Sheil-Small's covering problem: does a self-inversive polynomial cover a disc of radius $\max|a_k|$? (Problem 4.24)
Statement
Let $P(z)=\sum_{k=0}^{n}a_kz^k$ be a self-inversive polynomial: its zeros are invariant under $\zeta\mapsto 1/\bar\zeta$ (if $\zeta$ is a zero of multiplicity $m$, then so is $1/\bar\zeta$), equivalently $a_k=c\,\overline{a_{n-k}}$ for all $k$ and some fixed unimodular constant $c$. Put $A=\max_{0\le k\le n}|a_k|$. Is it true that $w=P(z)$ maps the open unit disc $D$ onto a domain containing some disc of radius $A$? (The centre of that disc is left unspecified.)
Acceptance. FULLY RESOLVES: EITHER a proof that every self-inversive polynomial $P$ of degree $n$ has $P(D)$ containing some open disc of radius $A=\max_k|a_k|$, OR an explicit self-inversive polynomial (exact algebraic coefficients) whose image $P(D)$ contains no disc of radius $A$ - a finite counterexample, certifiable by showing the omitted value nearest the image has modulus $<A$ via interval arithmetic. ADVANCES: a proof for all self-inversive $P$ of degree $\le n_0$ for an explicit $n_0$; a proof for the real-coefficient (self-reciprocal) subclass; or a sharp lower bound on the covering radius weaker than $A$.
Background
Posed by T. Sheil-Small; Problem 4.24 in W. K. Hayman & E. F. Lingham, Research Problems in Function Theory (Fiftieth Anniversary Edition, Springer 2019), source of record arXiv:1809.07200, whose Update 4.24 states 'No progress on this problem has been reported to us.' NOTE on the statement: the source asks only for a domain containing a disc of radius $A$ and leaves the centre unspecified (do not assume it is centred at $P(0)$). This covering statement is DISTINCT from the known Cordova-Ruscheweyh-type covering theorem (see 'Note on the covering theorem for complex polynomials', arXiv:1410.6772), which asserts that the image of $D$ under a general polynomial $q$ contains a disc of a Chebyshev-type radius $n(q)$ centred at $0$ - a different radius (not $A=\max|a_k|$) and not restricted to self-inversive $P$. Self-inversive polynomials are otherwise studied for zero-location (Cohn's theorem), not for this image-covering property; general context is Sheil-Small, Complex Polynomials (Cambridge University Press, 2002). The self-inversive symmetry forces $|P|$ to be a balanced trigonometric polynomial on $\partial D$; low-degree cases ($n\le4,5$) are finite real-algebraic systems. Vetted open as of 2026-07-06 (high confidence; 'No progress' marker, and the covering property is unresolved by the Cordova-Ruscheweyh line).
References
Investigations · 0
No published investigations yet. This problem is unclaimed territory.