Linear recurrence sequences and periodicity of multidimensional continued fractions

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1 arxiv: v1 [math.nt] 23 Dec 2017 Liear recurrece sequeces ad periodicity of multidimesioal cotiued fractios Nadir Murru Departmet of Mathematics Uiversity of Turi Turi, Italy Abstract Multidimesioal cotiued fractios geeralize classical cotiued fractios with the aim of providig periodic represetatios of algebraic irratioalities by meas of iteger sequeces. However, there does ot exist ay algorithm that provides a periodic multidimesioal cotiued fractio whe algebraic irratioalities are give as iputs. I this paper, we provide a characterizatio for periodicity of Jacobi Perro algorithm by meas of liear recurrece sequeces. I particular, we prove that partial quotiets of a multidimesioal cotiued fractio are periodic if ad oly if umerators ad deomiators of covergets are liear recurrece sequeces, geeralizig similar results that hold for classical cotiued fractios. 1 Itroductio Terary cotiued fractios geeralize classical cotiued fractios ad ca beusedtorepreset pairofrealumbers. Iparticular,apairofrealumbers 2010 Mathematics Subject Classificatio: Primary 11J70; Secodary 11B37. Key words ad phrases: algebraic irratioalities, Jacobi Perro algorithm, liear recurrece sequeces, multidimesioal cotiued fractios. 1

2 Periodic multidimesioal cotiued fractios 2 (α 0,β 0 ) is represeted by two sequeces (a ) =0 ad (b ) =0 as follows: α 0 = a 0 + b 1 + a 1 + a 2 + b 2 + a b a a 3 + b a 3 + ad β 0 = b 0 + a b 2 + a a 3 + b a 3 +. (1.1) Partial quotiets a i adb i cabeevaluatedbymeasofthejacobialgorithm: a = [α ] b = [β ] 1 α +1 = β [β ] β +1 = α a β b, = 0,1,2,... Terary cotiued fractios ad previous algorithm were itroduced by Jacobi [J] i order to aswer to a questio posed by Hermite. Ideed, Hermite asked for fidig a way to give a periodic represetatio for cubic irratioalities, simlarly to the case of quadratic irratioalities ad cotiued fractios [Her]. This is still a fasciatig ope problem i umber theory, sice periodicity of the Jacobi algorithm has bee ever proved whe iputs are cubic irratioals. For this reaso, terary cotiued fractios have bee widely studied durig the years. Covergece ad properties of terary cotiued fractios were studied by Daus ad Lehmer i several classical works as [D1],

3 Periodic multidimesioal cotiued fractios 3 [D2], [Leh]. More recetly, terary cotiued fractios ad Jacobi algorithm have bee studied for approachig the Hermite problem. I particular several authors tried to modify the Jacobi algorithm i order to obtai a periodic algorithm whe cubic irratioalities are give as iputs, [W], [T], [S]. Perro [P] geeralized the Jacobi approach to higher dimesios, represetig a m tuple of real umbers α (1) 0,...,α (m) 0 by meas of a multidimesioal cotiued fractio defied by m sequeces of itegers (a (1) ) =0,..., (a (m) ) =0. Partial quotiets a(1),..., a (m) ca be evaluated by meas of a (i) = [α (i) ], i = 1,...,m +1 = 1 α (1) α (m) α (i) +1 = α(i 1) α (m) a (m) a (i 1) a (m) i = 2,...,m, = 0,1,2,... (1.2) Multidimesioal cotiued fractios have bee itroduced with the aim of fidig a periodic represetatio for all algebraic irratioalities. However previous algorithm (ad modificatios) has ever bee proved periodic whe algebraic irratioals of order m are give as iputs. Further results ad modificatios of the Jacobi Perro algorithm ca be foud i[b2], [R], [He]. I this paper we show properties that could ope ew ways for approachig the Hermite problem ad provig the periodicity of the Jacobi Perro algorithm. These properties coect periodicity of partial quotiets with liear recurrece sequeces. I particular we will prove that sequeces (a (i) ) =0, for i = 1,...,m, are periodic if ad oly if sequeces of umerators ad deomiators of covergets of multidimesioal cotiued fractios are recurrece sequeces with costat coefficiets, geeralizig a similar result provided by Lestra ad Shallit for classical cotiued fractios [Le]. I sectio 2, we give some prelimiary properties ad otatio about multidimesioal cotiued fractios. Sectio 3 shows the mai result ad Sectio 4 describes further developmets of the preset work.

