty of available fixed point theorems [15] or results guaranteeing the existence of zeros, since periodic orbits are always solutions of equations of the form G(x) = x, where G is a return map in the continuous case, and G = F p for some p ∈ N in the case of a discrete system given by a map F . However when one tries to apply these results to a particular case it is not always easy to find effective ways to check the hypotheses. An example of this fact appears when trying to use the Newton-Kantorovich Theorem [19]. By using this approach, some bounds of the partial derivatives of the involved functions must be obtained. The work done in [3] exemplifies clearly the difficulties of this approach. In this work we present an effective procedure to prove the existence, determine the number and locate periodic orbits of dynamical systems of both discrete and continuous nature. This procedure is explained in detail in the next sections. As we will see, one of the main features of this procedure is the use of the Poincaré-Miranda theorem (PMT for short). We believe that one of the advantages of using PMT for finding fixed points of a given function is that only the signs of the components of it have to be controlled on some suitable sets, which is straightforward in the case that either the equations are polynomial or the problem can be polynomialized (see for instance the proof of Theorem 6 in Section 5). Recall that the use of Sturm sequences for polynomials in Q[x] allows to control their signs on intervals with rational endpoints ([32]). The PMT is the extension of the Bolzano theorem to higher dimensions. It was for- mulated and proved by H. Poincaré in 1883 and 1886 respectively, [29, 30]. C. Miranda re-obtained the result as an equivalent formulation of Brouwer fixed point theorem in 1940, [28]. Recent proofs are presented in [21, 33]. For completeness, we recall it. As usual, S and ∂S denote, respectively, the closure and the boundary of a set S ⊂ Rn . Theorem 1 (Poincaré-Miranda). Set B = {x = (x1 , . . . , xn ) ∈ Rn : Li < xi < Ui , 1 ≤ i ≤ n}. Suppose that f = (f1 , f2 , . . . , fn ) : B → Rn is continuous, f (x) 6= 0 for all x ∈ ∂B, and for 1 ≤ i ≤ n, fi (x1 , . . . , xi−1 , Li , xi+1 , . . . , xn ) ≤ 0 and fi (x1 , . . . , xi−1 , Ui , xi+1 , . . . , xn ) ≥ 0, Then, there exists s ∈ B such that f (s) = 0. For short, when given a map f we have a box B such that the hypotheses of the PMT hold we will say that B is a PM box. When we try to apply PMT to some f, sometimes it is better to consider some permutation of its components. 2 The paper is structured as follows: we start giving a new degree 6 counterexample of Kouchnirenko conjecture to illustrate the use and utility of our approach. In Section 3, we prove the existence of a 1-parameter family of rational counterexamples to a conjecture of La Salle (also known as discrete Markus-Yamabe conjecture) that extends the results of [6] providing also an alternative proof of them. In Section 4 we prove the existence of exactly two 5-periodic orbits and three 6-periodic orbits in a certain region for a Lotka-Volterra-type map correcting and complementing some results that appear in the literature. In Section 5 we provide another example of planar piecewise linear differential system with two zones having 3-limit cycles. Finally, in Section 6 we use PMT to give an alternative proof of the existence of a type of symmetric central configuration of the (1 + 4)-body problem. 2 A new counterexample to Kouchnirenko conjecture Descartes’ rule asserts that a 1-variable real polynomial with m monomials has at most m − 1 simple positive real roots. The Kouchnirenko conjecture was posed as an attempt to extend this rule to the several variables context. In the 2-variables case this conjecture said that a real polynomial system f1 (x, y) = f2 (x, y) = 0 would have at most (m1 − 1)(m2 − 1) simple solutions with positive coordinates, where mi is the number of monomials of each fi . This conjecture was stated by A. Kouchnirenko in the late 70’s, and published in the A. G. Khovanskiı̆’s paper [20]. In 2000, B. Haas ([17]) constructed a family of counterexam- ples given by two trimonomials, being the minimal degree of these counterexamples 106. In 2007 a much simpler family of counterexamples was presented in [9], being the simplest one again formed by two trimonomials, but of degree 6. Both examples have exactly 5 simple solutions with positive coordinates instead of the 4 predicted by the conjecture. In 2003, it was proved in [23] that any pair of bivariate trinomials has at most 5 simple solutions. We will prove in a very simple way, by using PMT, that system ( P (x, y) := x6 + ay 3 − y = 0, Q(x, y) := y 6 + ax3 − x = 0, with a = 61/43 ' 1.41860465 is a counterexample of the conjecture. We remark that in [9] it was given the counterexample with a = 44/31. The reason why we have changed this parameter is that it can be proved that when a = a = 7 × 124/5 /36 ' 1.14195168 the above system has the multiple solution (s, s) with s = 123/5 /6 ' 0.74021434 and a is quite close to 44/31 ' 1.4193548, making that, for that system, 3 of the its 5 solutions with positive entries are very close to each other. By using the approach developed in [12, 13], or the tools of [9], it can be proved that counterexamples to the conjecture only appear for a ∈ (a, a), where a ' 1.4176595. Both values are zeroes of some irreducible factor of the 3 polynomial ∆y (Res(P, Q; x)), where ∆y and Res denote, as usual, the discriminant and the resultant respectively. Hence, our value of a has also small numerator and denominator and, moreover, it is near the middle point of this interval, making that in the computations of our proof the rational numbers involved are simpler that the ones needed to use our approach when a = 44/31. We prove: Proposition 2. The bivariate trinomial system ( 61 3 P (x, y) = x6 + 43 y − y = 0, 61 3 (1) Q(x, y) = y 6 + 43 x − x = 0, has 5 real simple solutions with positive entries. Proof. It is not difficult to find numerically 5 approximated solutions of the system. They are (e e5 ), (e x1 , x e4 ), (e x2 , x e3 ), (e x3 , x e2 ), (e x4 , x e1 ), where x x5 , x e1 = 0.59679166, x e2 = 0.68913517, e3 = 0.74035310, x x e4 = 0.77980435 and x e5 = 0.81602099. We consider the following 5 ei ∈ Ii , intervals, with x       1 1619 1619 18 18 75857 I1 = , , I2 = , , I3 = , , 2 2500 2500 25 25 100000     75857 4 4 83 I4 = , , I5 = , . 