Newman Polynomials: Theory & Algorithms

• Newman polynomials are integer polynomials with coefficients in {0,1} and a constant term of 1, fundamental in algebra, analysis, and number theory.
• They exhibit confined root distributions and admit constructive bounds, connecting to Littlewood and Borwein polynomials through their extremal and divisibility properties.
• Algorithmic methods such as MILP and remainder-graph searches are employed to probe their divisibility, flatness properties, and open computational complexities.

A Newman polynomial is an integer polynomial all of whose coefficients lie in $\{0,1\}$, with the constant term set to 1. Formally, for degree $n$, it is written as $P(x) = 1 + \sum_{k=1}^n a_k x^k$, $a_k \in \{0,1\}$. The class is essential in the study of restricted-coefficient polynomials across algebra, analysis, and number theory, and features prominently in extremal root-count, divisibility, Mahler measure, flatness, and algorithmic investigations. The broader context includes relations to Littlewood polynomials (coefficients $\{\pm1\}$), Borwein polynomials (coefficients $\{0, \pm1\}$), and Salem numbers.

1. Definition, Notation, and Basic Properties

A Newman polynomial is a univariate integer polynomial $F(x)=x^d+a_{d-1}x^{d-1}+\cdots+a_1x+a_0$ where $a_i \in \{0,1\}$, with $a_0=1$; equivalently, $P(x)=1+\sum_{k=1}^{n} a_k x^k$, $a_k\in\{0,1\}$ (Drungilas et al., 2016, Idris et al., 16 Jan 2026). In analytic contexts, this extends to trigonometric polynomials $P(z)=\sum_{j=0}^{n-1} \epsilon_j z^j$, $\epsilon_j \in \{0,1\}$, $\epsilon_0=1$ (Abdalaoui, 2015).

Key properties:

• Real roots: No non-negative real root exists for any Newman polynomial since $P(0)=1$ and all other coefficients are non-negative.
• Root location: Roots are confined to the annulus $1/\tau < |x| < \tau$, with $\tau=(1+\sqrt{5})/2$ (Drungilas et al., 2016).
• Inclusion: Newman polynomials are Borwein polynomials ($\{-1,0,1\}$ coefficients with nonzero constant term).

Examples of low-degree Newman polynomials include $1+x$, $1+x+x^2$, $1+x^2+x^5$, and $1+x+x^3+x^7$ (Drungilas et al., 2016).

2. Real Root Counts and Constructive Bounds

Classical Jensen-type arguments restrict the number of real roots in $[0,1]$ for any $P_n(x)=1+\sum_{k=1}^n a_k x^k$, $|a_k|\le A$, to at most $C(A)\sqrt{n}$ (Jacob et al., 2024). For Newman polynomials ($A=1$), this gives a uniform $O(\sqrt{n})$ bound.

The Jacob–Nazarov constructive algorithm establishes that for any integer $r\geq1$, there exists a Newman polynomial of degree $n=O(r^2)$ with at least $r$ distinct real roots in $[0,1]$ (Jacob et al., 2024). The methodology involves:

• Prescribing $r$ target points in a fixed interval $I(a)=[1-2a,1-a]$.
• Employing a balanced coefficient condition to construct an infinite series solution $Q(x) = \sum_k E_k L^{-(L+k)} x^k$, $E_k \in \{0, (-1)^k\}$, with vanishing at prescribed nodes.
• Reducing the infinite system to a finite, controlled dynamical system ensuring the coefficient selection can be made algorithmically in polynomial time.

This matches the extremal $\sqrt{n}$ root count rate; the underlying constant $c$ currently appears suboptimal but of the correct order.

3. Divisibility, Multiples, and Algorithmic Methods

The problem of which integer polynomials divide Newman polynomials is central to understanding coefficient-restricted algebraic structures. The most general algorithmic approach is via an MILP (Mixed-Integer Linear Programming) framework (Idris et al., 16 Jan 2026):

• Given a monic $p(x)$, one seeks $q(x)$ such that $F(x) = p(x)q(x)$ has all coefficients in $\{0,1\}$.
• This translates to a linear convolution system in the $b_j$ (coefficients of $q$), subject to the constraints $c_k \in \{0,1\}$ for each coefficient in $F$.
• The absence of strictly positive real roots in $p$ is a necessary condition.
• MILP solvers (e.g., Gurobi) have enabled actual searches for multiples up to degrees beyond 150.

Results obtained:

• Among $8,438$ irreducible, reciprocal integer polynomials with $M(p)<1.3$ and degrees up to $180$, all but three have found Newman multiples of degree $\le1000$ (Idris et al., 16 Jan 2026).
• The polynomial $x^{10}-x^8-x^5+x+1$ (Mahler measure $\approx1.4194$) divides no Newman polynomial, tightening the universal threshold $\sigma$ conjecture to $\sigma\le 1.419404632$ (Idris et al., 16 Jan 2026).

For divisibility into other classes:

• Any Borwein polynomial (degree $\le8$) dividing a Newman polynomial also divides a Littlewood polynomial.
• Classifications exist for degrees up to $11$ (Drungilas et al., 2016, Drungilas et al., 2018).
• Specialized remainder-graph algorithms implement automaton-style searches for divisibility, relying on bounded remainder sets and derivative evaluations at roots off the unit circle.

Newman polynomial multiples of Lehmer’s Salem polynomial squared exist for degrees $20 \le d \le 150$, but none found for the cube of Lehmer’s polynomial up to degree $160$ (Idris et al., 16 Jan 2026).

