Littlewood Polynomial Research Articles

Step roots of Littlewood polynomials and the extrema of functions in the Takagi class

AbstractWe give a new approach to characterising and computing the set of global maximisers and minimisers of the functions in the Takagi class and, in particular, of the Takagi–Landsberg functions. The latter form a family of fractal functions $f_\alpha:[0,1]\to{\mathbb R}$ parameterised by $\alpha\in(-2,2)$ . We show that $f_\alpha$ has a unique maximiser in $[0,1/2]$ if and only if there does not exist a Littlewood polynomial that has $\alpha$ as a certain type of root, called step root. Our general results lead to explicit and closed-form expressions for the maxima of the Takagi–Landsberg functions with $\alpha\in(-2,1/2]\cup(1,2)$ . For $(1/2,1]$ , we show that the step roots are dense in that interval. If $\alpha\in (1/2,1]$ is a step root, then the set of maximisers of $f_\alpha$ is an explicitly given perfect set with Hausdorff dimension $1/(n+1)$ , where n is the degree of the minimal Littlewood polynomial that has $\alpha$ as its step root. In the same way, we determine explicitly the minima of all Takagi–Landsberg functions. As a corollary, we show that the closure of the set of all real roots of all Littlewood polynomials is equal to $[-2,-1/2]\cup[1/2,2]$ .

A combinatorial Schur expansion of triangle-free horizontal-strip LLT polynomials

In recent years, Alexandersson and others proved combinatorial formulas for the Schur function expansion of the horizontal-strip LLT polynomial \(G_{\boldsymbol\lambda}(\boldsymbol x;q)\) in some special cases. We associate a weighted graph \(\Pi\) to \(\boldsymbol\lambda\) and we use it to express a linear relation among LLT polynomials. We apply this relation to prove an explicit combinatorial Schur-positive expansion of \(G_{\boldsymbol\lambda}(\boldsymbol x;q)\) whenever \(\Pi\) is triangle-free. We also prove that the largest power of \(q\) in the LLT polynomial is the total edge weight of our graph.Keywords: Charge, chromatic symmetric function, cocharge, Hall--Littlewood polynomial, jeu de taquin, LLT polynomial, interval graph, Schur function, Schur-positive, symmetric function.Mathematics Subject Classifications: 05E05, 05E10, 05C15

Littlewood polynomials, spectral-null codes, and equipowerful partitions

Let $[n]$ denote $\{0,1, ... , n-1\}$. A polynomial $f(x) = \sum a_i x^i$ is a Littlewood polynomial (LP) of length $n$ if the $a_i$ are $\pm 1$ for $i \in [n]$, and $a_i = 0$ for $i \ge n$. Such an LP is said to have order $m$ if it is divisible by $(x-1)^m$. The problem of finding the set $L_m$ of lengths of LPs of order $m$ is equivalent to finding the lengths of spectral-null codes of order $m$, and to finding $n$ such that $[n]$ admits a partition into two subsets whose first $m$ moments are equal. Extending the techniques and results of Boyd and others, we completely determine $L_7$ and $L_8$ and prove that 192 is the smallest element of $L_9$. Our primary tools are the use of carefully targeted searches using integer linear programming (both to find LPs and to disprove their existence for specific $n$ and $m$), and an unexpected new concept (that arose out of observed symmetry properties of LPs) that we call regenerative pairs, which produce infinite arithmetic progressions in $L_m$. We prove that for $m \le$ 8, whenever there is an LP of length $n$ and order $m$, there is one of length $n$ and order $m$ that is symmetric (resp.~antisymmetric) if m is even (resp.~odd).

Approximately half of the roots of a random Littlewood polynomial are inside the disk

We prove that for large $n$, all but $o(2^{n})$ polynomials of the form $P(z) = \sum_{k=0}^{n-1}\pm z^k$ have $n/2 + o(n)$ roots inside the unit disk. This solves a problem from Hayman's book 'Research Problems in Function Theory' (1967).

Cyclic sieving, skew Macdonald polynomials and Schur positivity

When λ is a partition, the specialized non-symmetric Macdonald polynomial E λ (x;q;0) is symmetric and related to a modified Hall–Littlewood polynomial. We show that whenever all parts of the integer partition λ are multiples of n, the underlying set of fillings exhibit the cyclic sieving phenomenon (CSP) under an n-fold cyclic shift of the columns. The corresponding CSP polynomial is given by E λ (x;q;0). In addition, we prove a refined cyclic sieving phenomenon where the content of the fillings is fixed. This refinement is closely related to an earlier result by B. Rhoades.

Weak tangent and level sets of Takagi functions

In this paper, we study some properties of Takagi functions and their level sets. We show that for Takagi functions T_{a,b} with parameters a, b such that ab is a root of a Littlewood polynomial, there exist large level sets. As a consequence, we show that for some parameters a, b, the Assouad dimension of graphs of T_{a,b} is strictly larger than their upper box dimension. In particular, we can find weak tangents of those graphs with large Hausdorff dimension, larger than the upper box dimension of the graphs.