100000 5 5 100 Let us prove that system (1) has 5 actual solutions (x1 , x5 ), (x2 , x4 ), (x3 , x3 ), (x4 , x2 ), (x5 , x1 ), with xi ∈ Ii . Firstly, since P (x, x) = Q(x, x) = x6 + 61x3 /43 − x, by Descartes rule we know that there is exactly one simple positive real root of P (x, x). By Bolzano theorem it belongs to I3 . So there is a solution (x3 , x3 ) of the system in I3 × I3 . By the symmetry of the system, if (x∗ , y ∗ ) is one of its solutions then (y ∗ , x∗ ) also is. Hence, we only need to prove that there are two suitable different solutions. This will be proved by applying the PMT to the boxes I1 ×I5 , and I2 ×I4 , which are depicted in Figure 1. We start applying the PMT to the box I1 × I5 . Consider the polynomials       1 61 3 1 1619 61 3 1619 6 P ,y = y − y + 6 and P ,y = y −y+ . 2 43 2 2500 43 2500 By computing their corresponding Sturm sequences we get that both have no roots in [4/5, 83/100]. Moreover P (1/2, y) < 0 and P (1619/2500, y) > 0 on this interval. Similarly we get that    6    6 4 61 3 4 83 61 3 83 Q x, = x −x+ <0 and Q x, = x −x+ >0 5 43 5 100 43 100 on [1/2, 1619/2500]. Hence, I1 × I5 is under the hypotheses of the PMT, and system (1) has a solution (x1 , x5 ) in this box. 4 Figure 1: Intersection of the curves P (x, y) = 0 (in blue) and Q(x, y) = 0 (in magenta). The PM boxes I1 × I5 , I2 × I4 (left and right respectively, in red). By using the same arguments one gets that the box I2 × I4 , contains another solution (x2 , x4 ) of our system. In this case the polynomials involved are even simpler. In this occasion, for y ∈ [75857/100000, 4/5] it holds that     1619 176243010801 18 127998 Q , y = y6 − < 0 and Q , y = y6 − > 0, 2500 671875 × 106 25 671875 and for x ∈ [1619/2500, 18/25], that     75857 6 5991841917684627 4 810993 P x, 5 =x − 15 < 0 and P x, = x6 − > 0. 10 43 × 10 5 43 × 106 The above facts prove that in the boxes I1 × I5 , I2 × I4 and their symmetric ones, I5 × I1 and I4 × I2 , there are at least 4 solutions of the studied system. These solutions together with the solution in the diagonal give the 5 announced solutions with positive coordinates. To prove they are simple solutions we first compute 33489 2 2 183 2 183 2 J(x, y) := det D(P, Q) = 36x5 y 5 − x y + x + y − 1. 1849 43 43 Since Res(Res(P, Q; x), Res(P, J; x); y) 6= 0, J does not vanish on the solutions (real or complex) of system (1). Hence all their solutions are simple. In fact, by using that a bivariate trinomial system hay at most five different solutions ([23]) or the tools of the so-called discard procedure, that we will introduce in Section 4 we get that 5 is the exact number of solutions with positive entries and that these solutions together with (0, 0) are the only real solutions of the system. 5 3 A counterexample to the discrete Markus-Yamabe conjec- ture revisited In [22], J. P. La Salle proposed some possible sufficient conditions for discrete dynamical systems with a fixed point, xn+1 = F (xn ), x ∈ Rn , to be globally asymptotically stable (GAS). One of these conditions is: For all x ∈ Rn , ρ (DF (x)) < 1, (2) where ρ is the spectral radius of the differential matrix. This condition is known as a discrete Markus-Yamabe-type condition because of its similarity with the conditions of Markus-Yamabe conjecture for ordinary differential equations, stated by L. Markus and H. Yamabe in 1960 [27], that has been proved to be true in dimension two and false in superior dimensions, see for instance [7, 16]. In [6] the authors consider rational maps of the form   a F (x, y) = y, −bx + , (3) (1 + y 2 )2 and prove that there exist some real values, a = a∗ and b = b∗ , such that the map (3) satisfies the Markus-Yamabe condition (2) and it has the 3-periodic point (−0.1, 0.25). Moreover they show numerically that for a∗ = 1.8 and b∗ = 0.9 a 3-periodic orbit seems to exist. This example was proposed to simplify the previous one given by W. Szlenk, see [5, Appendix]; and to show that even for systems coming from rational difference equations the discrete Markus-Yamabe conjecture does not hold. In this section we apply the PMT to give a simple proof of the following result, that in particular fixes the numerical counterexample presented in [6]. Proposition 3. For b ∈ B := [113/128, 2916/3125] ' [0.883, 0.933] the map   2b F (x, y; b) = y, −bx + , (1 + y 2 )2 satisfies the Markus-Yamabe condition (2) and has a 3-periodic orbit. Prior to prove this proposition, we recall the following auxiliary lemma, that is a sim- plified version of a result given in [12]. Lemma 4. Let G(x; b) = gn (b)xn + gn−1 (b)xn−1 + · · · + g1 (b)x + g0 (b) be a family of real polynomials that depend continuously on one real parameter b ∈ B = [b1 , b2 ] ⊂ R. Fix J = [x, x] ⊂ R and assume that: (i) There exists b0 ∈ B such that G(x; b0 ) has no real roots in J. 6 (ii) For all b ∈ B, G(x; b) · G(x; b) · ∆x (Gb ) 6= 0, where ∆x (G(·; b)) is the discriminant of G(x; b) with respect to x. Then for all b ∈ B, G(x; b) has no real roots in J. Proof of Proposition 3. We start noticing that in [6], it is proved that the maps (3) satisfy p condition (2) if and only if |a| < 11664/3125 and b ∈ (3125a2 /11664, 1). When a = 2b these conditions reduce to b ∈ (0, 2916/3125) . The 3-periodic points are solutions of system F 2 (x, y; b) = F −1 (x, y; b), that can