4 Periodic multidimesioal cotiued fractios 4 2 Multidimesioal cotiued fractios I the followig, we write the multidimesioal cotiued fractio derived by equatios (1.2) as follows: (α (1) 0,...,α (m) 0 ) = [{a (1) 0,a (1) 1,a (1) 2,...},...,{a (m) 0,a (m) 1,a (m) 2,...}]. (2.1) This multidimesioal cotiued fractio geeralizes the structure of terary cotiued fractio (1.1) by meas of the followig relatios α (i 1) α (m) = a (i 1) + α(i) +1 α (1) +1 = a (m) + 1 α (1) +1, i = 2,...,m, = 0,1,2,..., which ca be derived by equatios (1.2). Let us observe that a multidimesioal cotiued fractio is fiite if ad oly if α (1) 0,...,α (m) 0 are ratioal umbers, similarly to classical cotiued fractios. I the followig, we focus o α (1) 0,...,α(m) 0 irratioal umbers. We ca itroduce the otio of covergets, like for classical ( cotiued ) (i) A fractios. I this case, we have m sequeces of covergets, for i = 1,...,m, defied by ( ) (1) A ad we have,..., A (m) lim =0 = [{a (1) 0,...,a (1) },...,{a (m) 0,...,a (m) }], 0 (2.2) A (i) = α (i) 0, i = 1,...,m. I [B1], the reader ca fid useful properties regardig covergets of multidimesioal cotiued fractios (i this paper we have used a little differet otatio). I particular, we highlight the followig properties. Propositio 2.1. Numerators ad deomiators of covergets of multidimesioal cotiued fractio (2.1) satisfy the followig recurreces of order

5 Periodic multidimesioal cotiued fractios 5 m + 1 with o costat coefficiets: A (i) = m j=1 a (j) A(i) j +A(i) m 1, i = 1,...,m+1, 1 (2.3) with iitial coditios A (i) 0 = a (i) 0, 0 = 1, A (i) i = 1, A(m+1) i = 0, i = 0, j = 1,...,m, j i A (i) j = 1,...,m. Propositio 2.2. Give the multidimesioal cotiued fractio (2.1), we have a (1) j a (2) j m m j=0 a (m 1) = j a (m) j A (m+1) 1... m ( A (1) for = 0,1,2,...,..., A (m) ) = [{a (1) 0,...,a(1) },...,{a(m) 0,...,a (m) }], Remark 2.3. Matrices i the previous propositio are (m + 1) (m + 1) matrices. Remark 2.4. I the followig, whe there is o possibility of cofusio, we oly write (a ) istead of (a ) =0. 3 Mai result I this sectio, we prove that periodicity of Jacobi Perro algorithm is strictly related to liear recurrece sequeces. We see that a multidimesioal cotiued fractio is periodic if ad oly if umerators ad deomiators of covergets are liear recurrece sequeces. I other words, we prove

6 Periodic multidimesioal cotiued fractios 6 that sequeces (a (i) ), for i = 1,...,m, are periodic if ad oly if sequeces (A (i) ), for i = 1,...,m+1, satisfy a recurrece relatio with costat coefficiets. Proofs follow ideas used i[le], but here the situatio is complicated by the presece of several sequeces of partial quotiets. Theorem 3.1. If sequeces of partial quotiets of (2.1) are periodic, the sequeces of umerators ad deomiators of its covergets are liear recurrece sequeces. Proof. Let us cosider the periodic multidimesioal cotiued fractio [{a (1) 0,...,a (1) p 1 1,b (1) 0,...,b (1) q 1 1},...,{a (m) 0,...,a (m) p m 1,b (m) 0,...,b (m) q m 1}]. We itroduce the followig quatities: u = lcm(q 1,...,q m ), v i = max(p 1,...,p m ) p i, i = 1,...,m. We defie u matrices M i = 0 j<u b (1) i+j+v b (2) i+j+v , i Z u. i+j+v m b (m) i+j+v m b (m 1) We ca easily observe that det(m i ) = ( 1) (m 1)u, furthermore tr(mi k) = tr(mj k) for all i,j Z u, k N. Ideed, give ay i,j Z u, it is always possible to fid two matrices P, Q such that M i = PQ ad M j = QP, ad cosequetly tr(m i ) = tr(m j ). Similar observatio holds for powers of matrices M i. Therefore, all matrices M i have same ivariats ad characteristic polyomial m 1 x m h i x i ( 1) (m 1)u, i=1 where h i are ivariats of these matrices.

