Jonas Jankauskas

On Littlewood Polynomials with Prescribed Number of Zeros Inside the Unit Disk

We investigate the numbers of complex zeros of Littlewood polynomials p(z) (polynomials with coefficientsf 1; 1g) inside or on the unit circlejzj = 1, denoted by N(p) and U (p), respectively. Two types of Littlewood polynomials are considered: Littlewood polynomials with one sign change in the sequence of coefficients and Littlewood polynomials with one negative coefficient. We obtain explicit formulas for N(p), U (p) for polynomials p(z) of these types. We show that if n + 1 is a prime number, then for each integer k, 0 6 k 6 n 1, there exists a Littlewood polynomial p(z) of degree n with N(p) = k and U (p) = 0. Furthermore, we describe some cases where the ratios N(p)=n and U (p)=n have limits as n!1 and find the corresponding limit values.

On Newman polynomials which divide no Littlewood polynomial

Recall that a polynomial P(x) G Z[x] with coefficients 0, 1 and constant term 1 is called a Newman polynomial, whereas a polynomial with coefficients -1, 1 is called a Littlewood polynomial. Is there an algebraic number ol which is a root of some Newman polynomial but is not a root of any Littlewood polynomial? In other words (but not equivalently), is there a Newman polynomial which divides no Littlewood polynomial? In this paper, for each Newman polynomial P of degree at most 8, we find a Littlewood polynomial divisible by P. Moreover, it is shown that every trinomial l + uxa +vxb, where a < b are positive integers and u, v € {1, 1}, so, in particular, every Newman trinomial 1 + xa + xb, divides some Littlewood polynomial. Nevertheless, we prove that there exist Newman polynomials which divide no Littlewood polynomial, e.g., x9+x6+x2+x+l. This example settles the problem 006:07 posed by the first named author at the 2006 West Coast Number Theory conference. It also shows that the sets of roots of Newman polynomials Vjv» Littlewood polynomials Vc and {-1,0,1} polynomials V are distinct in the sense that between them there are only trivial relations V/v C V and Vc C V. Moreover, V jz Vc U Vjsf. The proofs of several main results (after some preparation) are computational.

On a class of polynomials related to Barker sequences

For an odd integer n > 0, we introduce the class LPn of Laurent polynomials P (z) = (n+ 1) + n ∑ k=1 k odd ck(z k + z−k), with all coefficients ck equal to −1 or 1. Such polynomials arise in the study of Barker sequences of even length, i.e., integer sequences having minimal possible autocorrelations. We prove that polynomials P ∈ LPn have large Mahler measures, namely, M(P ) > (n + 1)/2. We conjecture that minimal Mahler measures in the class LPn are attained by the polynomials Rn(z) and Rn(−z), where Rn(z) = (n+ 1) + n ∑ k=−n k odd z is a polynomial with all the coefficients ck = 1. We prove that M(Rn) > n− 2 π logn+O(1). The results of experimental computations on polynomials in the class LPn suggest two conjectures which could shed light on the long-standing Barker problem.

No two non-real conjugates of a Pisot number have the same imaginary part

We show that the number α = (1 + √ 3 + 2 √ 5)/2 with minimal polynomial x4 − 2x3 + x − 1 is the only Pisot number whose four distinct conjugates α1, α2, α3, α4 satisfy the additive relation α1+α2 = α3+α4. This implies that there exists no two non-real conjugates of a Pisot number with the same imaginary part and also that at most two conjugates of a Pisot number can have the same real part. On the other hand, we prove that similar four term equations α1 = α2 + α3 + α4 or α1 + α2 + α3 + α4 = 0 cannot be solved in conjugates of a Pisot number α. We also show that the roots of the Siegel’s polynomial x3−x−1 are the only solutions to the three term equation α1+α2+α3 = 0 in conjugates of a Pisot number. Finally, we prove that there exists no Pisot number whose conjugates satisfy the relation α1 = α2 + α3.

THE MAXIMAL VALUE OF POLYNOMIALS WITH RESTRICTED COEFFICIENTS

Let be a fixed complex number. In this paper, we study the quantity , where is the set of all real polynomials of degree at most n-1 with coefficients in the interval [0, 1]. We first show how, in principle, for any given and , the quantity S(, n) can be calculated. Then we compute the limit for every of modulus 1. It is equal to 1/ if is not a root of unity. If , where and k [1, d-1] is an integer satisfying gcd(k, d) = 1, then the answer depends on the parity of d. More precisely, the limit is 1, 1/(d sin(/d)) and 1/(2d sin(/2d)) for d = 1, d even and d > 1 odd, respectively.

Extremal Mahler measures and _{} norms of polynomials related to Barker sequences

In the present paper, we study the class LPn which consists of Laurent polynomials P (z) = (n+ 1) + n ∑ k=1 k – odd ck(z k + z−k), with all coefficients ck equal to −1 or 1. Such polynomials arise in the study of Barker sequences of even length — binary sequences with minimal possible autocorrelations. By using an elementary (but not trivial) analytic argument, we prove that polynomials Rn(z) with all coefficients ck = 1 have minimal Mahler measures in the class LPn. In conjunction with an estimate M(Rn) > n − 2/π logn + O(1) proved in an earlier paper, we deduce that polynomials whose coefficients form a Barker sequence would possess unlikely large Mahler measures. A generalization of this result to Ls norms is also given.