Lehmer s problem for polynomials with odd coefficients

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1 Aals of Mathematics, 166 (2007), Lehmer s problem for polyomials with odd coefficiets By Peter Borwei, Edward Dobrowolski, ad Michael J. Mossighoff* Abstract We prove that if f(x) = 1 k=0 a kx k is a polyomial with o cyclotomic factors whose coefficiets satisfy a k 1 mod 2 for 0 k<, the Mahler s measure of f satisfies log M(f) log 5 ( 1 1 ). 4 This resolves a problem of D. H. Lehmer [12] for the class of polyomials with odd coefficiets. We also prove that if f has odd coefficiets, degree 1, ad at least oe ocyclotomic factor, the at least oe root α of f satisfies α > 1+ log 3 2, resolvig a cojecture of Schizel ad Zassehaus [21] for this class of polyomials. More geerally, we solve the problems of Lehmer ad Schizel ad Zassehaus for the class of polyomials where each coefficiet satisfies a k 1 mod m for a fixed iteger m 2. We also characterize the polyomials that appear as the ocyclotomic part of a polyomial whose coefficiets satisfy a k 1modp for each k, for a fixed prime p. Last, we prove that the smallest Pisot umber whose miimal polyomial has odd coefficiets is a limit poit, from both sides, of Salem [19] umbers whose miimal polyomials have coefficiets i { 1, 1}. 1. Itroductio Mahler s measure of a polyomial f, deoted M(f), is defied as the product of the absolute values of those roots of f that lie outside the uit disk, multiplied by the absolute value of the leadig coefficiet. Writig f(x) = *The first author was supported i part by NSERC of Caada ad MITACS. The authors thak the Baff Iteratioal Research Statio for hostig the workshop o The may aspects of Mahler s measure, where this research bega.

2 348 P. BORWEIN, E. DOBROWOLSKI, AND M. J. MOSSINGHOFF a d k=1 (x α k), we have (1.1) d M(f) = a max{1, α k }. k=1 For f Z[x], clearly M(f) 1, ad by a classical theorem of Kroecker, M(f) = 1 precisely whe f(x) is a product of cyclotomic polyomials ad the moomial x. I 1933, D. H. Lehmer [12] asked if for every ε>0 there exists a polyomial f Z[x] satisfyig 1 < M(f) < 1+ε. This is kow as Lehmer s problem. Lehmer oted that the polyomial l(x) =x 10 + x 9 x 7 x 6 x 5 x 4 x 3 + x +1 has M(l) = , ad this value remais the smallest kow measure larger tha 1 of a polyomial with iteger coefficiets. Let f deote the reciprocal polyomial of f, defied by f (x) = x deg f f(1/x); it is easy to verify that M(f )=M(f). We say a polyomial f is reciprocal if f = ±f. Lehmer s problem has bee solved for several special classes of polyomials. For example, Smyth [22] showed that if f Z[x] is oreciprocal ad f(0) 0, the M(f) M(x 3 x 1) = Also, Schizel [20] proved that if f is a moic, iteger polyomial with degree d satisfyig f(0) = ±1 ad f(±1) 0, ad all roots of f are real, the M(f) γ d/2, where γ deotes the golde ratio, γ =(1+ 5)/2. I additio, Amoroso ad Dvoricich [1] showed that if f is a irreducible, ocyclotomic polyomial of degree d whose splittig field is a abelia extesio of Q, the M(f) 5 d/12. The best geeral lower boud for Mahler s measure of a irreducible, ocyclotomic polyomial f Z[x] with degree d has the form ( ) log log d 3 log M(f) ; log d see [6] or [8]. I this paper, we solve Lehmer s problem for aother class of polyomials. Let D m deote the set of polyomials whose coefficiets are all cogruet to 1 mod m, { d } (1.2) D m = a k x k Z[x] :a k 1modm for 0 k d. k=0 The set D 2 thus cotais the set of Littlewood polyomials, defied as those polyomials f whose coefficiets a k satisfy a k = ±1 for 0 k deg f. We prove i Corollaries 3.4 ad 3.5 of Theorem 3.3 that if f D m has degree 1