7 Periodic multidimesioal cotiued fractios 7 By propositio 2.2 follows that +u +u u m +u +u u +u m.. = +u u m 1... m 1... m. for all max(p 1,...,p m ). Now, we ca write m 1 M u m i=1.. M u... m 1 h i M i u ( 1)(m 1)u I = O where I is the idetity matrix ad O is the zero matrix. Hece, multiplyig (o the left) both sides of previous equatio by 1... m we obtai +mu +mu 1... m. +mu mu mu m +mu m... +mu +mu mu m Thus, we have, e.g.,..... m 1 m 1 i=1 h i ( 1) (m 1)u +iu +iu iu +iu iu +iu m +iu m... +iu m 1... m. m 1 +mu h i +iu ( 1)m 1 = 0 i=1 +iu m +.. = O.... m 1 ad similar relatios for sequeces (A (i) ), for i = 2,...,m+1, i.e., sequeces (A (i) ) are all liear recurrece sequeces.

8 Periodic multidimesioal cotiued fractios 8 We eed some lemmas for provig the vice versa. Next lemma is kow as the Hadamard quotiet theorem ad it is due to Va der Poorte [V]. Lemma 3.2. Let i=0 a ix i ad i=0 b ix i be formal series represetig ra- a i i=0 x i is a b i tioal fuctios i C[x], if a i is a iteger for all i 0, the b i formal series represetig a ratioal fuctio. Remark 3.3. We recall that the geeratig fuctio of a liear recurrece sequece is a ratioal fuctio. We also eed some results about sums ad products of liear recurrece sequeces. The reader ca fid a iterestig overview o this topic i [C]. I the ext lemma, we summarize some results. Lemma 3.4. If (a ) ad (b ) are liear recurrece sequeces with characteristic polyomials p(x) ad q(x), the (a + b ) ad (a b ) are also liear recurrece sequeces whose characteristic polyomials are p(x) q(x) ad p(x) q(x), respectively, where p(x) q(x) is the characteristic polyomial of the matrix obtaied by the Kroecker product betwee compaio matrices of p(x) ad q(x). The followig lemma has bee also used i [Le]. Lemma 3.5. Let (a ) ad (b ) be liear recurrece sequeces such that characteristic polyomial of (b ) divides characteristic polyomial (of degree e) of (a ). The, there exists costats k ad 0 such that for ay 0. max( a, a 1,..., a e+2, a e+1 ) > k b, Fially, we prove the followig lemma. Lemma 3.6. With the above otatio, we have A (i) a (1) j, i = 1,...,m+1 for 1. j=1

9 Periodic multidimesioal cotiued fractios 9 Proof. The thesis ca easily proved by iductio. Let us observe that, by equatios (1.2), we have a (i) > 0, for i = 1,...,m ad 1. The basis is straightforward: For the iductive step, we have m A (m+1) = j=1 1 = a (1) 1, A(m+1) 2 = a (1) 2 a(1) 1 +a (2) 2. 1 a (j) A(m+1) j + m 1 a(1) j=1 a (1) j + m j=2 a (j) A(m+1) j + m 1. Sice i, for i = 1,2,..., are cocordat, the thesis follows. Similar cosideratios hold for sequeces (A (i) ), for i = 1,...,m. ( ) (i) A Theorem 3.7. Let be the sequeces of covergets, for i = 1,...,m, of the multidimesioal cotiued fractio (2.1). If (A (i) ), for i = 1,...,m+1, are liear recurrece sequeces, the (a (i) ), for i = 1,...,m, are periodic sequeces. Proof. For sufficietly large A (i) 0, for i = 1,...,m+1, ad by equatios (2.3), with a little bit of calculatio, it is possible to obtai a (m) = F G, where F ad G are sums ad products of some A (i) j, for certai values of i ad j. Sice (A (i) ) are all liear recurrece sequeces, by lemma 3.4, (F ) ad (G ) are liear recurrece sequeces. Moreover, (F ) ad (G ) satisfy hypothesis of lemma 3.2 ad cosequetly (a (m) ) is a liear recurrece sequece. Similarly, we ca prove that (a (i) ), for i = 1,...,m 1 are liear recurrece sequeces. Now, we prove that sequece (a (1) ) is bouded. Let us suppose that characteristic polyomial of the liear recurrece sequece (a (1) ) has degree e. By the Biet formula we ca write a (1) = e p i ()αi, = 0,1,2,... (3.1) i=1

