Problems Around Polynomials — Problem 4

Definitions and mathematical statement

Let $p,q\in\mathbb R[z]$ be nonzero polynomials, of arbitrary degrees, and put $s(z)=p(z)+iq(z)$. Their zero multisets include multiplicities and are invariant under complex conjugation. The problem asks for geometric localization and counting results for the complex zeros of $s$ in terms of these two multisets and the relative leading coefficient of $p$ and $q$.

The last datum matters: if $P,Q$ are the corresponding monic polynomials, multiplication of $s$ by a nonzero real constant reduces the question to \[ S_t(z)=P(z)+itQ(z),\qquad t\in\mathbb R\setminus\{0\}. \] Thus the input may be specified as two conjugation-invariant finite multisets and a nonzero real number $t$. Common roots of $P$ and $Q$ are allowed and must be accounted for, with the actual multiplicities they have in $S_t$.

Find sharp restrictions on which regions of the plane can contain these zeros and how many zeros can occur in such regions, counted with multiplicity. Seek conditions stated directly through the root geometry of $P$ and $Q$, for example their order or separation on the real line and their placement in half-planes. The classical case of simple interlacing real zeros motivates the question; the target allows nonreal and multiple zeros and unequal degrees. A complete answer should specify both its hypotheses and the region or root count it guarantees.

Short English statement

Extend root-location rules for a real polynomial plus an imaginary multiple of another real polynomial beyond the interlacing case.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2200008 (AMR-021-0008), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Shapiro - Problems Around Polynomials: The Good, The Bad and The Ugly (2015), source-order item 8 (Problem 4 as printed). Original source indexed by AMR-021-0008. https://arxiv.org/abs/1503.05295.