EXTRACTA - EXTRACTA MATHEMATICAE Vol. 35, Num. 2 (2020), 221 – 228 doi:10.17398/2605-5686.35.2.221 Available online June 19, 2020 MATHEMATICAE Around some extensions of Casas-Alvero conjecture for non-polynomial functions A. Cima, A. Gasull, F. Mañosas Departament de Matemàtiques, Universitat Autònoma de Barcelona Barcelona, Spain Volumen 33, Número 2, 2018 cima@mat.uab.cat , gasull@mat.uab.cat , manyosas@mat.uab.cat instituto de investigación de matemáticas de la universidad de extremadura Received April 21, 2020 Presented by Manuel Maestre Accepted May 2, 2020 Abstract: We show that two natural extensions of the real Casas-Alvero conjecture in the non- polynomial setting do not hold. Key words: polynomial, Casas-Alvero conjecture, zeroes of functions. AMS Subject Class. (2010): Primary: 30C15. Secondary: 12D10, 13P15, 26C10. 1. Introduction The Casas-Alvero conjecture affirms that if a complex polynomial P of degree n > 1 shares roots with all its derivatives, P (k) , k = 1, 2, . . . , n − 1, then there exist two complex numbers, a and b 6= 0, such that P (z) = b(z − a)n . Notice that, in principle, the common root between P and each P (k) might depend on k. Casas-Alvero arrived to this problem at the turn of this century, when he was working in his paper [1] trying to obtain an irreducibility criterion for two variable power series with complex coefficients. See [2] for an explanation of the problem in his own words. Although several authors have got partial answers, to the best of our knowl- edge the conjecture remains open. For n ≤ 4 the conjecture is a simple con- sequence of the wonderful Gauss-Lucas Theorem ([6]). In 2006 it was proved in [5], by using Maple, that it is true for n ≤ 8. Afterwards in [6, 7] it was proved that it holds when n is pm , 2pm , 3pm or 4pm , for some prime number p and m ∈ N. The first cases left open are those where n = 24, 28 or 30. See again [6] for a very interesting survey or [3, 8] for some recent contributions on this question. Adding the hypotheses that P is a real polynomial and all its n roots, taking into account their multiplicities, are real, the conjecture has a real ISSN: 0213-8743 (print), 2605-5686 (online) c The author(s) - Released under a Creative Commons Attribution License (CC BY-NC 3.0) 222 a. cima, a. gasull, f. mañosas counterpart, that also remains open. It says that P (x) = b(x − a)n for some real numbers a and b 6= 0. For this real case, the conjecture can be proved easily for n ≤ 4, simply by using Rolle’s Theorem. This tool does not suffice for n ≥ 5, see for instance [4] for more details, or next section. Also in the real case, in [6] it is proved that if the condition for one of the derivatives of P is removed, then there exist polynomials satisfying the remaining n − 2 conditions, different from b(x − a)n . The construction of some of these polynomials presented in that paper is very nice and is a consequence of the Brouwer’s fixed point Theorem in a suitable context. Finally, it is known that if the conjecture holds in C, then it is true over all fields of characteristic 0. On the other hand, it is not true over all fields of characteristic p, see again [7]. For instance, consider P (x) = x2 (x2 + 1) in characteristic 5 with roots 0, 0, 2 and 3. Then P 0 (x) = 2x(2x2 + 1), P 00 (x) = 12x2 + 2 = 2(x2 + 1) and P 000 (x) = 4x and all them share roots with P . The aim of this note is to present two natural extensions of the real Casas- Alvero conjecture to smooth functions and show that none of them holds. Question 1. Fix 1 < n ∈ N. Let F be a class C n real function such that F (n) (x) 6= 0 for all x ∈ R, and has n real zeroes, taking into account their multiplicities. Assume that F shares zeroes with all its derivatives, F (k) , k = 1, 2, . . . , n − 1. Is it true that F (x) = b(f (x))n for some 0 6= b ∈ R and some f , a class C n real function, that has exactly one simple zero? Notice that one of the hypotheses of the real Casas-Alvero conjecture can be reformulated as follows: The polynomial F shares roots with all its deriva- tives but one, precisely the one corresponding to its degree. Trivially, this is so, because all the derivatives of order higher than n are identically zero. The