bibkey: ostrovskii2025amplitude authors: Dmitrii M. Ostrovskii and Pavel S. Shcherbakov year: 2025 title: “Amplitude maximization in stable systems, Schur positivity, and some conjectures on polynomial interpolation” doi: 10.48550/arXiv.2508.13554 url: https://arxiv.org/abs/2508.13554v2 claim: “Conjecture 6.2 asserts the shifted Schur-ratio bound for self-conjugate grids in the closed unit disk, together with a hook-Schur bound under nonnegative real parts.” strata_touched:

• D5/S3/Analytic/Interpolation/SelfConjugateGridSchurRatioRefutation license: citation-only triage: anchor

Self-conjugate-grid interpolation

Verified locator

arXiv:2508.13554v2, https://arxiv.org/abs/2508.13554v2, DOI 10.48550/arXiv.2508.13554. The source archive at https://arxiv.org/e-print/2508.13554v2 contains shadrin.tex, partitions.tex and notation.tex. Definition 6.1 and Conjecture 6.2 are on PDF page 15; the definition of Q is on page 13; the elementary and complete homogeneous functions are equation (17), page 7; the tableau and Kostka definitions are on page 8, equation (19).

Source statements

Definition 6.1, page 15:

A list $z_{1:n}\in {\mathbb{C}}^n$ is self-conjugate if $(i)$ for any $z\in \{z_1,\dots ,z_n\}$ with $\Im (z)\ne 0$, the conjugate $\bar{z}$ is also contained in $\{z_1,\dots ,z_n\}$; $(ii)$ $z_{1:n}$ contains even number of copies of each $z\in \mathbb{R}$.

Conjecture 6.2 (Equivalent to Conjecture 6.1), page 15:

For all $0\le k<n\le t$ and self-conjugate $z_{1:n}\in {\mathbb{D}}^n$, $|Q_{t,n,k}(z_{1:n}+1_n)|\le 1$. Moreover, if the grid additionally satisfies $\Re (z_{1:n})\in {\mathbb{R}}_{+}^n$, then $|s_{(t-n|n-k-1)}(z_{1:n})|\le \left(\genfrac{}{}{0ex}{}{t}{n}\right)|e_{n-k}(z_{1:n})|$.

Page 13:

Let us define the rational multivariate function $Q_{t,n,k}:{\mathbb{C}}^n\to \mathbb{C}$, symmetric in its arguments, by

$$Q_{t,n,k}({\zeta }_{1:n}):=\sum _{d=0}^{t-n}(-1{)}^d\frac{\left(\genfrac{}{}{0ex}{}{t}{n+d}\right)s_{(d|n-k-1)}({\zeta }_{1:n})}{\left(\genfrac{}{}{0ex}{}{t}{n}\right)e_{n-k}({\zeta }_{1:n})}.$$

Page 8:

A semi-standard Young tableau (SSYT) with shape $\lambda \in \mathrm{Par}$ is a two-dimensional array $T$ that fills the cells of the Young diagram of $\lambda$ with positive integers, such that the entries (a) srtictly increase in each column; (b) do not decrease in each row.

Equation (19) is $s_{\lambda }={\sum }_{\mu \in \mathrm{Par}}{K}_{\lambda \mu }{m}_{\mu }$, where the Kostka number counts SSYTs of the specified shape and type. The spelling “srtictly” is in the source.

Encoding and scope

The source’s closed disk is expressed by the complex norm, and its nonnegative orthant by nonnegative real parts. Fin n relabels the source indices 1,…,n by 0,…,n-1. Elementary functions sum over d-element subsets. Complete homogeneous functions sum over Sym (Fin n) d, multisets of cardinality d, each with one sorted, weakly increasing tuple. A hook tableau consists of a corner c, an arm multiset of cardinality a whose entries are at least c, and a leg set of cardinality b whose entries exceed c. Sorting these gives the weak row and strict column. Grouping tableau monomials by weight and then permuted weights gives the source’s Kostka expansion; the finite tableau sum is its specialization, counted once per tableau.

The refutation retains both conjectured clauses. Complex division is totalized in Lean; the counterexample denominator is 3/5, so no singular-value convention is used. The source’s equivalence with Conjecture 6.1 is cited, rather than formalized here. The source’s independent proved results are outside this module’s conclusion.