be studied trough the equivalent system  gi (x, y; b) := Numer Fi2 (x, y; b) − Fi−1 (x, y) = 0, i = 1, 2, (4) where, as usual, Gi denotes the i-th component of a map G. Some computations give g1 (x, y; b) =−b2 x5 y 4 − 2b2 x5 y 2 − 2b2 x3 y 4 + x4 y 5 − b2 x5 − 4b2 x3 y 2 − b2 xy 4 + 2x4 y 3 + 2x2 y 5 + 2b2 x4 − 2b2 x3 − 2b2 xy 2 − 2by 4 + x4 y + 4x2 y 3 + y 5 + 4b2 x2 − b2 x − 4by 2 + 2x2 y + 2y 3 + 2b2 − 2b + y and that g2 (x, y; b) has degree 21 in (x, y) and degree 5 in b. We omit its expression. We claim, now, that for any b ∈ (b, b) = (113/128, 123/128) ' (0.883, 0.961) there is a solution of system (4) in the box B := [−0.2, 0] × [0, 0.5], corresponding with a 3-periodic point, where this interval of values of b is not optimal. Notice that the box B does not contain points on the diagonal line y = x and so, the found solution is not a fixed point, but a 3-periodic one. The curves g1 (x, y; 0.9) = 0 (in blue) and g2 (x, y; 0.9) = 0 (in magenta), together with the box B, are depicted in Figure 2. Figure 2: Intersection the curves g1 (x, y; 0.9) = 0 (in blue) and g2 (x, y; 0.9) = 0 (in ma- genta). It can be seen that there are seven intersection corresponding to one fixed point and two different 3-periodic orbits. In red, a PM box of one of the solutions of system (4). 7 In consequence, for b ∈ B the map satisfies the Markus-Yamabe condition (2) and has a 3-periodic point, as we wanted to prove. The claim will follow from PMT applied to the map f = (g1 , g2 ), once we prove for all b ∈ B: (I) h1 (y; b) := g1 (−0.2, y; b) · g1 (0, y; b) < 0 for y ∈ [0, 0.5], (II) h2 (x; b) := g2 (x, 0; b) · g1 (x, 0.5; b) < 0 for x ∈ [−0.2, 0]. Items (I) and (II) will be consequences of Lemma 4. We only give the details to prove item (I). Condition (i) of the lemma holds taking b∗ = 0.9, because 676 5 126936 4 1352 3 253872 2 676 9954 g1 (−0.2, y; 0.9) = y − y + y − y + y+ 625 78125 625 78125 625 78125 and g1 (0, y; 0.9) = y 5 − 95 y 4 + 2y 3 − 18 2 5 y +y− 9 50 do not vanish in [0, 0.5], as can be seen by computing their Strum sequences, and h1 (0; 0.9) < 0. To check condition (ii) we prove that the polynomial in the variable b, with rational coefficients and degree 71, h1 (0; b) · h1 (0.5; b) · ∆y (h1 (y, b)), has no roots for b ∈ B. This can be done again by computing its Sturm sequence. By using, the approach introduced in next section it is easy to prove for instance that the exact number of 3-periodic orbits of the map given in Proposition 3 when b = 0.9 is two, see again Figure 2. 4 Periodic orbits of a Lotka-Volterra map We consider the following Lotka-Volterra type map T (x, y) = (x(4 − x − y), xy) . (5) The interest for this map has grown after its consideration by A.N. Sharkovskiı̆ [31]. Notice that it unfolds the logistic map. It appears in many applications ([10]), being one of the most relevant ones, its relationship with some solutions of the Schrödinger equations modeling 1-dimensional quasi-cristalls with Thue-Morse sequence distributions, see [1]. This map is typically studied in the triangle 4 ⊂ R2 with vertices (0, 0), (4, 0), (0, 4), which is invariant. The low-period orbits of the map (5) were studied in [2, 26]. It is known that in Int(4) the fixed point (1, 2) is unique; there are not 2 and 3-periodic points; there is a unique 4-periodic orbit (which is explicitly known, [2]); and that there are 5 and 6-periodic points. The 5-periodic orbit is claimed to be unique in [2]. The following result completes and corrects those obtained in the above references. 8 Theorem 5. The following statements hold: (a) There exist exactly two different periodic orbits of minimal period 5 of T in Int(4). (b) There exist exactly three different periodic orbits of minimal period 6 of T in Int(4). Moreover one of them is (u, 1) → (1, u) → (3 − u, u) → (3 − u, 1) → (1, 3 − u) → (u, 3 − u) → (u, 1), √ where u = (3 − 5)/2 satisfies u(3 − u) = 1. √ Notice that taking as u = (3 + 5)/2 the other root of the same polynomial we obtain the same orbit. To prove the above result we use a methodology developed in [14], that can be summa- rized as: • We fix the period p. By using resultants, we include the solutions of T p (x, y) = (x, y), into the ones of an uncoupled system of equations given by two 1-variable polynomials. • We use the corresponding Sturm sequences for isolating the real roots of each 1- variable polynomials, and we apply a discard procedure in order to remove those solutions of the later system that do not correspond with the periodic points. • We apply the PMT to prove that the non discarded solutions are actual solutions of the first system of polynomial equations. Proof of Proposition 5. (a) We start noticing that imposing T 5 (x, y) = (x, y), one has the system of equations x · T5,1 (x, y) = 0, y · T5,2 (x, y) = 0, (6) where T5,1 and T5,2 are polynomials with degree 31, and 263 and 222 monomials respectively. We consider the resultants of these polynomials, and we remove the repeated factors and those factors corresponding to x = 0 and y = 0. Res(T5,1 , T5,2 ; y)  P (x) := 99 = (x − 2) (x − 1) x5 − 136x4 + 1784x3 − 5957x2 + 5850x − 1 100 x (x − 2) x10 − 41x9 + 482x8 − 2624x7 + 7847x6 − 13837x5 + 14655x4 − 9088x3 + 3019x2 − 414x +1) x15 − 178x14 + 7997x13 − 153777x12 + 1588330x11 − 9901048x10 + 39727694x9 −106108582x8 + 190846457x7 − 229400781x6 + 179062441x5 − 85605963x4 + 22367351x3  −2429213x2 + 6279x − 1 , 9 and Q(y) := Res(T5,1 , T5,2 ; x) = (y − 2) y 5 + 14520y 4 + 2662000y 3 + 121121000y 2 + 878460000y +1464100000) y 10 + 594y 9 + 16280y 8 + 56320y 7 − 567248y 6 + 220704y 5 + 2656192y 4  −2725888y 3 − 2385152y 2 + 4088832y − 1362944 y 15 + 15156y 14 + 11338084y 13 + 1961135256y 12 + 120710774176y 11 + 2862490382720y 10 + 25795669773184y 9 + 52844703170304y 8 − 280355579032320y 7 − 811324992569856y 6 + 760407187850240y 5 +2215201573881856y 4 − 1452783687979008y 3 − 1660265095602176y 