second question that we consider is: Question 2. Fix 1 < n ∈ N. Let F be a real analytic function that shares zeroes with all its derivatives but one, say F (n) . Is it true that F (x) = b(f (x))n for some 0 6= b ∈ R and some real analytic function f , that has exactly one simple zero? Theorem A. (i) The answer to the Question 1 is “yes” for n ≤ 4 and “no” for n = 5. (ii) The answer to the Question 2 is already “no” for n = 2. Our result reinforces the intuitive idea that Casas-Alvero conjecture is mainly a question related with the rigid structure of the polynomials. extensions of casas-alvero conjecture 223 2. Proof of Theorem A (i) The answer to Question 1 is “yes” for n = 2, 3, 4 because the proof of the real Casas-Alvero conjecture for the same values of n, based on the Rolle’s Theorem and given in [4], does not uses at all that P is a polynomial. Let us adapt it to our setting. Since F (n) does not vanish we know that F has exactly n real zeroes, taking into account their multiplicites. Moreover we know that F has to have at least a double zero, that without loss of generality can be taken as 0. Next we can do a case by case study to discard all situations except that F has only a zero and it is of multiplicity n. For the sake of brevity, we give all the details only in the most difficult case, n = 4. Assume, to arrive to a contradiction, that n = 4, F is under the hypotheses of Question 1 and x = 0 is not a zero of multiplicity four. Notice that by Rolle’s theorem, for k = 1, 2, 3, each F (k) has exactly 4−k zeroes, taking into account their multiplicities. Moreover, the only zero of F 000 must be one of the zeroes of F . If F 00 (0) = 0 and F 000 (0) 6= 0 then F has only another zero at x = a and, without loss of generality, we can assume that a > 0. Applying three times Rolle’s theorem we get that F 000 (b) = 0 for some b ∈ (0, a) which is a contradiction with the hypotheses. If F 00 (0) 6= 0 then F has two more zeroes counting multiplicities. There are three possibilities. The first one is that there is a > 0 such that F (a) = F 0 (a) = 0. In this case, applying two times Rolle’s theorem we obtain that there exist b, c ∈ (0, a) with F 00 (b) = F 00 (c) = 0 and they are the only zeroes of F 00 . This fact gives again a contradiction because none of them is a zero of F . The second one is that there exist a1 , a2 ∈ R with 0 ∈ (a1 , a2 ) such that F (a1 ) = F (a2 ) = 0. Also in this case, by applying two times Rolle’s theorem we obtain that there exist b, c ∈ (a1 , a2 ) such that 0 ∈ (b, c) and F 00 (b) = F 00 (c) = 0 giving us the desired contradiction. Lastly, assume that the other two zeroes of F are a1 and a2 , with 0 < a1 < a2 . By Rolle’s Theorem the zeroes of F 0 are 0, b1 and b2 and satisfy 0 < b1 < a1 < b2 < a2 . Then, since F 00 has to have two zeroes, say c1 , c2 , and they satisfy 0 < c1 < b1 < c2 < b2 , the hypotheses force that c2 = a1 . Hence the zero of F 000 has to be between c1 and c2 = a1 , that is in particular in (0, a1 ), interval that contains no zero of F , arriving once more to the desired contradiction. In short, we have proved for n ≤ 4, that F (x) = xn G(x), for some class C n function G, that does not vanish. Hence s !n G(x) F (x) = sign(G(0)) x n = b(f (x))n , sign(G(0)) 224 a. cima, a. gasull, f. mañosas where f has only one zero, x = 0, that is simple, as we wanted to prove. To find a map F for which the answer to Question 1 is “no” we consider n = 5 and a configuration of zeroes of F and its derivatives proposed in [4] as the simplest one, compatible with the hypotheses of the Casas-Alvero conjecture and Rolle’s Theorem. Specifically, we will search for a function F , of class at least C 5 , with the five zeroes 0, 0, 1, c, d, to be fixed, satisfying 0 < 1 < c < d, and moreover F 0 (0) = 0, F 00 (1) = 0, F 000 (c) = 0, F (4) (1) = 0, (2.1) and such that F (5) does not vanish. Notice that F 0 (0) = 0 is not a new restriction. We start assuming that F (5) (x) = r − sin(x), for some r > 1 to be deter- mined. By imposing that conditions (2.1) hold, together with F (0) = 0, we get that Z xZ uZ wZ zZ y  F (x) = r − sin(t) dt dy dz dw du. 0 0 1 c 1 Some straightforward computations give that r 5 r + cos(1) 4 2rc − 2 sin(c) + 2 cos(1)c − rc2 3 F (x) = x − x + x 120 12 12 6 sin(c) + 2r + 9 cos(1) − 6rc + 3rc2 − 6 cos(1)c 2 + x − 1 + cos(x). 