The Mahler measure of trinomials of height 1 Valérie Flammang To cite this version: Valérie Flammang. The Mahler measure of trinomials of height 1. Journal of the Australian Mathematical Society 14 9 pp.1-4. <1.117/S14478871>. <hal-1597> HAL Id: hal-1597 https://hal.archives-ouvertes.fr/hal-1597 Submitted on 7 Nov 15 HAL is a multi-disciplinary open access archive for the deposit dissemination of scientific research documents whether they are published or not. The documents may come from teaching research institutions in France or abroad or from public or private research centers. L archive ouverte pluridisciplinaire HAL est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche publiés ou non émanant des établissements d enseignement et de recherche français ou étrangers des laboratoires publics ou privés.
THE MAHLER MEASURE OF TRINOMIALS OF HEIGHT 1 V. FLAMMANG In Memory Of My Mother Abstract We study the Mahler measure of the trinomials n ± k ± 1.We give two criteria to identify those whose Mahler measure is less than 1.815... =M1 + 1 +. We prove that these criteria are true for n sufficiently large. 1 Introduction The Mahler measure of a polynomial P = a n +... + a n = a as defined by D. H. Lehmer [L] in 19 is MP = a n max1 α j. In 19 K. Mahler [Ma] gave the following definition 1 MP = exp log P e it dt which is equivalent to Lehmer s definition by Jensen s formula [J] 1 j=1 log e it α dt = log max1 α. n j=1 α j C[X] a The polynomial P is reciprocal if n P 1/ = P an algebraic number is reciprocal if its minimal polynomial is reciprocal. C.J. Smyth [Sm1] has proved that if the algebraic number α 1 is nonreciprocal then Mα θ where θ = 1.4717... is the smallest Pisot number which is the real root of the polynomial 1. Concerning the Mahler measures of reciprocal polynomials D. Boyd [B1] [B] computed all irreducible noncyclotomic integer polynomials P with degree D having MP < 1.. M.J. Mossinghoff [M] using the same algorithm extended the computation to D 4. The author G. Rhin J-M Sac-Epée [FRSE] employed a new method that uses a large family of explicit auxiliary functions to produce improved bounds on the coefficients of polynomials with small Mahler measure determined all irreducible polynomials P with MP < θ D polynomials P with MP < 1.1 D = 8 or 4. More recently M.J. Mossinghoff G. Rhin Q. Wu [MRW] computed all primitive irreducible noncyclotomic polynomials P with degree at most 44 MP < 1.. C.J. Smyth [Sm] has also shown that θ is an isolated point in the spectrum of Mahler measures of nonreciprocal algebraic integers. We have done an exhaustive search of nonreciprocal polynomials of height 1 Mahler measure less than 1.815... up to degree 1. We observed that the smallest points of the spectrum are either trinomials of the type P = n ± k ± 1 or their irreducible factors. As the number of nonreciprocal polynomials grows quickly we have studied then trinomials of height 1 with Mahler measure less than 1.815... from degree 1 up to degree. They are either irreducible give a new point of the spectrum or are divisible by +. In the latter case their quotient gives a new point of the spectrum. The results of these computations are in the Appendix. For example the six smallest known points are more points of the spectrum are given in the Appendix: 1
1.4717... = M 1 1.4971... = M 5 4 + = M 7 + +1 ++1 1.59914... = M 5 + = M 8 ++1 ++1 1.4199... = M 5 1.7854... = M 9 8 + 5 + = M 11 + 4 +1 ++1 1.7... = M 9 8 + 5 + = M 11 ++1 ++1 The four first points were found by D. Boyd. D. Boyd M. Mossinghoff [BM] have computed a set of 48 small Mahler measures of twovariable polynomials less than 1.7. These measures are limit points of Mahler measures of one-variable polynomials. This set contains the five first points of our list. For any trinomial P we have MP = M±P = M n P 1/ = MP l for all integer l so we can assume that gcdn k = 1 k < n/ we can restrict the trinomials n ± k ± 1 to three families: 1 n + k n k with n odd n k 1 with n even. Put λ = M 1 +. C.J. Smyth proved that λ = 1.815... for more details see [B]. C.J. Smyth private communication claims that the following conjecture is true: Conjecture 1. 1 M n + k < λ iff divides n + k M n k < λ with n odd iff does not divide n + k M n k 1 < λ with n even iff does not divide n + k.. This criterion generalies a result of D. Boyd. In [B] he has shown the criterion for the first family when k is equal to 1 n is sufficiently large. A first generaliation for the first family has been done by W. Duke in [D] in 7. He showed: for < k < n with n k = 1 log Mx n + x k cn k k = log Mx + y + n + where cn k = / if divides n + k cn k = /18 otherwise. Then in 1 J. Condon [Co] has studied the quantities µ n P = MP x x n MP x y for a large set of bivariate polynomials but uniquely irreducible. He has obtained some considerably more general results. However some polynomials of our families are reducible. So the results that we need are easier to obtain by following Duke s proof because it does not depend on the factoriation of the polynomial. Although Condon s formula is more precise it is not feasible in practice to use it to obtain a general formula for polynomials of our families whose factoriation depends on n k. It makes sense to recall here a result of W. Ljunggren [Lj] on the irreducibility of trinomials of height 1. He proved: if n = n 1 d k = k 1 d n 1 k 1 = 1 n k then the polynomial gx = x n + ɛx k + ɛ ɛ = ±1 ɛ = ±1 is irreducible apart from the following three cases where n 1 + k 1 mod : n 1 k 1 both odd ɛ = 1; n 1 even ɛ = 1; k 1 even ɛ = ɛ gx then being a product of the polynomial a second irreducible polynomial. x d + ɛ k ɛ n x d In Section we establish Conjecture 1 when n is sufficiently large relative to k for the second family of trinomials n k. Sectio deals with the third family. We give the main elements of the proof that differ from those of Section. In Section 4 we give a second criterion equivalent to the first one involving resultants of trinomials of the three families with some cyclotomic polynomials.