3 LEHMER S PROBLEM FOR POLYNOMIALS WITH ODD COEFFICIENTS 349 ad o cyclotomic factors, the ( log M(f) c m 1 1 ), with c 2 = (log 5)/4 ad c m = log( m 2 +1/2) for m>2. We provide i Theorem 2.4 a characterizatio of polyomials f Z[x] for which there exists a polyomial F D p with f F ad M(f) =M(F ), where p is a prime umber. The proof i fact specifies a explicit costructio for such a polyomial F whe it exists. I [21], Schizel ad Zassehaus cojectured that there exists a costat c>0 such that for ay moic, irreducible polyomial f of degree d, there exists arootα of f satisfyig α > 1 + c/d. Certaily, solvig Lehmer s problem resolves this cojecture as well: If M(f) M 0 for every member f of a class of moic, irreducible polyomials, the it is easy to see that the cojecture of Schizel ad Zassehaus holds for this class with c = log M 0. We prove some further results o this cojecture for polyomials i D m. I Theorem 5.1, we show that if f D m is moic with degree 1 ad M(f) > 1, the there exists arootα of f satisfyig α > 1+c m /, with c 2 = log 3 ad c m = log(m 1) for m>2. We also prove (Theorem 5.3) that oe caot replace the costat c m i this result with ay umber larger tha log(2m 1). Recall that a Pisot umber is a real algebraic iteger α>1, all of whose cojugates lie iside the ope uit disk, ad a Salem umber is a real algebraic iteger α>1, all of whose cojugates lie iside the closed uit disk, with at least oe cojugate o the uit circle. (I fact, all the cojugates of a Salem umber except its reciprocal lie o the uit circle.) I Theorem 6.1, we obtai a lower boud o a Salem umber whose miimal polyomial lies i D 2. This boud is slightly stroger tha that obtaied from our boud o Mahler s measure of a polyomial i this set. The smallest Pisot umber is the miimal value of Mahler s measure of a oreciprocal polyomial, M(x 3 x 1) = I [4], it is show that the smallest measure of a oreciprocal polyomial i D 2 is the golde ratio, M(x 2 x 1) = γ, ad therefore this value is the smallest Pisot umber whose miimal polyomial lies i D 2. Salem [19] proved that every Pisot umber is a limit poit, from both sides, of Salem umbers. We prove i Theorem 6.2 that the golde ratio is i fact a limit poit, from both sides, of Salem umbers whose miimal polyomials are also i D 2 ; i fact, they are Littlewood polyomials. This paper is orgaized as follows. Sectio 2 obtais some prelimiary results o factors of cyclotomic polyomials modulo a prime, ad describes factors of polyomials i D p. Sectio 3 derives our results o Lehmer s problem for polyomials i D m. The method here requires the use of a auxiliary polyomial, ad Sectio 4 describes two methods for searchig for favorable auxiliary polyomials i a particularly promisig family. Sectio 5 proves our

4 350 P. BORWEIN, E. DOBROWOLSKI, AND M. J. MOSSINGHOFF bouds i the problem of Schizel ad Zassehaus for polyomials i D m, ad Sectio 6 cotais our results o Salem umbers whose miimal polyomials are i D 2. Throughout this paper, the th cyclotomic polyomial is deoted by Φ. Also, for a polyomial f(x) = d k=0 a kx k, the legth of f, deoted L(f), is defied as the sum of the absolute values of the coefficiets of f, (1.3) L(f) = d a k, k=0 ad f deotes the supremum of f(x) over the uit circle. 2. Factors of polyomials i D p Let p be a prime umber. We describe some facts about factors of cyclotomic polyomials modulo p, ad the prove some results about cyclotomic ad ocyclotomic parts of polyomials whose coefficiets are all cogruet to1modp. We begi by recordig a factorizatio of the biomial x 1 modulo p. Lemma 2.1. Suppose p is a prime umber, ad = p k m with p m. The x 1 d m Φ pk d (x) mod p. Proof. Usig the stadard formula Φ (x) = ( d x d 1 ) μ(/d), where μ( ) deotes the Möbius fuctio, oe obtais the well-kow relatios Φ q (x p ), if p q, Φ pq (x) = Φ q (x p ), if p q. Φ q (x) Thus, if = p k m with p m, the Φ (x) Φ ϕ(pk ) m (x) modp, where ϕ( ) deotes Euler s totiet fuctio. Therefore, x 1= Φ d (x) Φ d d m establishig the result. k i=0 ϕ(pi ) d (x) = d m Φ pk d (x) mod p, Let F p deote the field with p elemets, where p is a prime umber. Cyclotomic polyomials are of course irreducible i Q[x], but this is ot ecessarily the case i F p [x]. However, cyclotomic polyomials whose idices are relatively prime ad ot divisible by p have o commo factors i F p [x].

5 LEHMER S PROBLEM FOR POLYNOMIALS WITH ODD COEFFICIENTS 351 Lemma 2.2. Suppose m ad are distict, relatively prime positive itegers, ad suppose p is a prime umber that does ot divide m. The Φ (x) ad Φ m (x) are relatively prime i F p [x]. Proof. Let e deote the multiplicative order of p modulo. IF p [x], the polyomial Φ (x) factors as the product of all moic irreducible polyomials with degree e ad order (see [13, Ch. 3]). Sice their factors i F p [x] have differet orders, we coclude that Φ ad Φ m are relatively prime modulo p. We ext describe the cyclotomic factors that may appear i a polyomial whose coefficiets are all cogruet to 1 modulo p. Lemma 2.3. Suppose f(x) Z[x] has degree 1 ad Φ r f. Iff D 2, the r 2; if f D p for a odd prime p, the r. Proof. Suppose f D p with p prime. Write = p k m with p m. By Lemma 2.1, we have (2.1) (x 1)f(x) d m Φ pk d (x) mod p. Write r = p l s with p s. Ifl = 0, the i view of Lemma 2.2, the polyomial Φ r must appear amog the factors Φ d o the right side of (2.1), so that r m. If l > 0, the Φ r Φ ϕ(pl ) s mod p, sos m. If s>1 the we also have p k p l p l 1, ad so if p>2 the k l ad thus r ; ifp = 2 the k l 1 ad cosequetly r 2. Last, if s = 1 the p k p l p l ad thus k l ad r. We ow state a simple characterizatio of polyomials f Z[x] that divide a polyomial with the same measure havig all its coefficiets cogruet to 1 modulo p. Theorem 2.4. Let p be a prime umber, ad let f(x) be a polyomial with iteger coefficiets. There exists a polyomial F D p with f F ad M(f) =M(F ) if ad oly if f is cogruet modulo p to a product of cyclotomic polyomials. Proof. Suppose first that F D p factors as F (x)=f(x)φ(x) with M(Φ)=1, so that Φ(x) is a product of cyclotomic polyomials. Sice F D p, it is cogruet modulo p to a product of cyclotomic polyomials. Usig Lemma 2.2 ad the fact that F p [x] is a uique factorizatio domai, we coclude that the polyomial f must also be cogruet modulo p to a product of cyclotomic polyomials.