2 + 1449013276164096y −281389965541376) . By using the Sturm approach we obtain that P (x) has 32 different real roots (all of them positive), and Q(x) has 31 different real roots, 11 of them positive. Hence, each solution in the positive quadrant of system (6) is contained in isolation in one of the 352 = 32 × 11 boxes Ii,j = Ii × Jj , i = 1, · · · , 32; j = 1, . . . , 11, where all Ii and Jj are intervals with positive rational endpoints such that each one of them contains a positive root of P and Q, respectively, in isolation. As we have already explained, to discard those sets Ii,j that do not contain any solution of system (6), we apply the discard method presented in [14]. We consider all boxes Ii,j . For each one we want to know whether the function f (x, y) = P ` M` (x, y), where f can be either T5,1 or T5,2 , has or not a fixed sign. Setting Ii,j = [x, x] × [y, y] ⊂ (R+ )2 , for each monomial M` (x, y) = a` x`1 y `2 one has M ` ≤ M (x, y) ≤ M ` , where M ` = a` x`1 y `2 and M ` = a` x`1 y `2 if a` > 0, or M ` = a` x`1 y `2 and M ` = a` x`1 y `2 if a` < 0. P P P P If either 0 < ` M ` < ` M` (x, y) = f (x, y) or f (x, y) = ` M` (x, y) < ` M ` < 0 then we can discard the box Ii,j . If not, but we suspect (by our previous numerical computations) that it should be discarded, we substitute it by one of smaller size. To apply the discard procedure efficiently we need to compute the intervals Ii and Jj with maximum length 10−40 which are given in the appendix. It gives that each solution of system (6) must be contained in one of the following 11 non-discarded boxes I5,7 , I6,11 , I7,9 , I8,6 , I9,10 , I10,2 , I14,3 , I20,1 , I23,5 , I24,4 (7) and I11,8 = [1, 1] × [2, 2] which, obviously corresponds with the unique fixed point of T (x, y) = (1, 2), so we discard it. To prove that there is a (unique) solution of system (6) in each box, and therefore there are 2-periodic orbits with 10 periodic points of minimal period 5 we apply the PMT. To 10 illustrate the type of computations we deal with, we only show one of the computations. We prove that there is a unique solution in the box I9,10 . To obtain simpler expressions and work more comfortably we will show that the hy- potheses of the PMT are verified for a bigger box B =:= [0.6, 1] × [2.3, 2.9] , which has been obtained by visual inspection, see Figure 3, instead of using the actual box. It is easy to check that the only box of (7) contained in B is I9,10 , and therefore if there is a solution of system (6) in B then it must be in I9,10 , and be unique by construction. Figure 3: The PM box of a solution of system (6) used in the proof of Theorem 5 (in red). It corresponds to the intersection of the curves defined by the curves T5,1 (x, y) = 0 (in blue) and T5,2 (x, y) = 0 (in magenta). We take, g1 (y) := T5,1 (0.6, y) · T5,1 (1, y) where 16679880978201y 11 1167478199895987y 10 6483464101392579y 9 T5,1 (0.6, y) = − + 95367431640625 476837158203125 476837158203125 86310474195914829y 8 447004829071396386y 7 8389294048378453266y 6 − + + 2384185791015625 11920928955078125 298023223876953125 148429346693234077422y 5 80721993185246247282y 4 317995739735111299953y 3 − + + 1490116119384765625 1490116119384765625 7450580596923828125 1653153957629818831467y 2 4929661459397407475239y 3868538667523044137292 − + + . 37252902984619140625 931322574615478515625 4656612873077392578125 and T5,1 (1, y) = (y − 2) ×  y 10 − 15y 9 + 97y 8 − 353y 7 + 792y 6 − 1130y 5 + 1022y 4 − 566y 3 + 177y 2 − 27y + 1 . and prove that g1 it is negative for y ∈ [2.3, 2.9]. This can be done by using the Sturm sequences of both polynomials. Proceeding in an analogous way we obtain that g2 (y) := T5,2 (x, 2.3) · T5,2 (x, 2.9) is a polynomial of degree 62 and it is negative for x ∈ [0.6, 1]. Hence the map f = (T5,1 , T5,2 ) satisfies the hypothesis of the PMT and there exists a solution of system (6) in B. 11 (b) By imposing T 6 (x, y) = (x, y), we get x · T6,1 (x, y) = 0, y · T6,2 (x, y) = 0, (8) where T6,1 and T6,2 are polynomials with degree 63, and 967 and 910 monomials respectively. We compute the resultants of these polynomials, and remove the repeated factors and those factors corresponding to x = 0 and y = 0. Res(T6,1 , T6,2 ; y)   P (x) := = A x3 − 10x2 + 17x − 1 x2 − 3x + 1 x420 (x2 − 3x + 1) (x − 1)2 (x − 2)405  x6 − 25x5 + 184x4 − 547x3 + 669x2 − 254x + 1 128x12 − 2816x11 + 25280x10 − 124256x9  +372768x8 − 713136x7 + 876616x6 − 677024x5 + 309828x4 − 74692x3 + 7552x2 − 272x + 1 x12 − 40x11 + 638x10 − 5436x9 + 27664x8 − 88424x7 + 181016x6 − 237152x5 + 195072x4   −96608x3 + 26624x2 − 3456x + 128 x6 − 11x5 + 44x4 − 78x3 + 60x2 − 16x + 1 (x − 1)  (x − 2) x3 − 9x2 + 14x − 1 , and Res(T6,1 , T6,2 ; x)   Q(y) := = B (y − 2) y 3 + 26y 2 + 104y + 104 y 3 + 36y 2 + 180y + 216 y 6 (y − 1)(y 2 − 3y + 1)  y 6 + 182y 5 + 3136y 4 + 16072y 3 + 25872y 2 + 15680y + 3136 128y 12 + 23808y 11 + 602304y 10 + 2820832y 9 − 4126176y 8 − 29841552y 7 + 8077160y 6 + 52324032y 5 − 24120108y 4 − 9219772y 3  +3690240y 2 − 133920y + 837 16384y 12 + 671744y 11 + 5943296y 10 + 1502208y 9 − 62922752y 8 −53763840y 7 + 165704768y 6 + 167848384y 5 − 52858224y 4 − 48703232y 3   +5309928y 2 − 133920y + 837 (y − 1) y 2 − 3y + 1 , where A and B are non-zero constants. By using the Sturm method we obtain that P (x) has 46 different real roots (all of them positive), and Q(x) has 40 different real roots (16 of them positive). Hence, each solution in the positive quadrant of system (8) is contained in isolation in one of the 736 = 46 × 16 sets of the form Ii,j = Ii × Jj , i = 1, · · · , 46; j = 1, . . . , 16. where {Ii ⊂ R+ , i = 1, . . . , 46} and {Jj ⊂ R+ , j = 1, . . . , 16} are intervals with rational ends such that each one of them contains a positive root of P and Q, respectively, in isolation. In our computations we have obtained these intervals, with rational ends and maximum length bounded by 10−100 (in order to apply the discard procedure efficiently). We don’t give these intervals in this paper. But in order to facilitate the reproduction of our results and allow the reader to determine and locate the 6-periodic orbits, we indicate that these intervals (and therefore the roots) are ordered, in the sense that if ` < m then I` (respectively J` ) is completely to the left of Im (respectively Jm ). 