On Littlewood and Newman polynomial multiples of Borwein polynomials

On Littlewood and Newman polynomial multiples of Borwein polynomials

Lq Norms of Fekete and Related Polynomials

AbstractA Littlewood polynomial is a polynomial in ℂ[z] having all of its coefficients in {−1, 1}. There are various old unsolved problems, mostly due to Littlewood and Erdos, that ask for Littlewood polynomials that provide a good approximation to a function that is constant on the complex unit circle, and in particular have small Lq normon the complex unit circle. We consider the Fekete polynomialswhere p is an odd prime and (· |p) is the Legendre symbol (so that z-1fp(z) is a Littlewood polynomial). We give explicit and recursive formulas for the limit of the ratio of Lq and L2 norm of fp when q is an even positive integer and p → ∞. To our knowledge, these are the first results that give these limiting values for specific sequences of nontrivial Littlewood polynomials and infinitely many q. Similar results are given for polynomials obtained by cyclically permuting the coefficients of Fekete polynomials and for Littlewood polynomials whose coefficients are obtained from additive characters of finite fields. These results vastly generalise earlier results on the L4 norm of these polynomials.

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

AbstractWe investigate the numbers of complex zeros of Littlewood polynomials p(z) (polynomials with coefficients {−1, 1}) inside or on the unit circle |z| = 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 ≤ k ≤ 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 → ∞ and find the corresponding limit values.

A three shuffle case of the compositional parking function conjecture

We prove here that the polynomial 〈∇Cp1,eahbhc〉q,t-enumerates, by the statistics dinv and area, the parking functions whose supporting Dyck path touches the main diagonal according to the composition p⊨a+b+c and has a reading word which is a shuffle of one decreasing word and two increasing words of respective sizes a, b, c. Here Cp1 is a rescaled Hall–Littlewood polynomial and ∇ is the Macdonald eigen-operator introduced by Bergeron and Garsia. This is our latest progress in a continued effort to settle the decade old shuffle conjecture of Haglund et al. This result includes as special cases all previous results connected with the shuffle conjecture such as the q,t-Catalan, Schröder and two shuffle results of Haglund as well as their compositional refinements recently obtained by the authors. It also confirms the possibility that the approach adopted by the authors has the potential to yield a resolution of the shuffle parking function conjecture as well as its compositional refinement more recently proposed by Haglund, Morse and Zabrocki.

A Compositional Shuffle Conjecture Specifying Touch Points of the Dyck Path

AbstractWe introduce aq,t-enumeration of Dyck paths that are forced to touch the main diagonal at specific points and forbidden to touch elsewhere and conjecture that it describes the action of the Macdonald theory ∇ operator applied to a Hall–Littlewood polynomial. Our conjecture refines several earlier conjectures concerning the space of diagonal harmonics including the “shuffle conjecture” (Duke J. Math.126(2005), pp. 195 − 232) for ∇ en[X]. We bring to light that certain generalized Hall–Littlewood polynomials indexed by compositions are the building blocks for the algebraic combinatorial theory ofq,t-Catalan sequences, and we prove a number of identities involving these functions.

Infinite Barker series

We say a polynomial p ( z ) = a n z n + a n − 1 z n − 1 + ⋯ + a 0 is a Littlewood polynomial if a k = ± 1 for 0 ⩽ k ⩽ n . Let p ( z ) p ( 1 / z ) = c n z n + c n − 1 z n − 1 + ⋯ + c − n z − n . It is easy to show that c 0 = n + 1 . We say that p ( z ) is a Barker polynomial if | c k | ⩽ 1 for k ≠ 0 . There are only 8 known Barker polynomials (normalized to have a n = a n − 1 = 1 ). There are many results known about the existence and non-existence of Barker polynomials for various degrees. This paper deals with the infinite case, when f ( z ) = ± 1 ± z ± z 2 ± ⋯ is a power series with ±1 coefficients. We give a complete description of all Barker series.

UNIMODULAR ROOTS OF RECIPROCAL LITTLEWOOD POLYNOMIALS

The main result of this paper shows that every reciprocal Littlewood polynomial, one with {−1, 1} coefficients, of odd degree at least 7 has at least five unimodular roots, and every reciprocal Littlewood polynomial of even degree at least 14 has at least four unimodular roots, thus improving the result of Mukunda. We also give a sketch of alternative proof of the well-known theorem characterizing Pisot numbers whose minimal polynomials are in

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.

Unimodular Roots of Special Littlewood Polynomials

AbstractWe call α(z) = a0 + a1z + · · · + an–1zn–1 a Littlewood polynomial if aj = ±1 for all j. We call α(z) self-reciprocal if α(z) = zn–1α(1/z), and call α(z) skewsymmetric if n = 2m + 1 and am+j = (–1)jam–j for all j. It has been observed that Littlewood polynomials with particularly high minimum modulus on the unit circle in ℂ tend to be skewsymmetric. In this paper, we prove that a skewsymmetric Littlewood polynomial cannot have any zeros on the unit circle, as well as providing a new proof of the known result that a self-reciprocal Littlewood polynomial must have a zero on the unit circle.

Littlewood Pisot numbers

A Pisot number is a real algebraic integer, all of whose conjugates lie strictly inside the open unit disk; a Salem number is a real algebraic integer, all of whose conjugate roots are inside the closed unit disk, with at least one of them of modulus exactly 1. Pisot numbers have been studied extensively, and an algorithm to generate them is well known. Our main result characterises all Pisot numbers whose minimal polynomial is a Littlewood polynomial, one with { + 1 ,- 1 } -coefficients, and shows that they form an increasing sequence with limit 2. It is known that every Pisot number is a limit point, from both sides, of sequences of Salem numbers. We show that this remains true, from at least one side, for the restricted sets of Pisot and Salem numbers that are generated by Littlewood polynomials. Finally, we prove that every reciprocal Littlewood polynomial of odd degree n ⩾ 3 has at least three unimodular roots.