6 352 P. BORWEIN, E. DOBROWOLSKI, AND M. J. MOSSINGHOFF For the coverse, suppose f(x) Φ ed d (x) mod p, p d with each e d 0. Let k = log p (max{e 1 +1, max{e d : d>1,p d}}), m = lcm{d : e d > 0,p d}, = mp k + 1, ad The Φ(x) =(x 1) pk e 1 1 Φ pk e d d (x). d m d>1 (x 1)f(x)Φ(x) d m Φ pk d (x) x 1 mod p, ad so F (x) =f(x)φ(x) has the required properties. Theorem 2.4 suggests a algorithm for determiig if a give polyomial f with degree d divides a polyomial F i D p with the same measure: Costruct all possible products of cyclotomic polyomials with degree d, ad test if ay of these are cogruet to f mod p. Usig this strategy, we verify that oe of the 100 irreducible, ocyclotomic polyomials from [15] represetig the smallest kow values of Mahler s measure divides a Littlewood polyomial with the same measure. This does ot imply, however, that o Littlewood polyomials exist with these measures, sice measures are ot ecessarily represeted uiquely by irreducible iteger polyomials, eve discoutig the simple symmetries M(f) =M(±f(±x k )). See [7] for more iformatio o the values of Mahler s measure. The requiremet i Theorem 2.4 that F (x) cotai o ocyclotomic factors besides f is certaily ecessary. For example, the polyomial x 10 x 7 x 5 x 3 +1 is ot cogruet to a product of cyclotomic polyomials mod 2, so o Littlewood polyomial exists havig this polyomial as its oly ocyclotomic factor. However, the product (x 10 x 7 x 5 x 3 + 1)(x 10 x 9 + x 5 x +1) is cogruet to Φ 33 mod 2, ad our costructio idicates that multiplyig this product by Φ 1 Φ 2 3 Φ2 11 Φ 33 yields a polyomial with all odd coefficiets. (I fact, usig the factors Φ 2 Φ 3 Φ 6 Φ 33 Φ 44 istead yields a Littlewood polyomial.) We close this sectio by otig that oe may demad stroger coditios o the polyomial F of Theorem 2.4 i certai situatios. Corollary 2.5. Suppose f Z[x] has o cyclotomic factors, ad there exists a polyomial F D 2 with eve degree 2m havig f F ad M(f) = M(F ). The there exists a polyomial G D 2 with deg G = 2m, f G, M(f) =M(G), ad the additioal property that G(x) ad 1+x+x 2 + +x 2m have o commo factors.

7 LEHMER S PROBLEM FOR POLYNOMIALS WITH ODD COEFFICIENTS 353 Proof. Suppose Φ d F. By Lemma 2.3, we have d (4m + 2). If d is odd ad d 3, so that Φ d (x) is a factor of 1 + x + + x 2m, the we ca replace the factor Φ d i F with Φ 2d without disturbig the required properties of F, sice Φ 2d (x) =Φ d ( x). Let G be the polyomial obtaied from F by makig this substitutio for each factor Φ d of F with d 3 odd. 3. Lehmer s problem We derive a lower boud o Mahler s measure of a polyomial that has o cyclotomic factors ad whose coefficiets are all cogruet to 1 modulo m for some fixed iteger m 2. Our results deped o the bouds o the resultats appearig i the followig lemma. Lemma 3.1. Suppose f D m with degree 1, ad let g be a factor of f. If gcd(g(x),x 1) = 1, the (3.1) Res(g(x),x 1) m deg g. Further, if m =2,k is a oegative iteger, ad gcd(g(x),x 2k +1)=1,the (3.2) Res(g(x),x 2k +1) 2 deg g. Proof. Defie the polyomial s(x) by (3.3) ms(x) =(x 1)+(1 x)f(x), ad ote that s(x) Z[x] sice f D m.ifg has o commo factor with x 1, the gcd(g, s) = 1, so Res(g, s) 1. Thus, by computig the resultat of both sides of (3.3) with g, we obtai (3.1). Suppose m =2. Fork 0, defie the polyomial t k (x) by 2 k 1 2t k (x) =(x 2k +1)+(1+x)f(x) x j. Now, (3.2) follows by a similar argumet. We also require the followig result regardig the legth of a power of a polyomial. j=0 ap- Lemma 3.2. For ay polyomial f C[x], the value of L(f k ) 1/k proaches f from above as k. Proof. From the triagle ad Cauchy-Schwarz iequalities, we have f k L(f k ) 1+k deg f f k, ad sice f k = f k, the result follows immediately.