10 Periodic multidimesioal cotiued fractios 10 where α i s arerootsof the characteristic polyomial adp i () polyomials of degree t i i, for i = 1,...,e. By lemma 3.6 we have A qe (m+1) 1 j qe a(1) j, for q 1. Moreover, for a give idex i, sequeces (a (1) ) ad ( t i αi ) satisfy hypothesis of lemma 3.5. Hecebylemma3.5,fixed1 h m, wehavemsequecesa (1) (h 1)e+1,...,a(1) he, such that the maximum elemet is greater tha k(he) t i αi e h. Said k a costat greater tha the product of all other elemets, we have qe > k k m (m!) t i e mt i α e i m(m+1)/2. A similar iequality was fid i [Le] for deomiators of covergets of classical cotiued fractios. Sice (A (m+1) ) is a liear recurrece sequece, it ca ot growtoo rapidly. Ideed, it is well kow that loga (m+1) = O() ad cosequetly α i < 1 or α i = 1 ad t i = 0. Thus, by equatio (3.1), we have that (a (1) ) is bouded. Sice (a (1) ) is a bouded liear recurrece sequece of itegers, we have proved that (a (1) ) is periodic. Fially, by equatios (1.2) we have that if (a (1) ) is periodic the sequeces (a (i) ), for i = 2,...,m, are periodic. 4 Future developmets We have foud a characterizatio of periodicity for the Jacobi Perro algorithm by meas of liear recurrece sequeces. This result could be exploited for approachig the Hermite problem by ew ways. I [M], the author foud a periodic represetatio for ay cubic irratioal usig a terary cotiued fractio with ratioal partial quotiets, whose covergets have umerators ad deomiators that are liear recurrece sequeces. However, this represetatio depeds o the kowledge of the miimal polyomial of the ivolved cubic irratioal ad cosequetly oe algorithm ca be foud for geeratig such a represetatio. I particular, give the cubic irratioal α, whose miimal polyomial is x 3 px 2 qx r,

11 Periodic multidimesioal cotiued fractios 11 we have [{z, 2z +p2 +q, pq +r (pq +r)tr(n),tr(n), tr(n) det(n) pq +r }, {p, z2 +qz +p 2 z pr, I 1(N) pq +r det(n), (pq +r)i 1(N), I 1(N) det(n) pq +r }] = (r α,α) (4.1) where I 1 is the secod ivariat of a 3 3 matrix. Moreover, z r pr N = 0 q +z pq+r 1 p p 2 +q +z for a give iteger z. Let us defie [{a 0,a 1,...},{b 0,b 1,...}] = ( r,α) (4.2) α the terary cotiued fractio whose partial quotiets are derived by the Jacobi algorithm. Sice terary cotiued fractios (4.1) ad (4.2) are equal, it could be iterestig to compare covergets of (4.1) with covergets of (4.2) with the aim of provig that umerators ad deomiators of covergets of (4.2) are liear recurrece sequeces too. 5 Ackowledgmets I would like to thak Prof. Umberto Cerruti for itroducig me to these beautiful problems. Refereces [B1] L. Berstei, The Jacobi Perro algorithm its theory ad applicatio, Lectures Notes i Mathematics 207, (1971). [B2] L. Berstei, Uits from periodic Jacobi Perro algorithms i algebraic umber fields of degree > 2, Mauscripta Mathematica 14(3), , (1974).

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