4. Zero Distribution, Unit Disk Roots, and Pisot Obstructions

For degree $n \geq 7$ and any integer $3\leq k\leq n-3$, there exists a Newman polynomial $f(z)$ of degree $n$ with exactly $k$ zeros strictly inside the unit disk $\mathcal{D}=\{z\in\mathbb{C}:|z|<1\}$, and none on the boundary $|z|=1$ (Hare et al., 2019). The constructive mechanism:

• Use a large-minimum-modulus base polynomial (e.g., $P_{38}$ of degree $38$ with modulus $>2$ on the unit circle) and an addition lemma to adjust the zero count.
• Explicit coefficient-pattern constructions suffice for the small and large $k$ regime.

Obstructions and exceptional cases:

• $(k,n)=(3,6)$ is uniquely inadmissible for $k=3$.
• Endpoint cases $(1,n)$, $(n-1,n)$ correlate directly to minimal polynomials of real Pisot numbers for odd $n$, and are prohibited for even $n$.
• Next-to-endpoint cases are governed by complex Pisot numbers, with infinite families and sporadic exceptions determined by precise congruences modulo $30$.

Statistical distribution conjecture: Empirical evidence indicates the normalized root count $N(f)$ for random Newman polynomials approaches a standard normal law (CLT behavior) as $n\to\infty$ (Hare et al., 2019).

5. Flatness Properties, Mahler Measure, and Spectral Applications

A sequence of normalized Newman polynomials can attain $L^\alpha$-flatness in $0<\alpha<2$ (Abdalaoui, 2015):

• Construction via Singer sets in $Z/qZ$, yielding polynomials $P_q(z)$ for which $|P_q(e^{2\pi i r/q})| \to 1$ for all $r\neq 0$.
• Marcinkiewicz-Zygmund inequalities link finite root evaluations to $L^\alpha$-norms, implying pointwise flatness except for a single spike at $z=1$.
• For $\alpha\geq 4$, it is proven that no sequence of normalized Newman polynomials is $L^\alpha$-flat ($\|P_n\|_4\to\infty$).

Corollaries:

• Mahler’s problem receives a positive answer in the Newman class: sequences with Mahler measure tending to $1$ are constructible.
• The maximal $L^1$-norm for normalized idempotent polynomials with $0$-$1$ coefficients is exactly $1$, answering Bourgain’s question.
• The same flatness constructions yield ergodic, $\sigma$-finite, measure-preserving transformations with simple Lebesgue spectrum, resolving Banach’s Scottish Book problem.

6. Extremal Minimum Modulus and Finite Search Problems

Let $\mu(n)$ denote the largest possible minimal modulus on $|z|=1$ for a length-$n$ Newman polynomial, i.e., $$\mu(n)=\sup_{\{a_j\}} \min_{|z|=1}|z^{a_1}+\cdots+z^{a_n}|$$ (Mercer, 2017). Known results:

• $\mu(3)=M(0,1,3)\approx0.607346$; $\mu(4)=M(0,1,2,4)\approx0.752394$.
• For $n=5$, $1\leq \mu(5)\leq 1+\pi/5 \approx1.628$.

The main challenge is the transition from the infinite set of exponent patterns to a finite, computable structure. Reductions involve GCD normalization, palindromic symmetry, and equispaced sets. Exceptional gap patterns are handled by direct linear algebra and checking finite systems. Beyond $n=5$, the combinatorial explosion of cases remains an obstacle; whether $\mu(n)$ can be determined algorithmically in general is open.

Boyd conjectured that $\mu(n)>1$ for all $n\geq 6$ and possibly unbounded growth $\mu(n)\to\infty$ as $n\to\infty$; current constructions show $\mu(n)\gtrsim n^{0.14}$ for some infinite subsequence.

7. Structural Comparison, Open Questions, and Algorithmic Complexity

Structural relations:

• $N\subset B$, $L\subset B$, but neither $N\subset L$ nor $L\subset N$; explicit enumerations confirm this up to degree $9$ for Borwein polynomials and up to degree $11$ for Newman polynomials (Drungilas et al., 2016, Drungilas et al., 2018).
• Periodic coefficient sets necessitate balance (alternation of nonzero values) to admit arbitrarily many roots; otherwise, Descartes’ Rule implies $O(M)$ root limitation (Jacob et al., 2024).

Algorithmic complexity:

• Deciding Newman divisibility is algorithmically hard (NP-hard MILP, automaton graph search size exponential in degree) but tractable for moderate $n$ (Idris et al., 16 Jan 2026, Drungilas et al., 2018).
• Deciding Littlewood multiples is NP$\cap$coNP via automaton-based reduction; P status is unresolved.
• Exact classification of Newman quadrinomials admitting Littlewood multiples extends up to degree $15$; in a few cases, the minimal multiple degree can exceed $32,000$ (Drungilas et al., 2018).

Open questions:

• Existence of a universal Mahler measure threshold $\sigma>1$ guaranteeing Newman multiples remains unsettled; the best current bound is $\sigma\le 1.419404632$ (Idris et al., 16 Jan 2026).
• Whether any Newman quadrinomial fails to divide a Littlewood polynomial at some degree.
• Saturation and optimality of constants in root count bounds—a plausible implication is further improvement might remain possible within coefficient restrictions.
• Flatness and distribution conjectures (asymptotic CLT for zero count, ultraflat sequences).

In summary, Newman polynomials serve as a focal point connecting extremal root-count theory, computational divisibility, algebraic number theory, and harmonic analysis. Recent algorithmic, constructive, and statistical advances have elucidated their depth, while open problems persist at the interface of root location, divisibility, and computational complexity.