8 354 P. BORWEIN, E. DOBROWOLSKI, AND M. J. MOSSINGHOFF Our mai theorem i this sectio provides a lower boud o the measure of a polyomial i D m that depeds o certai properties of a auxiliary polyomial. For a polyomial g Z[x], let ν k (g) deote the multiplicity of the cyclotomic polyomial Φ 2 k(x) ig(x), ad let ν(g) = k 0 ν k(g). Theorem 3.3. Suppose f D m with degree 1, ad suppose F Z[x] satisfies gcd(f(x),f(x )) = 1. The ( ν(f ) log 2 log F 1 1 ), if m =2, deg F log M(f) ( ν 0 (F ) log m log F 1 1 ), if m>2. deg F Proof. Suppose m = 2. Sice f(x) ad F (x ) have o commo factors, by Lemma 3.1 each cyclotomic factor Φ 2 k of F cotributes a factor of 2 1 to their resultat. Thus If α isarootoff, the so that Therefore or (3.4) Res(f(x),F(x )) 2 ν(f )( 1). F (α ) L(F ) max { 1, α deg F }, Res(f(x),F(x )) L(F ) 1 M(f) deg F. 2 ν(f )( 1) L(F ) 1 M(f) deg F, log M(f) ν(f ) log 2 log L(F ) deg F ( 1 1 ). Let k be a positive iteger. Sice ν(f k )=kν(f ) ad deg F k = k deg F,we obtai log M(f) ν(f ) log m log L(F k ) 1/k ( 1 1 ). deg F The theorem follows by lettig k ad usig Lemma 3.2. The proof for m>2 is similar, with ν 0 (F ) i place of ν(f ). For example, if f has all odd coefficiets ad o cyclotomic factors, the we may use F (x) =x 2 1 i Theorem 3.3 to obtai log M(f) log 2 ( 1 1 ) (3.5). 2

9 LEHMER S PROBLEM FOR POLYNOMIALS WITH ODD COEFFICIENTS 355 For m>2, if f D m has o cyclotomic factors, the we may use F (x) =x 1 to obtai ( log M(f) log(m/2) 1 1 ) (3.6). Sectio 4 describes a class of polyomials that oe might expect to cotai some choices for F that improve the bouds (3.5) ad (3.6), ad describes some algorithms developed to search this set for better auxiliary polyomials. We record here some improved bouds that arose from these searches. Corollary 3.4. Let f be a polyomial with degree 1 havig odd coefficiets ad o cyclotomic factors. The log M(f) log 5 ( 1 1 ) (3.7), 4 with equality if ad oly if f(x) =±1. Proof. Let F (x) = ( 1+x 2)( 1 x 2) 4. Sice ν(f ) = 9, deg F = 10, ad F = (1 + y)(1 y) 4 =2 5 max cos(πt) si 4 (πt) 2 9 = 0 t , usig Theorem 3.3 we establish (3.7). Last, if the leadig or costat coefficiet of f is greater tha 1 i absolute value, the M(f) 3; if >1 ad these coefficiets are ±1, the M(f) is a uit. Aother auxiliary polyomial yieldig the lower boud (3.7) appears i Sectio 4. We remark that the boud of 5 1/4 = is ot far from the smallest kow measure of a polyomial with odd coefficiets ad o cyclotomic factors: M(1 + x x 2 x 3 x 4 + x 5 + x 6 )= This umber is i fact the smallest measure of a reciprocal polyomial with ±1 coefficiets havig o cyclotomic factors ad degree at most 72; see [4]. Sectio 6 provides more iformatio o the structure of kow small values of Mahler s measure of these polyomials. For the case m>2, a auxiliary polyomial similar to the oe employed i Corollary 3.4 improves (3.6) slightly. Corollary 3.5. Let f D m have degree 1 ad o cyclotomic factors. The ( ) ( m log M(f) log ) (3.8), 2 with equality if ad oly if f(x) =±1.

10 356 P. BORWEIN, E. DOBROWOLSKI, AND M. J. MOSSINGHOFF Proof. Let F (x) =(1+x)(1 x) m2. Sice ν 0 (F )=m 2, deg F = m 2 +1, ad F =2 m2 +1 max cos(πt) si m2 (πt) 0 t 1 2 m2 +1 m m2 = (m 2 +1) (m2 +1)/2, usig Theorem 3.3 we verify (3.8). The argumet for the case of equality is similar to that of Corollary 3.4. Sectio 4.3 shows that the boud of 10/2 = for m = 3 may be replaced by by usig the auxiliary polyomial (1 x) 425 (1 x 2 ) 50 (1 x 5 ). No improvemets are kow for m>3. 4. Auxiliary polyomials We obtai otrivial bouds o the measure of a polyomial f D m from Theorem 3.3 by usig auxiliary polyomials havig small degree, small supremum orm, ad a high order of vaishig at 1. I this sectio, we ivestigate a family of polyomials havig precisely these properties ad search for auxiliary polyomials yieldig good lower bouds Pure product polyomials. A pure product of size is a polyomial of the form (1 x ek ), k=1 with each e k a positive iteger. Let A() deote the miimal supremum over the uit disk amog all pure products of size, { } A() = mi (1 x ek ) : e k 1 for 1 k. k=1 Erdős ad Szekeres studied this quatity i [10], provig that the growth rate of A() is subexpoetial: lim A()1/ =1. The upper boud o the asymptotic growth rate of log A() has sice bee greatly improved. Atkiso [2] obtaied O( log ), Odlyzko [17] proved O( 1/3 log 4/3 ), Koloutzakis [11] demostrated O( 1/3 log ), ad Belov ad Koyagi [3] showed O(log 4 ). The best kow geeral lower boud o A()