12 Imposing now that F (1) = 0 we obtain that  5 8 cos(1)c − 41 cos(1) − 8 sin(c) + 24 r= = R(c). 4(5c2 − 10c + 4) Next we have to impose that F (c) = 0. By replacing the above expression of r in F we obtain that G(c) F (c) = , 96(5c2 − 10c + 4) where G(c) = − c2 12 c4 − 369 c3 + 1437 c2 − 1708 c + 532 cos (1)  − 8 c2 (c − 1) (c − 2)2 sin (c) + 480 c2 − 960 c + 384 cos (c)  − 24 (c − 1) 9 c4 − 36 c3 + 24 c2 + 24 c − 16 .  extensions of casas-alvero conjecture 225 A carefully study shows that G has exactly one real zero c1 ∈ (17/10, 19/10) = I, with c1 ≈ 1.79343096. To prove its existence it suffices to show that     17 99211099 18207 17 G =− cos (1) − sin 10 500000 12500 10   696 17 1583211 + cos + > 0, 5 10 12500     19 180110481 3249 19 G =− cos (1) − sin 10 500000 12500 10   1464 19 3616677 + cos + < 0. 5 10 12500 By using Taylor’s formula we know that for any c > 0, S − (c) < sin(c) < S + (c) and C − (c) < cos(c) < C + (c) where c3 c5 c7 c9 c11 S ± (c) = c − + − + ± 3! 5! 7! 9! 11! and c2 c4 c6 c8 c10 C ± (c) = 1 − + − + ± . 2! 4! 6! 8! 10! Hence we can replace the values of the trigonometric functions in G by rational numbers to have upper or lower bounds of this function evaluated at 1, 17/10 or 19/10. For instance, 1960649 280093 0.5403023 ≈ = C − (1) < cos(1) < C + (1) = ≈ 0.5403028. 3628800 518400 We obtain that       17 99211099 + 18207 + 17 696 − 17 G >− C (1) − S + C 10 500000 12500 10 5 10 1583211 3444600099561969856969 + = >0 12500 49896000000000000000000 and       19 180110481 − 3249 − 19 1464 + 19 G <− C (1) − S + C 10 500000 12500 10 5 10 3616677 1689627895469649855823 + =− < 0. 12500 16632000000000000000000 226 a. cima, a. gasull, f. mañosas To show the uniqueness of the zero in I, we will prove that G is strictly decreasing in this interval. It holds that G0 (c) = T (c) cos (1) + U (c) sin (c) + V (c cos (c) + W (c), with T (c) = − c 72 c4 − 1845 c3 + 5748 c2 − 5124 c + 1064 ,  U (c) = − 8 5 c2 − 10 c + 4 c2 − 2 c + 12 ,   V (c) = − 8 (c − 1) c4 − 4 c3 + 4 c2 − 120 ,  W (c) = − 120(9c4 − 36c3 + 36c2 − 8). By computing the Sturm sequences of T, U and V we can prove that T (c) < 0, U (c) < 0 and V (c) > 0 for all c ∈ I. Hence, for c ∈ I, G0 (c) < T (c)C − (c) + U (c)S − (c) + V (c)C + (c) + W (c) = Q(c), where 72469 669211 2 18852329 3 8854991 4 Q(c) = c− c + c − c 64800 43200 302400 80640 4732471 5 532 6 8 7 191 8 + c − c + c + c 50400 15 7 70 797 9 34 10 1651 11 3533 12 − c − c + c + c 1890 405 103950 2494800 193 13 1 1 − c + c14 − c15 . 623700 142560 831600 The Sturm sequence of Q shows that it has no zeroes in I. Moreover, it is negative in this interval, and as a consequence, G0 is also negative, as we wanted to prove. We fix c = c1 . Then, r = R(c1 ) and F is also totally fixed. Moreover, by using the same techniques we get that r = R(c1 ) > R(19/10) > 1 and as a consequence F (5) does not vanish. In fact, r = R(c1 ) ≈ 1.04591089. Finally, F has one more real zero d ∈ (33/10, 34/10). In fact, d ≈ 3.32178369. This F gives our desired example, see Figure 1. (ii) Consider F (x) = 4x2 + π 2 (cos(x) − 1) that has a double zero at 0 and also vanishes at ±π/2. Moreover, F 0 (x) = 8x − π 2 sin(x) vanishes at x = 0, F 00 (x) = 8 − π 2 cos(x) has no common zeroes with F and, for any k > 1, extensions of casas-alvero conjecture 227 |F (2k) (x)| = π 2 | cos(x)| vanishes at x = π/2 and |F (2k−1) (x)| = π 2 | sin(x)| vanishes at x = 0. A similar example for n = 3 is F (x) = 4x3 − 6πx2 + π 3 (1 − cos(x)), that vanishes at 0, π (double zeroes) and π/2. Figure 1: Plot of a map F for which the answer to Question 1 for n = 5 is “no”. Acknowledgements The authors are supported by Ministerio de Ciencia, Innovación y Universidades of the Spanish Government through grants MTM2016- 77278-P (MINECO/AEI/FEDER, UE, first and second authors) and MTM2017-86795-C3-1-P (third author). The three authors are also supported by the grant 2017-SGR-1617 from AGAUR, Generalitat de Catalunya. References [1] E. Casas-Alvero, Higher order polar germs, J. Algebra 240 (2001), 326 – 337. [2] Interview to E. 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