12 To discard those sets Ii,j that do not contain any solution of system (8), we apply the discard method. The procedure allows to eliminate 717 boxes. Moreover, the box I17,12 = [1, 1] × [2, 2] corresponds with the fixed point (1, 2) and the boxes I11,10 , I17,7 , I31,7 , I31,10 , I17,15 and I11,15 correspond to the explicit 6-periodic orbit given in the statement. Hence the remaining solutions of system (8) must be contained in one of the following 12 non-discarded boxes: I8,6 , I9,3 , I12,14 , I13,11 , I16,16 , I18,1 , I20,13 , I21,8 , I30,4 , I34,9 , I35,2 , I37,5 . (9) Again, the PMT can be used to prove that in each of them there is a solution of system (8). Since the solution must be unique, we prove that in total there are 18 periodic points of minimal period 6. Since the computation are quite similar to the ones used to study the 5-periodic points we skip them. 4.1 Determination of the 5-periodic orbits By using the boxes computed in the proof of the above result, it is easy to determine which points correspond to each orbit. Indeed, first we concentrate on the 5-periodic orbits. Let us denote Pi,j = (x, y) the (unique) 5-periodic point lying in the box Ii,j of (7). We notice that the two 5-periodic orbits of T in Int(4) are given by P9,10 → P7,9 → P8,6 → P14,3 → P23,5 , and P5,7 → P10,2 → P20,1 → P24,4 → P6,1 , where Orbit 1 Orbit 2 P9,10 ' (0.8581419568, 2.587834436) P5,7 ' (0.3667104103, 1.192698099) P7,9 ' (0.4754309022, 2.220729307) P10,2 ' (0.8949903070, 0.4373748091) P8,6 ' (0.6198857282, 1.055803338) P20,1 ' (2.387507364, 0.3914462147) P14,3 ' (1.440807176, 0.6544774210) P24,4 ' (2.915257323, 0.9345807202) P23,5 ' (2.744327621, 0.9429757645) P6,11 ' (0.4377607446, 2.724543288) These decimal approximations have obtained using the intervals given in the appendix. Remember that they give an approximation with a maximum error of 10−40 . The points are depicted in Figure 4. The above assertions can be proved, by using the fact that taking Ii,j = [x, x] × [y, y], and setting (e x, ye) = T (Pi,j ), since ye = xy it must satisfy x y < ye < x y, and from these inequalities is easy to identify in which box of (7) is (e x, ye). For instance, for the point P9,10 ∈ I9,10 , and setting ye = T (P9,10 )2 , one has that a := x y < ye < b := x y where 16852116181629561083016478047392649752537328677768476306665633800018763276864769965 a= , 7588550360256754183279148073529370729071901715047420004889892225542594864082845696 4213029045407390270754119511848162438134407216162333392848558471828944039611220943 b= . 1897137590064188545819787018382342682267975428761855001222473056385648716020711424 13 Figure 4: Two 5-periodic orbits of the map (5) in Int(4) (Orbit 1 in red and Orbit 2 in green). They correspond to the intersection of the curves defined by the curves T5,1 (x, y) = 0 (in blue) and T5,2 (x, y) = 0 (in magenta). The fixed point (1, 2) (in brown). The PM box containing the point P9,10 used in the proof of Theorem 5 (in red). Now, it is easy to check that the only interval Jj for j ∈ {1, 2, . . . , 10, 11} with nonempty intersection with (a, b) is J9 , hence P7,9 = T (P9,10 ). 4.2 Determination of the 6-periodic orbits Proceeding as in the previous section, we determine the points of two of the 6-periodic orbits. The third one is explicit. Again we denote Pi,j = (x, y) the (unique) 6-periodic point lying in the box Ii,j of (9). The points are depicted in Figure 5. We have, Orbit 1: P20,13 → P16,16 → P12,14 → P13,11 → P21,8 → P34,9 ; Orbit 2: P8,6 → P9,3 → P18,1 → P35,2 → P30,4 → P37,5 ; and Orbit 3: P11,10 → P17,7 → P31,7 → P31,10 → P17,15 → P11,15 , where the points of Orbit 3 are the ones of the statement and Orbit 1 Orbit 2 P20,13 ' (1.300802119, 2.018868702) P8,6 ' (0.09022635321, 0.2685661106) P16,16 ' (0.8849736378, 2.626148686) P9,3 ' (0.3285328773, 0.02423174076) P12,14 ' (0.4326438557, 2.324072356) P18,1 ' (1.198236734, 0.007960923513) P13,11 ' (0.5378990919, 1.005495625) P35,2 ' (3.347636595, 0.009539070990) P21,8 ' (1.321405751, 0.5408551836) P30,4 ' (2.151942266, 0.03193334313) P34,9 ' (2.824820695, 0.7146891501) P37,5 ' (3.908194837, 0.06871871076) 14 Figure 5: Three 6-periodic orbits of the map (5) in Int(4) (Orbit 1,2 and 3 in brown, red and green, respectively). They correspond to the intersection of the curves defined by the curves T6,1 (x, y) = 0 (in blue) and T6,2 (x, y) = 0 (in magenta). The fixed point (1, 2) (in black). The decimal approximations have obtained using the intervals computed in the proof of Theorem 5. They gave an approximation with a maximum error of 10−100 . The points are depicted in Figure 5. 5 Limit cycles of piecewise linear differential systems The study of the number of limit cycles for planar differential systems is a classical topic in the theory of dynamical systems. In the last years, many attention has been devoted to the study of nested limit cycles of piecewise linear systems, steered by the applicability of these systems in the modelling of biological and mechanical applications. In 2012, S.M. Huan and X.S. Yang gave numerical evidences of a piecewise linear system with two zones and a discontinuity straight line, having three nested limit cycles ([18]). A proof based on the Newton–Kantorovich theorem of the existence of these limit cycles for this example and a nearby one, was given by J. Llibre and E. Ponce ([25]). A different proof, from a bifurcation viewpoint, was presented by E. Freire, E. Ponce and F. Torres in [11]. Until now, as far as we know, three is the maximum observed number of limit cycles in piecewise linear differential systems with two zones and a discontinuity straight line, but it is not known if this is the maximum number that such type of systems can have. 15 In this section we present a new example, again with 3 limit cycles, inspired on the ones given in [18, 25]. The main difference is that our proof of their existence is based on the PMT. Theorem 6. The two-zones piecewise linear differential system ( A+ x if x ≥ 1, ẋ = (10) A− x if x ≤ 1, where x = (x, y)t , ! ! 