11 LEHMER S PROBLEM FOR POLYNOMIALS WITH ODD COEFFICIENTS 357 is simply 2; stregtheig this would provide iformatio o the Diophatie problem of Prouhet, Tarry, ad Escott (see for istace [14]). Erdős cojectured [9, p. 55] that i fact A() c for ay c>0. Sice ν 0 (A()) = ad log A() =o(), it follows that there exist pure product polyomials F (x) that yield otrivial lower bouds i Theorem 3.3. The article [5] exhibits some pure products of size 20 with very small legth ad degree, ad these polyomials yield otrivial lower bouds i Theorem 3.3. However, these polyomials arise as optimal examples of polyomials with { 1, 0, 1} coefficiets havig a root of prescribed order at 1 ad miimal degree. We obtai better bouds by desigig some more specialized searches. We describe two such searches Hill-climbig. Our first method employs a modified hill-climbig strategy to search for good auxiliary polyomials F (x), replacig the objective fuctio appearig i Theorem 3.3 with the computatioally more attractive fuctio from (3.4). So for each m we wish to fid large values of ν(f ) log 2 log L(F ), if m =2, B m (F )= deg F ν 0 (F ) log m log L(F ), if m>2. deg F Algorithm 4.1. Modified hill-climbig for auxiliary polyomials. Iput. A iteger m 2, a set E of positive itegers, ad for each e E, a oegative iteger r e. Output. A sequece of pure products {F k } with F k 1 F k for each k. Step 1. Let F 0 (x) = e E (1 xe ) re, let b 0 = B m (F 0 ), ad set k =1. Step 2. Step 3. For each e E, compute B m ((1 x e )F k 1 (x)). If the largest of these E values is greater tha b k 1, the set F k (x) =(1 x e )F k 1 (x) for the optimal choice of e, set b k = B m (F k ), prit F k ad b k, icremet k, ad repeat Step 2. Otherwise, cotiue with Step 3. For each subset {e 1,e 2 } of E, compute B m ((1 x e1 )(1 x e2 )F k 1 (x)). If the largest of these ( ) E 2 values exceeds bk 1, the set F k (x) = (1 x e1 )(1 x e2 )F k 1 (x) for the optimal choice {e 1,e 2 }, set b k = B m (F k ), prit F k ad b k, icremet k, ad repeat Step 3. Otherwise, set b k 1 = 0 ad perform Step 2. Several criteria may be used for termiatio, for example, a prescribed boud o k or deg F k, or the appearace of a decreasig sequece of values of b k of a particular legth.

12 358 P. BORWEIN, E. DOBROWOLSKI, AND M. J. MOSSINGHOFF We remark that a pure hill-climbig method would omit the resettig of b k 1 to 0 at the ed of Step 3 ad would termiate as soo as oe of the adjustmets of Steps 2 or 3 improves the boud. By addig this assigmet, we eed ot stop at local maxima ad istead allow our objective value to decrease temporarily i order to cotiue searchig for better values. We use the revolvig door algorithm [16] to eumerate the ( ) E 2 combiatios of factors to test i Step 3. This way, each polyomial we test ca be costructed from the previous polyomial cosidered with just oe divisio by a biomial ad oe multiplicatio. We implemeted Algorithm 4.1 i C++ ad ra it o a Apple PowerPC G4. For m = 2, lettig F 0 (x) =1 x 2 ad choosig E to be a set of small positive itegers like {1, 2,...,8}, we see that Algorithm 4.1 produces a sequece of polyomials of the form (1 x 2 ) a (1 x 4 ) b with a 3b. This suggests the sequece F k (x) = ((1 x 2 ) 3 (1 x 4 )) k ad hece Corollary 3.4. Despite several variatios o the iitial values, o better sequece was foud with Algorithm 4.1 for m =2. For several values of m greater tha 2, Algorithm 4.1, startig with F 0 (x) = 1 x, produces a sequece of auxiliary polyomials of the form (1 x) a (1 x 2 ) b with a (m 2 1)b, suggestig the polyomial employed i Corollary 3.5. Our method also idicates a further improvemet for the case m = 3. With E = {1, 2, 3, 4, 5}, Algorithm 4.1 costructs the polyomial (4.1) F (x) =(1 x) 1078 (1 x 2 ) 127 (1 x 5 ) 3, which has B 3 (F )= > 10/2. This example is ivestigated further i our secod search method Special families. A secod method of searchig for good auxiliary polyomials i Theorem 3.3 computes F directly for certai families of pure products rather tha usig the quatity L(F ) as a boud. For a polyomial F, let β m (F ) deote the expressio appearig i Theorem 3.3: ν(f ) log 2 log F, if m =2, β m (F )= deg F ν 0 (F ) log m log F, if m>2. deg F Give m ad fixig a set of positive itegers E, we evaluate ( ) β m (1 x e ) re e E for a umber of selectios for the expoets r e, subject to gcd{r e : e E} =1. For example, for m = 2 ad E = {2, 4, 8}, we fid o polyomials that yield a boud as good as that of Corollary 3.4. However, usig E = {2, 4, 6},

