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Statement

Setting. $\mathbb U_n$, $M(F)$, $m(F)$, $W(F)$, $M_n$, $m_n$, $W_n$ are as on the page for the conjecture on p. 4, and $M_n^*$, $m_n^*$, $W_n^*$ are the skew-symmetric analogues of the conjecture on p. 5.

Computation (pp. 3--5). The search examined all $F\in\mathbb U_n$ for $n\le52$ (p. 3), and skew-symmetric polynomials of even degree up to $n=104$ (p. 5). Fig. 1 (p. 3) plots $M_n$, $m_n$, $W_n$ for $10\le n\le50$, with $M_n^*$, $m_n^*$, $W_n^*$ for even $n$ in that range as dots; Fig. 2 (p. 5) plots $M_n^*$, $m_n^*$, $W_n^*$ for even $10\le n\le100$. The numerical values and attaining polynomials are said to be on the author's home page (p. 4) and, for the skew-symmetric search, in online tables (p. 5); they are not printed in the paper.

Values the paper reports.

• Among all polynomials tested, the degree-10 Barker polynomial has the smallest $M(F)$, $1.1464$; the degree-12 Barker polynomial has the largest $m(F)$, $0.8375$, and the smallest $W(F)$, $0.5493$ (p. 7).
• $M_{102}^*=1.2633\ldots$ (Fig. 4, pp. 7--8), the smallest $M(F)$ among all skew-symmetric polynomials of degrees $84\le n\le104$; the attaining polynomial has $G(F)=5.7973$ and $m(F)=0.0985$ (p. 8). Another skew-symmetric polynomial of degree 102 has $M(F)=1.2647$, and the tenth smallest value of $M(F)$ is $1.2876$ (p. 8).
• $W_{24}=0.8344$, against $0.9528$ for the best skew-symmetric polynomial of degree 24 (pp. 4--5).
• The skew-symmetric polynomial of degree 94 shown in Fig. 5 has $W(F)=0.733\ldots$, the smallest annulus of all skew-symmetric polynomials of degrees $72\le n\le104$, with $M(F)=1.3162\ldots$ and $m(F)=0.5830\ldots$ (p. 11).

Method (pp. 4, 11--12). The search used the symmetries $F(z)\mapsto z^nF(1/z)$, $F\mapsto-F$, $F(z)\mapsto F(-z)$, which leave $M(F)$ and $m(F)$ unchanged (equation (11), p. 4). Writing $F=F_1+F_2$ with, for example, $F_1=\sum_{k=0}^{15}a_kz^k$, the values of every $F_1$ were precomputed at a small set of points on the upper half of the unit circle (typically 32), and combinations giving values too large or too small there were discarded; survivors were examined more carefully (pp. 11--12). Total run time was on the order of 30 years on a single core (p. 12).

Completeness (p. 12, section 8). The paper says the reported values of $m(F)$, $M(F)$, $W(F)$ are trustworthy, having been recomputed for the candidates with a straightforward program using the trivial bounds on the first and second derivatives. It says it is not completely certain that all extremal polynomials were found: network or storage faults during the months-long distributed run may have gone undetected, a probability it calls slight.

Scope

Finite computation. Nothing here bounds $M_n$, $m_n$ or $W_n$ for $n>52$; the skew-symmetric search covers a proper subfamily, and $M_n\le M_n^*$, $m_n\ge m_n^*$. No code or data files are printed in the paper, so the values cannot be reproduced from it alone.

Read depth

Claims checked: the reported ranges, values and the completeness statement were read on the page images of the print. No value was recomputed.

Dependencies

None.

Source. Andrew Odlyzko, "Search for Ultraflat Polynomials with Plus and Minus One Coefficients," in Connections in Discrete Mathematics, pp. 39--55, Cambridge University Press, 2018, doi:10.1017/9781316650295.004; the version read, the author's revised version of 18 May 2017, and its page numbering are named on the source card.

Bears on

• Problem 1150: the search computes $M_n$ for each $n\le52$, and the smallest $M(F)$ it reports among all polynomials tested is $1.1464$, at degree 10; a finite range cannot settle the problem's all-large-$n$ question, and the paper offers the data as support for the conjecture on p. 4.