67 833 3 − 125 −1 A− := 50 1 and A+ := 8 , 2 − 87 50 1 3 8 has at least three nested hyperbolic limit cycles surrounding the origin. To prove the above result, we will use systematically the following lemma, that is a straightforward consequence of Taylor’s formula. Lemma 7. Set h(x) = A cos(αx) + B sin(αx) + Ceβx + De−βx , with A, B, C, D ∈ R, α 6= 0, Pn β > 0 and x ∈ [x, x] ⊂ R+ . Then for each n ≥ 0 we have h(x) = aj xj + mn (x)xn+1 , j=0 where 1  jh  π  π i   aj = α A cos j + B sin j + β j C + (−1)j D , j! 2 2 n+1  |α| (|A| + |B|) + |β|n+1 |C|eβx + |D|e−βx |mn (x)| ≤ m = . (11) (n + 1)! Proof of Theorem 6. Let ϕ± (t; p) = (x± (t; p), x± (t; p)) denote the flows associated to the linear systems ẋ = A± x. Observe that if there exists a limit cycle then it must lie on both sides of the line x = 1, so let t− > 0 be the smaller time such that x− (t− ; (1, y)) = 1 for a point (1, y) with y > 0, and let t+ > 0 be the smaller time such that x+ (−t+ ; (1, y)) = 1. Then any limit cycle must satisfy x+ (−t+ ; (1, y)) − 1 = 0, x− (t− ; (1, y)) − 1 = 0, y + (−t+ ; (1, y)) − y − (t− ; (1, y)) = 0, or equivalently 3 e− 8 u (cos (u) + y sin (u)) − 1 = 0, (12)      1 49 49 1 35 cos v + (−238y + 55) sin v e− 5 v − 1 = 0, (13) 35 50 50      1 49 49 v 3 −49 cos v y + (77y − 25) sin v e− 5 + e− 8 u (cos (u) y − sin (u)) = 0, (14) 49 50 50 where u = t+ > 0 and v = t− > 0. 16 3  By solving equation (12) we get y = e− 8 u − cos(u) /sin(u). By substituting this ex- pression in equations (13) and (14), we obtain 3 g1 (u, v) := a(v) cos(u) + b(v) sin(u) − a(v)e 8 u = 0, 3 3 (15) g2 (u, v) := c(v) cos(u) + d(v) sin(u) + e(v)e 8 u + f (v)e− 8 u = 0, where v  v  v  a(v) = 238 e− 5 sin 49 50 v , b(v) = 55 e− 5 sin 49 50 v + 35 e− 5 cos 49 50 v − 35, and v  v  v  c(v) = 49 e− 5 cos 49 50 v − 77 e− 5 sin 49 50 v + 49, d(v) = −25 e− 5 sin 49 50 v , v  v  e(v) = 77 e− 5 sin 49 50 v − 49 e− 5 cos 49 50 v , f (v) = −49. Figure 6: Left part: Intersection points between g1 (u, v) = 0 (in blue) and g2 (u, v) = 0 (in magenta) and some PM boxes containing them. Right part: the 3 limit cycles of system (10). Numerically it is easy to guess that there are 3 different solutions of system (15), see Fig- ure 6. Their approximate values in (u, v) variables are (0.441441, 4.554696), (0.639391, 4.105752) and (1.686596, 3.458345). Once we prove that near these values there are actual solutions of system (15), each one of them will correspond to a solution of the system of equations (12)–(14) and, consequently, all them will give rise to 3 limit cycles of system (10), see again Figure 6. To prove the existence of 3 solutions of system (15), we consider the 3 boxes:             9 1 219 26 1 7 71 219 7 17 71 B1 := , × , , B2 := , × , B3 := ,2 × , 25 2 50 5 2 5 20 50 5 5 20 and prove that they are PM boxes for (g1 , g2 ). 17 To see that we are under the hypotheses of the PMT, in all the cases we proceed systematically in the following form: We suppose that we want to prove that a function h(x) of the form of Lemma 7 is positive (resp. negative) in [x, x] ⊂ R+ . Firstly, we use this lemma to have its Taylor polynomial at x = 0 up to a certain order n. Secondly, we minorize (resp. majorize) the polynomial by a polynomial with rational coefficients, obtained by truncating the decimal expression up to some suitable order k, and subtracting (resp. adding) 10−k to the obtained quantity, that is a± k j := Trunc(aj · 10 ) · 10 −k ± 10−k ∈ Q, where Trunc stands for the truncation to the next nearest integer towards 0. Finally we ± P n consider Pn,k (x) = a± j j x ± Mx n+1 , where M ∈ Q is a suitable upper bound of the j=0 right-hand side expression in (11), so that − + Pn,k (x) − M xn+1 ≤ h(x) ≤ Pn,k (x) + M xn+1 . − + Now we only have to check if Pn,k (x) > 0 (resp. Pn,k (x) < 0) in [x, x]. To do this we use the Sturm sequences of these polynomials. Applying this approach we prove that B1 , B2 and B3 are PM boxes by setting the following parameters n, k and M in each face (we use the notation B = [u, u] × [v, v]): Box B1 : Face u=u u=u v=v v=v Target function h and sign g1 < 0 g1 > 0 g2 > 0 g2 < 0 + − − + Polynomial Pn,k Pn,k Pn,k Pn,k Parameters: n 16 16 4 4 k 15 15 3 3 7 4 Bound M 10−13 10−12 10 5 Box B2 : Face u=u u=u v=v v=v v v Target function h and sign e · g2 > 0 e · g2 < 0 g1 < 0 g1 > 0 5 5 − + + − Polynomial Pn,k Pn,k Pn,k Pn,k Parameters: n 16 16 6 4 k 14 10 2 2 1 13 Bound M 10−13 10−12 50 10 18 Box B3 : Face u=u u=u v=v v=v v v Target function h and sign e · g1 < 0 e · g1 > 0 g2 > 0 g2 < 0 5 5 + − − + Polynomial Pn,k Pn,k Pn,k Pn,k Parameters: n 13 11 6 7 k 8 8 3 2 Bound M 10−8 10−6 10−2 10−2 We only give details of some of the computations for B1 . For instance we show that g2 (u, v) > 0 for all u ∈ [u, u]. All the other computations can be reproduced using the information given above. Indeed, we take B1 and we observe that         219 219 219 3 u 219 − 3 u g2 (u, v) = c cos(u) + d sin(u) + e e8 + f e 8 , 50 50 50 50 which has the form of the function h in Lemma 7, where 219  219 − 250  − 219  A=c 50 = 49 e cos 10731 2500 − 77 e 250 sin 10731 2500 + 49, 219  219 − 250 10731  B=d 50 = −25 e sin 2500 , 219  10731  10731  − 219 C=e 50 = −49 cos 2500 + 77 sin 2500 e 250 , 219  D=f 50 = −49. P4 By applying this lemma with n = 4, we obtain P4 (u) = j j=0 aj u . After taking a− j := Trunc(aj · 103 ) · 10−3 − 10−3 for each j = 0, . . . , 4 we get − 1 1001 39899 2 669 3 357 4 P4,3 (u) = − + u− u − u + u . 