A393039 Maximum Mahler measure of all quartic polynomials with coefficients in {-1, 0, 1}.

2, 1, 5, 3, 7, 2, 1, 3, 7, 5, 5, 4, 1, 7, 6, 7, 9, 0, 0, 8, 6, 5, 9, 9, 2, 7, 4, 8, 7, 6, 3, 8, 6, 4, 0, 0, 1, 7, 7, 5, 2, 3, 7, 9, 8, 8, 6, 9, 9, 6, 5, 7, 1, 9, 7, 1, 4, 4, 6, 7, 1, 2, 6, 9, 4, 8, 4, 1, 8, 9, 4, 0, 5, 9, 1, 6, 8, 3, 2, 5, 2, 5, 9, 9, 9, 2, 9, 1, 4, 5, 1, 0, 0, 6, 6, 5, 6, 5, 6, 6, 9, 2, 8, 8, 5

Offset

1,1

Comments

Decimal expansion of the Mahler measure of x^4 - x^3 + x^2 + x + 1 (or x^4 + x^3 + x^2 - x + 1).

Maximum Mahler measure M(d) of all polynomials degree d with coefficients in {-1, 0, 1}:

d=1 1.0000000000000000...

d=2 1.6180339887498948...

d=3 1.8392867552141611...

d=4 2.1537213755417679...

d=5 2.2276901986994872...

d=6 2.4966982051626220...

d=7 2.6420852480514064...

d=8 2.72501233853003345...

d=9 2.920761693039644267...

d=10 3.166249388941174755...

d=11 3.334625385907742119...

d=12 3.556393149272513432...

d=13 3.558023079320484749... 3.57536188126420052136431... (reducible)

d=14 3.644853435884449341... 3.74089481847209666720106... (reducible)

d=15 3.776452805541376875...

d=16 3.878475003367087683...

in first column are records for irreducible polynomials.

Conjecture 1: For every d M(d+1) > M(d).

Conjecture 2: M(d) < sqrt(d+1). - Charles R Greathouse IV, Feb 05 2026

Links

Table of n, a(n) for n=1..105.

P. Borwein and T. Erdélyi, Questions about polynomials with {0, -1, +1} coefficients: Research problems 96-3, Constructive Approximation, Vol. 12 (1996), pp. 439-442.

Musbahu Idris and Jean-Marc Sac-Épée, Algorithmic aspects of Newman polynomials and their divisors, arXiv:2601.11486 [math.NT], 2026.

Michael J. Mossinghoff, Polynomials with restricted coefficients and prescribed noncyclotomic factors, LMS Journal of Computation and Mathematics, Volume 6 (2003), pp. 314-325.

Wikipedia, Mahler measure.

Index entries for algebraic numbers, degree 4.

Formula

Larger of the real roots of x^4 - 3*x^3 + 3*x^2 - 3*x + 1.

Example

2.1537213755417679...

Mathematica

pol = x^4 - x^3 + x^2 + x + 1; vec = NSolve[pol == 0, x, WorkingPrecision -> 110];
prod = 1; Do[max = Max[1, Abs[x /. vec[[n]]]]; prod = prod max, {n, 1, Length[vec]}]; RealDigits[prod, 10, 105][[1]]

Prog

(PARI) Mahler(P)=vecprod(apply(k->max(abs(k), 1), polroots(P)))
topol(n)=Pol(apply(k->k-1, digits(n, 3))) \\ coefficients in {-1, 0, 1}
do(d)=my(r=1); for(n=2*3^d, 3^(d+1)-1, my(P=topol(n), t=Mahler(P)); if(t>r, r=t)); r
do(4) \\ Charles R Greathouse IV, Jan 30 2026
(PARI) polrootsreal(x^4 - 3*x^3 + 3*x^2 - 3*x + 1)[2] \\ Charles R Greathouse IV, Feb 05 2026

Crossrefs

Analogs for other degrees: A000007 (d=1), A001622 (d=2), A058265 (d=3), this sequence (d=4), A393038 (d=5), A392802 (d=6), A392808 (d=7).

Cf. A073011, A219300, A306078, A306079, A316605-A316610, A390906.

Sequence in context: A262211 A345967 A094512 * A182650 A127367 A054084

Adjacent sequences: A393036 A393037 A393038 * A393040 A393041 A393042

Keyword

nonn,cons

Author

Artur Jasinski, Jan 31 2026

Status

approved