13 LEHMER S PROBLEM FOR POLYNOMIALS WITH ODD COEFFICIENTS Figure 1: β 2 (G k ) for k 100. we detect a auxiliary polyomial that does just as well. After computig β 2 ((1 x 2 ) a (1 x 4 ) b (1 x 6 )) for 1 a, b 50, we fid that the polyomials G k (x) = ( 1 x 2) 2k+1 ( 1 x 4 ) k ( 1 x 6 ) produce values rather close to (log 5)/4, ad further that these values are icreasig i k over this rage. We obtai Figure 1 by computig β 2 for additioal polyomials i this sequece. The maximum value occurs at k = 34, ad G 34 = (1 y) 104 (1 + y) 34 (1 + y + y 2 ) =2 138 max si 104 (πt) cos 34 (πt)(2 cos(2πt)+1) 0 t 1 = Sice ν(g 34 ) = 242 ad deg G 34 = 280, we agai obtai (3.7). For m = 3, give (4.1) we ivestigate polyomials with E = {1, 2, 5} ad fid that we obtai good bouds usig r 1 8.5r 2 ad r 3 = 1. The maximum value of β 3 ((1 x) 17k (1 x 2 ) 2k (1 x 5 )) occurs at k = 25, yieldig the value cited at the ed of Sectio 3. Similar ivestigatios for larger values of m have ot improved the boud of Corollary 3.5. For example, choosig E = {1, 2, 5}, r 1 =(2m 2 1)k, r 2 =2k, k 1, ad r 3 = 1, we obtai a sequece of values uder β m approachig log( m 2 +1/2) from below, but o polyomial tested improves this boud. 5. The cojecture of Schizel ad Zassehaus The lower bouds o log M(f) for f D m of Corollaries 3.4 ad 3.5 automatically yield lower bouds o max{ α : f(α) =0} for polyomials f D m havig o cyclotomic factors. The followig theorem improves these results i the Schizel-Zassehaus problem i two ways: weakeig the hypotheses ad improvig the costats.

14 360 P. BORWEIN, E. DOBROWOLSKI, AND M. J. MOSSINGHOFF Theorem 5.1. Suppose f D m is moic with degree 1 havig at least oe ocyclotomic factor. The there exists a root α of f satisfyig 1+ log 3, if m =2, (5.1) α > 2 log(m 1) 1+, if m>2. Proof. Let g deote the ocyclotomic part of f, let d = deg g, ad let α 1,..., α d deote the roots of g. Suppose that max{ α k :1 k d} < 1+ c for a positive costat c, so that αk <ec for each k. Suppose m = 2. Sice the maximum value of 1 z 2 for complex umbers z lyig i the disk {z : z r} is 1 + r 2, with the maximum value occurrig at z = ±ir, we have 1 α 2 k < 1+e 2c for each k. Cosequetly, usig Lemma 3.1 with both x + 1 ad x 1, we fid (5.2) 2 2d Res(g(x), 1 x 2 ) < ( 1+e 2c ) d. Therefore 1 + e 2c > 4, ad the iequality for m = 2 follows. If m>2, i a similar way we obtai (5.3) m d Res(g(x), 1 x ) < (1 + e c ) d, ad the theorem follows. No better bouds were foud by usig other auxiliary polyomials i place of 1 x 2 ad 1 x i (5.2) ad (5.3). However, for some m we fid that the polyomials employed i Corollaries 3.4 ad 3.5 do just as well. For example, let F a,b (x) =(1 x 2 ) a (1 + x 2 ) b, with a ad b positive itegers. The supremum of F a,b o the disk {z C : z = r} is ( 2(1 + r F a,b z =r = a a/2 b b/2 4 ) (a+b)/2 ), a + b ad we obtai a lower boud o c from the iequality 2 2a+b < F a,b z =e c. The optimal choice of parameters is a = 4 ad b = 1, as i Corollary 3.4, yieldig c (log 3)/2. Likewise, for m>1 the optimal choice for a ad b i the auxiliary polyomial (1 x) a (1+x) b is a = m 2 ad b = 1, but this selectio achieves c log(m 1) oly for m =3.