Proof of Conjecture 1 for n large We prove the following result: Theorem 1. 1. For the second family of trinomials we have: log M n k cn k k = log M 1 + + n + where cn k = / if does not divide n + k cn k = /1 otherwise.. For the third family of trinomials we have: log M n k 1 = log M 1 + + cn k k n + where cn k = /18 if does not divide n + k cn k = / otherwise. The constants involved in O are effective. The proof follows the same scheme as Duke s one in [D]. Corollary 1. There exists a computable constant c such that if n > c k then M n k ± 1 M 1 + < if does not divide n + k > otherwise. We choose to present first the proof in detail for the second family of trinomials n k. k 1 + l Let = e it 5 + l for t. When t belongs to i.e. 1 k > 1 we have l= log n k = log 1 k + 1 m 1 n m m 1 k m 1 when t belongs to k 5 + k 1 5 + l l= 1 we have 7 + l i.e. 1 k < log n k = log n + 1 m 1 1 k m m n. m 1 Put λ nk = log M n k = 1 λ = log M 1 + = 1 log n k dt log + dt. Putting u = tk we have k 1 l= 5+l +l log 1 e itk dt = 1 k k 1 l= 5 +l = +l 5 log 1 e iu du = log 1 e iu du = log + 1 e iu du = log + 1 + e iu du.
Hence where λ nk λ = 1 Re m 1 k 1 c 1 m = l= 5+l +l 1 m 1 m c 1m + c m e inmt 1 e itk m dt c m = k e inmt 1 e itk m dt+ l= 7+l 5+l Put u = tk then c 1 m = 1 k 1 5 +l e inmu k 1 e iu m du k l= +l = 1 k 1 1+l e inmu k l= +l = 1 k 1 e lnm k k l= Thus Re c 1 m = Re 5 k 1 e iu m du + e inmt 1 e itk m dt+ e inmt 1 e itk m dt. 5+k 1 e inmu k 1 e iu m du By the same argument we obtain Re c m = Re Re c m = otherwise. +l 1+l. e inmu k 1 e iu m du e inmt k 1 e it m dt if k divides m Re c 1 m = otherwise. e inmt k 1 e it m dt if k divides m Put m = kq. In order to estimate c 1 kq + c kq we need to integrate by parts three times the integrals e inqt 1 e it kq dt e inqt 1 e it kq dt. We obtain for each integral four types of terms that we have to study. [ e inqt 1 e it kq The first type of term is inq ] [ e inqt 1 e it kq inq ]. It is easy to see that the sum of such terms is not real thus does not occur in Rec 1 kq + c kq. [ ] [ kqkq e itnq e it kq The second type of term is inqnq nq + These terms are in modulus K 1k. Now we have to estimate the modulus of the third type of term coming from the integration of c 1 kq. In the integral I 1 = Thus I 1 = kqkq kq + I = e itnq e it kq dt nqnq nq + dv sin v kq+. 1 e it kq dt put v = t. kqkq 1e itnq 1 e it kq inqnq 1nq ]. 4
For any v [ ] 1 sin v v + 5 4 so that I 1 v + 5 kq+ dv 4 Finally we get I K k. In the same way we have to estimate the modulus of due to the integration of c kq. For q > in the integral J 1 = Thus J 1 = 4 q + 4. kqkq 1kq J = e itnq 1 e it kq dt nqnq 1nq sin v kq dv. For any v [ ] sin v v Thus we get J K J 1 for m 1. 1 e it kq dt put v = t. so that kq v dv kq. Finally the only terms that occur in Rec 1 kq + c kq are those which contain in c 1 kq We obtain kq inqnq 1 in c kq. kq inqnq Rec 1 kq + c kq = Re kq in q e i nq+1 1 e i kq 1 + e i nq 1 1 e i kq 1 k + = kq k n q cos qn + k + Hence i. e. λ nk λ = λ nk λ = n q 1 1 qk 1 cos qn + k k q + k n + k 1n + if does not divide n + k if divides n + k 5