1000 50 1000 500 125 By using (11) we also obtain that 219 1377 !   49 e− 250 3969 e− 2000 10731 m4 = + cos + 120 1310720 2500 219 1377 !   27 17 e− 250 6237 e− 2000 10731 49 3969 e− 200 7 − − sin + + ' 0.6664 < = M. 20 1310720 2500 120 1310720 10 − By using the Sturm sequence of P4,3 (u)−M u5 we prove that it has no roots in [u, u] and, − moreover, it is positive in this interval. Hence 0 < P4,3 (u)−M u5 < g2 (u, v) for all u ∈ [u, u]. To prove the hyperbolicity of the limit cycles we can follow the same ideas that in [25]. 19 6 On the existence of a symmetric central configuration Central configurations are a very special type of solutions of the N -body problem in Celestial Mechanics, in which the acceleration of every body is proportional to the position vector of the body with respect the center of mass of the system. They play an important role in practical applications and there is a vast literature on the topic, both classical and recent. An account of known facts and open problems can be found in [24]. In the (1 + n)-body problem it is supposed that there is one body with a large mass and n bodies whose masses can be neglected in comparison with the large one. These bodies are named as infinitesimal masses. With our approach we prove in a very simple way the existence of a special planar central configurations of the (1 + 4)-body problem, already given in [8]. According to the results in [4, 8], all planar central configurations in the (1 + n)-body problem lie on a circle centered at the position of the large mass. Furthermore, denoting by αi ∈ S1 , with α1 < α2 < . . . < αn , the angles defined by the position of the ith infinitesimal masses on a circle centered at the origin, central configurations must satisfy the system of n equations, i = 1, . . . , n, n  √  X 2 f (αj − αi ) = 0, with f (θ) := 1 − p sin(θ). j=1 4 (1 − cos(θ))3 Notice that f (−θ) = −f (θ). When n = 4, we introduce the variables u = α2 − α1 , v = α3 − α1 and w = α4 − α1 . Then the above system is equivalent to the system (with only 3 equations): f (u) + f (v) + f (w) = 0, −f (u) + f (v − u) + f (w − u) = 0, −f (v) + f (u − v) + f (w − v) = 0. Although our point of view could be applied to prove the existence of solutions of the above system (and so of central configurations), for simplicity we will look for a symmetric one, the one satisfying tat α4 − α3 = α2 − α1 . In our coordinates this implies that w = u + v and hence the system reduces to the system with 2 equations g1 (u, v) := f (u) + f (v) + f (u + v) = 0, g2 (u, v) := f (u) − f (v) − f (v − u) = 0. Let us prove that we can apply PMT to the box [0.7, 0.8] × [1.3, 1.5], see Figure 7. To simplify the notation we denote by gi (I) + gj (J) the set of all values gi (x) + gj (y), with x ∈ I and y ∈ J. Then, simply using that f is increasing between in (0, θ∗ ) and decreasing 20 Figure 7: Left figure: graph of f (θ) in (0, π]. Right figure: intersection of the curves g1 (u, v) = 0 (in blue) and g2 (u, v) = 0 (in magenta). In red, a PM box. in (θ∗ , π), where θ∗ ' 1.891 ∈ (1.8, 1.9), see again Figure 7, we get: g1 (0.7, v)|v∈[1.3,1.5] = f (0.7) + f (v) + f (0.7 + v)|v∈[1.3,1.5] = f (0.7) + f ([1.3, 1.5]) + f ([2, 2.2]) ≤ f (0.7) + f (1.5) + f (2) ' −0.031 < 0. Similarly, g1 (0.8, v)|v∈[1.3,1.5] = f (0.8)+f ([1.3, 1.5])+f ([2.1, 2.3]) ≥ f (0.8)+f (1.3)+f (2.3) ' 0.24 > 0, g2 (u, 1.3)|u∈[0.7,0.8] = f ([0.7, 0.8]) − f (1.3) − f ([0.5, 0.6]) ≥ f (0.7) − f (1.3) − f (0.6) ' 0.40 > 0, and g2 (u, 1.5)|u∈[0.7,0.8] = f ([0.7, 0.8]) − f (1.5) − f ([0.7, 0.8]) ≤ f (0.8) − f (1.5) − f (0.7) ' −0.052 < 0. Hence we have proved the existence of a symmetric central configuration for this problem. In fact, numerically this solution is u = u∗ ' 0.7242718590, v = v ∗ ' 1.376255451 and it corresponds with the values x∗ = cos((u∗ +v ∗ )/4) ' 0.86525786 and y ∗ = sin((v ∗ − u∗ )/4) ' 0.162275119 given in the proof of [8, Prop 13] and found using the variables x = cos((α2 + α3 − 2α1 )/4) and y = sin((α3 − α2 )/4). 21 Appendix. The 32 intervals of maximum length 10−40 containing in isolation the positive roots of the polynomial P (x) that appear in the proof of Theorem 1 are:   30416172743817963645774671148052190959399 60832345487635927291549342296104381918799 I1 = , , 178405961588244985132285746181186892047843328 356811923176489970264571492362373784095686656   61004111234951360792324423318169046946281 30502055617475680396162211659084523473141 I2 =, , , 356811923176489970264571492362373784095686656 178405961588244985132285746181186892047843328   27421400457977260353638941607050001933877 54842800915954520707277883214100003867755 I3 =, , , 11150372599265311570767859136324180752990208 22300745198530623141535718272648361505980416   55177596464914226273769532370430091204151 6897199558114278284221191546303761400519 I4 =, , , 22300745198530623141535718272648361505980416 2787593149816327892691964784081045188247552   63889964233534261969205802666073326201313 31944982116767130984602901333036663100657 I5 =, , , 174224571863520493293247799005065324265472 87112285931760246646623899502532662132736   76268678313703396955788049048149760899577 38134339156851698477894024524074880449789 I6 =, , , 174224571863520493293247799005065324265472 87112285931760246646623899502532662132736   82831745394319077306596661336176030777141 41415872697159538653298330668088015388571 I7 =, , , 174224571863520493293247799005065324265472 87112285931760246646623899502532662132736   26999831398533091510037286533787056284381 53999662797066183020074573067574112568763 I8 =, , , 43556142965880123323311949751266331066368 87112285931760246646623899502532662132736   74754707508083115080188235040108949111941 37377353754041557540094117520054474555971 I9 =, , , 87112285931760246646623899502532662132736 