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Statement

Setting (pp. 1--5). $\mathbb U_n$, $M(F)$, $m(F)$, $W(F)$ and $M_n$, $m_n$, $W_n$ are as on the page for the conjecture on p. 4. For even $n$, the skew-symmetric polynomials are the $F\in\mathbb U_n$ with $F(z)=\pm z^nF(-1/z)$, where the sign has to be $(-1)^{n/2}$ (p. 4); they have $n/2+1$ free coefficients (p. 5). $M_n^*$, $m_n^*$ and $W_n^*$ are defined as $M_n$, $m_n$ and $W_n$ but over skew-symmetric polynomials of degree $n$ only (p. 5).

Conjecture (p. 5). As $n\to\infty$ through even values, $M_n^*\to M$, $m_n^*\to m$ and $W_n^*\to W$, where $M$, $m$, $W$ are the limits of $M_n$, $m_n$, $W_n$ conjectured on p. 4.

The paper presents this as suggested by Fig. 1 (p. 3), where for even $10\le n\le50$ the skew-symmetric optimum usually coincides with the unrestricted one and otherwise usually differs only slightly; the largest exception it reports is $W_{24}=0.8344$ against $0.9528$ for the best skew-symmetric polynomial (pp. 4--5).

Scope

A conjecture, not a theorem. Since $M_n\le M_n^*$ and $m_n\ge m_n^*$ by definition, the skew-symmetric values bound the unrestricted extremes from one side only.

Read depth

Claims checked: the definitions and the conjecture were read on the page images of the print. Nothing here is independently reviewed.

Dependencies

Conjecture, p. 4, which supplies the limits named here.

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 skew-symmetric minima $M_n^*$ are upper bounds for $M_n$, so they cannot give the lower bound the problem asks for; this conjecture bears on the problem only together with the conjecture on p. 4.