15 LEHMER S PROBLEM FOR POLYNOMIALS WITH ODD COEFFICIENTS 361 We ow show that the costat i Theorem 5.1 for f D m caot be replaced with ay umber larger tha log(2m 1). We first require the followig iequality. Lemma 5.2. Suppose f(z) = z + a 1 z a 1 z + a 0, ad let K = {k :0 k 1 ad a k 0}. For each k K, let c k be a positive umber, ad suppose that k K c k 1. Ifα is a root of f, the { ( ak ) } 1 k α max : k K. c k Proof. See [18, part III, problem 20]. Theorem 5.3. For each m 2, ay ε>0, ad all 0 (m, ε), there exists a polyomial f D m with degree satisfyig 1+ log m ε 1) + ε < max α < 1+log(2m. f(α)=0 Proof. Fix m 2. Let f (x) =x + x x +1 m, so that (x 1)f (x) =x +1 mx + m 1. By Rouché s theorem, f has exactly oe zero iside the uit disk, ad this root is a real umber approachig (m 1)/m for large. Thus f has a sigle root outside the uit disk ear m/(m 1), so that M(f )=M(f) m as. If f has a reciprocal factor g, the g divides f as well, ad so g f f = m(x 1). However, f (1) = +1 m ad f (ζ) =1 m for ay complex th root of uity ζ; sof has o reciprocal factor, ad hece o roots o the uit circle, if m 1. Give ε>0. For sufficietly large, the polyomial f has 1 roots outside the uit circle, ad at least oe of them must have modulus at least as large as the geometric mea of these roots. Thus, max α M(f ) 1/( 1) > ( e ε m ) 1/( 1) log m ε > 1+. f (α)=0 For the upper boud, we apply Lemma 5.2 usig f(z) =z +1 mz+m 1, c 0 =(m 1)/(2m 1), ad c 1 = m/(2m 1) to obtai max α (2m f 1)1/. (α)=0 Takig sufficietly large completes the proof. For m = 2, the positive real root of the polyomial f appearig i the proof of Theorem 5.3 is a Pisot umber, sice all its cojugates lie iside the ope uit disk. I the ext sectio we study some properties of Pisot ad Salem umbers that appear as roots of Littlewood polyomials.

16 362 P. BORWEIN, E. DOBROWOLSKI, AND M. J. MOSSINGHOFF 6. Pisot ad Salem umbers We say a real umber α>1isalittlewood-pisot umber if it is a Pisot umber ad its miimal polyomial is a Littlewood polyomial, ad defie a Littlewood-Salem umber i the same way. The article [4] proves that the miimal value of Mahler s measure of a oreciprocal polyomial i D 2 is the golde ratio. Thus, this value is the smallest Littlewood-Pisot umber. We first improve Theorem 3.3 slightly for Salem umbers. Sice we focus o Littlewood polyomials i this sectio, we preset oly the case of polyomials with odd coefficiets; a aalogous argumet improves the boud for Salem umbers whose miimal polyomial lies i D m with m>2. Theorem 6.1. Suppose f is a moic, irreducible polyomial i D 2 with degree 1 havig exactly oe root α outside the uit disk. The log α > log 5 ( 1+ 1 ) Proof. If F (x) is a polyomial with f(x) F (x ), the usig Lemma 3.1 ad the fact that the complex roots of f lie o the uit circle, we obtai 2 ν(f )( 1) Res(f(x),F(x )) F 3 F (α )F (α ). Choose F (x) =(1 x 2 ) 4 (1 + x 2 ). The F (α )F (α ) = α 10 F (α ) 2 = α 10 ( α 2 1 ) 6 ( α 4 1 ) 2 <α 10, so that Cosequetly Thus where log α > log 5 4 c = 2 9( 1) <α 10 ( 2 9 α > 5 1/4 ( /2 5 5/2 ) 3. ) 1/ log log 5 4 = log 5 ( 1+ c ), 4 36 log 2 5 log 5 3= > Our mai result of this sectio cocers a limit poit of Littlewood-Salem umbers. It is well-kow that every Pisot umber is a two-sided limit poit of Salem umbers. We prove that more is true for the smallest Littlewood-Pisot umber.

17 LEHMER S PROBLEM FOR POLYNOMIALS WITH ODD COEFFICIENTS 363 Theorem 6.2. The smallest Littlewood-Pisot umber is a limit poit, from both sides, of Littlewood-Salem umbers. Proof. We first defie two sequeces of Littlewood polyomials which have exactly oe root outside the uit circle. Let P (x) deote the cyclotomic product ad let P (x) =Φ 6 (x) 2 k=0 x 3k =Φ 6 (x) x6+3 1 x 3 1 p (t) =e( (3 +1)t)P (e(t)), =Φ 6 (x) d 6+3 d>3 Φ d (x), where e(t) =e 2πit. Thus, p (t) is a real-valued, periodic fuctio with period 1 havig simple zeros i the iterval (0, 1/2) at the poits { } 1 { } k :1 k 3 +1,k Let a (t) = 2 cos((6 +2)πt) ad Sice b (t) = 4 si(πt) si((6 +1)πt). 2k < k 2k +1 < < k for 1 k 3, it follows that betwee two cosecutive zeros of a (t)i(0, 1/2) there exists exactly oe zero of p (t), with two exceptios correspodig to the absece of a zero of p (t) att = 1/3 ad the extra zero at t = 1/6. Cosequetly, the fuctio a (t) p (t) has at least 3 2 zeros i (0, 1/2). A similar computatio verifies that p (t) b (t) has at least 3 2 zeros i (0, 1/2). For 1, defie the Littlewood polyomials A (x) ad B (x) by 2 1 (6.1) A (x) =x 6 +(1 x x 2 ) x 3k ad (6.2) k=0 2 1 B (x) =x 6 1 +(1+x x 2 ) x 3k, so that deg A =6 ad deg B =6 2. Sice k=0 P (x)+xa (x) =x