The family n k 1 Here we have λ nk = logm n k 1 = 1 log n k 1 dt = 1 1 dt = logm n + k so we work with log n + k. If t belongs to if t belongs to Hence 4 + k 1l k 1 + l l= λ nk λ = 1 Re m 1 where c 1 m = 4 + l 1 m 1 m c 1m + c m k e inmt 1+e itk m dt+ l= k 1 c m = 8+l l= 4+l k 4 + l l= then 1 + e itk < 1. 4+l +l log n + k + 8 + l then 1 + e itk > 1 e inmt 1+e itk m dt+ e inmt 1+e itk m dt 4+k 1 e inmt 1 + e itk m dt. By the same argument as in Section 1 we obtain Re c 1 m = Re divides m Re c 1 m = otherwise Re c m = Re m Re c m = otherwise. e inmt/k 1 + e it m if k e inmt/k 1+e it m dt if k divides As before we put m = kq. We integrate three times by parts keep only the terms with We get Rec 1 kq + c kq = 1kq kq n q cos qn k i.e. λ nk λ = n q + Therefore q 1 λ nk λ = cos k k 18n + k n + qn k + k. if does not divide n + k if divides n + k m in q.
4 An equivalent criterion We claim that the following conjecture is true: Conjecture. Let ɛ η be equal to ±1. Put r 1 = resultant n + ɛ k + η + r = resultant n + ɛ k + η. 1 M n + k < λ iff + divides n + k. M n k < λ with n odd iff {r 1 r } = {11} or {17}. M n k 1 < λ with n even iff {r 1 r } = {11} or {17}. In this section we prove that: Theorem. Conjectures 1 are equivalent. Proof Put j = e i. 1. divides n + k n 1 mod k mod or n mod k 1 mod j n + j k = j + j =.. We give the proof for the family n k. The argument is the same for the family n k 1. Suppose that divides n + k. If n 1 mod k mod then r 1 = j n j k = j j = 4 r = j n j k = j j = 4. If n mod k 1 mod then r 1 = j n j k = j j = 4 r = j n j k = j j = 4. Thus the situations {r 1 r } = {11} or {17} are not possible. Suppose that does not divide n + k. It is easy to see that this is equivalent to divides nkn k. If divides n then r 1 = j k j k = 5 j k + j k = 7 r = j k j k = 1. If divides k does not divide n then r 1 = j n j n = 1. When k is even r = j n j n = 1 when k is odd r = j n + j n + = 5 j n + j n = 7. If divides n k does not divide nk then r 1 = j k j n k 1 j k j n k 1 = 1. When k is even r = j k j n k j k j n k = j k j k = 5 j k + j k = 7 when k is odd r = j k j n k 1 j k j n k 1 = 1. Thus we have {r 1 r } = {11} or {17}. Acknowledgements The author wishes to thank C. J. Smyth for his communication concerning the first criterion for the reference of D. Boyd s paper G. Rhin for his precious help. The author also thanks the referee for his useful recommendations. 7
Appendix Smallest known Mahler measures of nonreciprocal polynomials with coefficients -1 1 up to degree. 