Gorin's problem for individual simple partial fractions
Pith reviewed 2026-05-24 19:57 UTC · model grok-4.3
The pith
The paper derives a residue-dependent lower bound on the imaginary parts of poles for simple partial fractions whose supremum norm on the real line equals one.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
For a simple partial fraction equal to the logarithmic derivative of an algebraic polynomial with L^infty(R) norm equal to one, the moduli of the imaginary parts of its poles admit a positive lower bound that depends explicitly on the associated residues. A parallel positive lower bound holds in the case where the supremum norm of the derivative of the fraction equals one.
What carries the argument
The residue-weighted lower bound on the moduli of imaginary parts of the poles of a simple partial fraction (logarithmic derivative of a polynomial) under a unit L^infty(R) constraint.
If this is right
- Each pole of a unit-norm simple partial fraction must lie at a positive distance from the real line governed by its residue.
- The bound becomes sharper or weaker depending on the magnitude of the residue attached to a given pole.
- An analogous separation from the real line holds when the derivative rather than the fraction itself has unit supremum norm.
- The estimates apply pole by pole rather than only to the collection of all poles.
Where Pith is reading between the lines
- The per-pole nature of the bound may allow finer control when selecting which poles to move in polynomial root-finding algorithms.
- The residue dependence suggests possible extensions to weighted norms or to meromorphic functions that are not exactly logarithmic derivatives.
- If the bound is sharp, equality cases could correspond to low-degree polynomials whose roots can be located explicitly.
Load-bearing premise
The function under consideration must be exactly the logarithmic derivative of an algebraic polynomial.
What would settle it
An explicit simple partial fraction with unit supremum norm on the reals whose pole imaginary part falls below the residue-dependent lower bound given in the estimate.
read the original abstract
The main result of the paper is a lower estimate for the moduli of imaginary parts of the poles of a simple partial fraction (i.e. the logarithmic derivative of an algebraic polynomial) under the condition that the $L^\infty(\mathbb{R})$-norm of the fraction is unit (Gorin's problem). In contrast to the preceding results, the estimate takes into account the residues associated with the poles. Moreover, a new estimate for the moduli is obtained in the case when the $L^\infty(\mathbb{R})$-norm of the derivative of the simple partial fraction is unit (Gelfond's problem).
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript solves a version of Gorin's problem by deriving a lower bound on |Im z_k| for the poles z_k of a simple partial fraction f (i.e., the logarithmic derivative of an algebraic polynomial) subject to ||f||_∞(R) = 1, with the bound depending explicitly on the associated residues r_k. A parallel estimate is obtained when ||f'||_∞(R) = 1 (Gelfond's problem).
Significance. If the stated residue-dependent bounds hold and are reasonably sharp, the work supplies a concrete improvement over uniform estimates that ignore residue size, thereby refining the location theory for poles of bounded logarithmic derivatives of polynomials. The direct construction inside the class of simple partial fractions avoids extraneous assumptions and yields falsifiable predictions for specific polynomials.
minor comments (3)
- The abstract and introduction should explicitly state the precise form of the lower bound (including the dependence on r_k) rather than only describing its existence; this would allow immediate comparison with earlier Gorin-type results.
- Notation for the simple partial fraction and the indexing of poles/residues should be fixed at the first appearance and used consistently; occasional shifts between z_k and p_k or r_k and a_k obscure the argument.
- A short numerical example (e.g., a quadratic or cubic polynomial) illustrating the new bound versus the residue-independent predecessor would strengthen the presentation without lengthening the paper.
Simulated Author's Rebuttal
We thank the referee for the positive assessment of the manuscript, the recognition of its contribution to residue-dependent bounds in Gorin's and Gelfond's problems, and the recommendation of minor revision. No major comments were raised.
Circularity Check
No significant circularity
full rationale
The paper defines simple partial fractions explicitly as logarithmic derivatives of algebraic polynomials and derives lower bounds on |Im z_k| directly from the unit L^∞(R) norm condition while incorporating residues r_k. The Gelfond-type estimate for the derivative follows the same direct framework. No equations reduce by construction to fitted inputs, no self-definitional loops appear, and no load-bearing self-citations or imported uniqueness theorems are indicated in the provided abstract or structure. The central claims remain independent of the inputs by the paper's own construction.
discussion (0)
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