43556142965880123323311949751266331066368   77964651533856060199939131867310019005089 38982325766928030099969565933655009502545 I10 =, , , 87112285931760246646623899502532662132736 43556142965880123323311949751266331066368 I11 =, [1, 1] ,   238239449906197613139925190460054741169 121978598351973177927641697515548027478529 I12 =, , , 170141183460469231731687303715884105728 87112285931760246646623899502532662132736   61150748376490059476117465769766977687027 122301496752980118952234931539533955374055 I13 =, , , 43556142965880123323311949751266331066368 87112285931760246646623899502532662132736   31378001665582463990512894476418249863733 125512006662329855962051577905672999454933 I14 =, , , 21778071482940061661655974875633165533184 87112285931760246646623899502532662132736   173371830902734537060512513441447388046677 86685915451367268530256256720723694023339 I15 =, , , 87112285931760246646623899502532662132736 43556142965880123323311949751266331066368   43530335013270399357309383085115875521389 174121340053081597429237532340463502085557 I16 =, , , 21778071482940061661655974875633165533184 87112285931760246646623899502532662132736 I17 =, [2, 2] ,   43581896065820670510185474007765782536433 174327584263282682040741896031063130145733 I18 =, , , 21778071482940061661655974875633165533184 87112285931760246646623899502532662132736   175074047096038860833682791052723868617523 43768511774009715208420697763180967154381 I19 =, , , 87112285931760246646623899502532662132736 21778071482940061661655974875633165533184   207981224135133170727752198124246427433217 103990612067566585363876099062123213716609 I20 =, , , 87112285931760246646623899502532662132736 43556142965880123323311949751266331066368   53389351149121164414177573637629460658781 213557404596484657656710294550517842635125 I21 =, , , 21778071482940061661655974875633165533184 87112285931760246646623899502532662132736 22   213578718968144293553725465775995745176425 106789359484072146776862732887997872588213 I22 =, , , 87112285931760246646623899502532662132736 43556142965880123323311949751266331066368   59766163097780263153587646066230635923515 239064652391121052614350584264922543694061 I23 =, , , 21778071482940061661655974875633165533184 87112285931760246646623899502532662132736   253954729472389360941133632461626408129629 126977364736194680470566816230813204064815 I24 =, , , 87112285931760246646623899502532662132736 43556142965880123323311949751266331066368   243357617176921003857605925220978518350987 486715234353842007715211850441957036701975 I25 =, , , 43556142965880123323311949751266331066368 87112285931760246646623899502532662132736   487996318307203475975406100394067486420489 243998159153601737987703050197033743210245 I26 =, , , 87112285931760246646623899502532662132736 43556142965880123323311949751266331066368   854083577046230668182843971813839810572423 106760447130778833522855496476729976321553 I27 =, , , 87112285931760246646623899502532662132736 10889035741470030830827987437816582766592   854169231180689628527098761743375311769429 427084615590344814263549380871687655884715 I28 =, , , 87112285931760246646623899502532662132736 43556142965880123323311949751266331066368   562325002245556480093549131285848700772183 2249300008982225920374196525143394803088733 I29 =, , , 21778071482940061661655974875633165533184 87112285931760246646623899502532662132736   2252874711653478373358567435415938516285773 1126437355826739186679283717707969258142887 I30 =, , , 87112285931760246646623899502532662132736 43556142965880123323311949751266331066368   10605493671441752150124990095059709232394013 5302746835720876075062495047529854616197007 I31 =, , , 87112285931760246646623899502532662132736 43556142965880123323311949751266331066368   10620731390948027186722372703422173139084449 5310365695474013593361186351711086569542225 I32 =, , . 87112285931760246646623899502532662132736 43556142965880123323311949751266331066368 The 11 intervals of maximum length 10−40 containing in isolation the positive roots of the polynomial Q(y) are:   34099774584992488192779339819330125093373 68199549169984976385558679638660250186747 J1 = , , 87112285931760246646623899502532662132736 174224571863520493293247799005065324265472   76201438867882234589322968824595491762585 38100719433941117294661484412297745881293 J2 = , , 174224571863520493293247799005065324265472 87112285931760246646623899502532662132736   7126628029028087048991427050514157728201 57013024232224696391931416404113261825609 J3 = , , 10889035741470030830827987437816582766592 87112285931760246646623899502532662132736   20353365730967257062570091063747267693705 81413462923869028250280364254989070774821 J4 = , , 21778071482940061661655974875633165533184 87112285931760246646623899502532662132736   20536193605572751128435745571834887603857 82144774422291004513742982287339550415429 J5 = , , 21778071482940061661655974875633165533184 87112285931760246646623899502532662132736   5748340141743633609606285424423240727937 91973442267898137753700566790771851646993 J6 = , , 5444517870735015415413993718908291383296 87112285931760246646623899502532662132736   103898657804950550140822640990534612319235 25974664451237637535205660247633653079809 J7 = , , 87112285931760246646623899502532662132736 21778071482940061661655974875633165533184 J8 = [2, 2] ,   193452806356507885400952564030524665947305 96726403178253942700476282015262332973653 J9 = , , 87112285931760246646623899502532662132736 43556142965880123323311949751266331066368   225432173349181613519899061972772631001865 112716086674590806759949530986386315500933 J10 = , , 87112285931760246646623899502532662132736 43556142965880123323311949751266331066368   237341193967033134227594463359390906202991 14833824622939570889224653959961931637687 J11 = , . 87112285931760246646623899502532662132736 5444517870735015415413993718908291383296 23 References [1] Y. 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