1.4717 = M 1.4971 = M 5 4 + 7 + = M + 1.59914 = M 5 + 8 + = M + 1.4199 = M 5 1.7854 = M 9 8 + 5 + 11 + 4 = M + 1.7 = M 9 8 + 5 + 11 + = M + 1.7957 = M 5 1 1.711 = M 11 1 + 8 7 + 4 + 1 + 8 = M + 1.791 = M 11 1 + 8 7 + 5 4 + 1 + = M + 1.7895 = M 7 4 1.74571 = M 1 1 + 9 7 + 4 + 14 + 1 = M + + 1 1.7518 = M 14 1 + 11 1 + 9 7 + 4 + 1 + 11 = M + 1.7519 = M 15 14 + 1 11 + 9 7 + 4 + 17 + 1 = M + 1.787 = M 15 14 + 1 1 + 9 7 + 4 + 17 + 1 = M + 1.7755 = M 17 1 + 14 1 + 11 1 + 9 7 + 4 + 19 + 11 +1 = M + 1.7795 = M 15 14 + 1 11 + 9 8 + 5 + 17 + = M + 1.7759 = M 17 1 + 14 1 + 1 1 + 9 7 + 4 + 19 + 14 +1 = M + 1.778 = M 9 5 1 1.7799 = M 18 17 + 15 14 + 1 1 + 9 7 + 4 + + 1 +1 = M + 1.774 = M 9 8 1.7754 = M 17 1 + 15 1 + 1 1 + 9 7 + 4 + 19 + 17 +1 = m + 1.7794 = M 1 7 1 1.784 = M 8 5 1 1.787 = M 11 9 1 1.7878 = M 11 1.79 = M 1 11 1 1.797 = M 7 1.79458 = M 1 1 1.79545 = M 9 7 1 1.7519 = M 15 14 + 1 11 + 9 8 + 4 + = M 1.7957 = M 1 7 1 1.79 = M 1 7 1 8 17 + 7 +
1.797 = M 14 9 1 1.79849 = M 11 8 1.79954 = M 15 11 1 1.788 = M 18 1 + 15 1 + 1 1 + 9 7 + 4 + +1 = M 1.79849 = M + 1 1 1.81 = M 1 9 1 1.84 = M 15 8 1.811 = M 15 14 1.811 = M 1 9 1 1.8175 = M 1 1 1 1.877 = M 4 1 1.88 = M 17 1 1.81 = M 18 1 1.8 = M 1 9 1 1.85 = M 17 9 1 1.89 = M 17 15 1 1.8418 = M 14 11 1 1.8418 = M 18 11 1 1.84 = M 1 1 1.8598 = M 19 1 1 1.857 = M 18 17 1 1.8541 = M 17 11 1 1.8557 = M 19 1 1.8558 = M 19 1 1.859 = M 15 1 1 1.88 = M 19 1 1.884 = M 11 1 1.891 = M 18 1 1 1.877 = M 17 14 1.8719 = M 1 15 1 1.8799 = M 19 15 1 1.8879 = M 17 1 1.888 = M 19 18 + 19 + 9
References [B1] D. W. Boyd. Reciprocal polynomials having small measure Math. Comp. 5 198 n 15 11 177. [B] D. W. Boyd. Speculations concerning the range of Mahler s measure Canad. Math. Bull. 4 4 1981 45 49. [B] D. W. Boyd. Reciprocal polynomials having small measure II Math. Comp. 5 1989 n 187 55 57 S1 S5. [BM] D. W. Boyd M. J. Mossinghoff. Small limit points of Mahler s measure Exp. Math. 14 5 n 4 4 414. [Co] John D. Condon. Asymptotic expansion of the difference of two Mahler J. Number Theory 1 1 n 9 19 198. [D] W. Duke. A combinatorial problem related to Mahler measure Bull. Lond. Math. Soc. 9 7 n 5 741 748. [FRSE] V. Flammang G. Rhin J-M Sac-Epée. Integer transfinite diameter polynomials with small Mahler measure Math. Comp 75 n 55 157 154. [J] J.L.V.W. Jensen. Sur un nouvel et important théorème de la théorie des fonctions Acta Math. 1899 59 4. [L] D. H. Lehmer. Factoriation of certain cyclotomic functions Ann. of Math. 4 19 n 41 479. [Lj] W. Ljunggren.On the irreducibility of certain trinomials quadrinomials Math. Sc. 8 19 5 7. [Ma] K. Mahler. On some inequalities for polynomials in several variables J. London Math. Soc. 7 19 41 44. [M] M. J. Mossinghoff.Polynomials with small Mahler measure Math. Comp. 7 1998 n 4 197 175 S11 S14. [MRW] M. J. Mossinghoff G. Rhin Q. Wu.Minimal Mahler measures Experiment. Math. 17 8 n 4 451 458. [Sm1] C. J. Smyth.On the product of the conjugates outside the unit circle of an algebraic integer Bull. London Math. Soc. 1971 19 175. [Sm] C.J. Smyth.Topics in the theory of numbers PhD thesis Univ. of Cambridge 197. [Sm] C. J. Smyth. An inequality for polynomials Number theory Ottawa ON 199 15-1 CRM Proc. Lecture Notes 19 Amer. Math. Soc. Providence RI 1999. UMR CNRS 75 IECL Université de Lorraine Département de Mathématiques UFR MIM Ile du Saulcy 5745 METZ cedex 1 FRANCE E-mail address :valerie.flammang@univ-lorraine.fr 1