18 364 P. BORWEIN, E. DOBROWOLSKI, AND M. J. MOSSINGHOFF ad P (x) x 2 B (x) =(x 1)(x 6+1 1), it follows that e( 3t)A (e(t)) ad e( (3 1)t)B (e(t)) each have at least 6 4 zeros i (0, 1), ad thus that A (x) ad B (x) each have at least 6 4 zeros o the uit circle. Sice A ( 1) = 1, A (1) = 1 2, B ( 1) = 1, ad B (1) = 1 + 2, it follows that A (x) ad B (x) have oe real root i the iterval ( 1, 1), ad, sice these polyomials are reciprocal, oe real root outside the uit disk as well. This accouts for all roots of B (x), ad all but two roots of A (x). These last two roots caot be real, sice evidetly A (x) has a odd umber of roots i [ 1, 1] ad hece its total umber of real roots is cogruet to 2 mod 4. Also, they must have modulus 1, for otherwise a root α would have distict cojugates 1/α, α, ad 1/α. We ext prove that A (x) ad B (x) are irreducible. Let α deote the real root of A (x) outside the uit disk, ad let β deote that of B (x). If f A ad f(α ) 0, the by Kroecker s theorem f is a product of cyclotomic polyomials. Suppose the that Φ d A. By Lemma 2.3, we have d Ifd 6 + 1, the Φ d divides A (x)+ x6+1 1 x 1 =2 x6+3 1 x 3 1, so that d ad thus d = 1, but A (1) 0. Ifd is a eve divisor of , the Φ d/2 (x) A ( x). Sice x x 1 A ( x) =2x 2 Φ 3 (x) x6 1 x 6 1, we have d 12 as well ad agai arrive at a cotradictio. The proof that B (x) is irreducible is similar. Let γ deote the golde ratio, γ =(1+ 5)/2. Sice A (1)=1 2 ad A (γ) =1,wehaveα <γfor each. Similarly, we compute B ( 2) = ( )/9 ad B ( γ) =1 γ, so that β < γ for each. Further, A +1 (α )=α 6+6 α 6 +(1 α α 2 )(α 6 + α 6+3 ) = α 6+1 (α 3 + 1)(α 2 α 1), ad sice x 2 x 1 < 0o(1,γ), we coclude A +1 (α ) < 0 ad thus α +1 >α. Similarly, B +1 (β )=β 6 1 (β 3 + 1)(β 2 + β 1) > 0, ad so β +1 >β. Thus, the sequeces {α } ad {β } coverge. Fially, sice A (x) coverges uiformly to (1 x x 2 )/(1 x 3 )oay compact subset of ( 1, 1), it follows that lim 1/α =1/γ, ad so {α } coverges to γ. Similarly, {β } coverges to γ. Thus, A (x) ad B ( x) provide the required Littlewood-Salem umbers.

19 LEHMER S PROBLEM FOR POLYNOMIALS WITH ODD COEFFICIENTS 365 The root α 1 of A 1 (x) i the proof of Theorem 6.2 is the smallest kow measure of a irreducible Littlewood polyomial. Moreover, the sequeces {α } ad { β } ecompass all kow values of Mahler s measure below of reciprocal, irreducible Littlewood polyomials (see [15]). Fially, it is likely that the method of the proof exteds, at least i part, to the other Littlewood-Pisot umbers appearig i the proof of Theorem 5.3. Let m 2 f m (x) =x m 1 x k, ad let γ m deote the Pisot umber havig f m (x) as its miimal polyomial. Followig (6.1) ad (6.2), for each m 3 ad 1 defie the Littlewood polyomials 2 1 A m, (x) =x 2m + fm(x) x mk ad k=0 k=0 2 1 B m, (x) =x 2m 1 f m (x) x mk. For each m, it appears that A m, (x) yields a sequece of Salem umbers approachig γ m from below. However, while M(B m, ) approaches γ m as, evidetly B m, (x) has m 2 roots outside the uit circle. Added i proof. Sice this article was writte, some other papers have appeared o the topics treated i this paper. Lower bouds i Lehmer s problem ad the Schizel-Zassehaus problem for polyomials with coefficiets cogruet to 1 mod m are developed further i A. Dubickas ad M. J. Mossighoff, Auxiliary polyomials for some problems regardig Mahler s measure, Acta Arith. 119 (2005), Results o Lehmer s problem are geeralized i C. L. Samuels, The Weil height i terms of a auxiliary polyomial, Acta Arith. 128 (2007), More iformatio o Littlewood-Pisot ad Salem umbers may be foud i K. Mukuda, Littlewood Pisot umbers, J. Number Theory 117 (2006), k=0 Simo Fraser Uiversity, Buraby, B.C., Caada address: pborwei@cecm.sfu.ca College of New Caledoia, Price George, B.C., Caada address: dobrowolski@cc.bc.ca Davidso College, Davidso, N.C., USA address: mjm@member.ams.org

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