Local Single-Pole Approximation

• Local single-pole approximation is a method that models a complex object as dominated by a single pole in a specific local window.
• It is applied across domains—from polynomial homotopy continuation and feedforward OTA reduction to waveguide QED—demonstrating varied techniques to isolate dominant behavior.
• The approach simplifies analysis in control and spectral theories but remains sensitive to nearby competing singularities, emphasizing the need for precise local criteria.

Searching arXiv for papers on "local single-pole approximation" and closely related usages across domains. Local single-pole approximation is a domain-dependent analytical reduction in which a complicated object is treated, within a restricted neighborhood of interest, as if its behavior were controlled by one dominant pole. In recent literature this idea appears in several technically distinct forms: as a local asymptotic model for Taylor coefficients of singular homotopy paths, as a recursive pole-cancellation scheme for multistage feedforward OTAs, and as a local resonance reduction of a multi-pole scattering amplitude to a Fano form. Closely related but not identical constructions include simple-pole bases for system level synthesis, local linearizations of rational matrices, and reduction-compatible single-pole reference models in many-body screening theory (Verschelde et al., 2024, Lee, 2024, Xiang et al., 8 Jul 2026, Fisher et al., 2022, Dopico et al., 2019, Imoto, 24 Mar 2026).

1. Terminological scope and shared structure

The phrase is not used as a single universal term of art. Instead, the common structure is that a higher-complexity analytic, dynamical, or spectral object is reduced locally to a one-pole description whose validity is tied to a specific window: an index range, a frequency interval, a stage of a circuit, or a target spectral set.

Domain	Locality notion	One-pole mechanism
Polynomial homotopy continuation	Finite index window of Taylor coefficients	Dominance of one nearby singularity in coefficient ratios
Feedforward OTA stability	Stage-by-stage recursive reduction	Feedforward zero nearly cancels preceding pole
Waveguide QED frequency conversion	Narrow window around target inelastic peak	One resonant pole plus smooth background
SLS in control	Pole set chosen near unknown optimal poles	Simple-pole basis functions $1/(z-p)$
Rational-matrix theory	Target set $E\subset\mathbb F$	Local preservation of pole-zero structure
UEG screened exchange	Reference kernel with constant pole scale $\mu$	Exactly reducible one-pole screening model

A plausible unifying characterization is that local single-pole approximations are not primarily about replacing an entire model by a globally one-pole transfer function. They are instead about isolating a regime in which one pole, or one pole-canceling pair, dominates the behavior relevant to computation, synthesis, or interpretation.

2. Singular homotopy paths and coefficient-ratio asymptotics

In polynomial homotopy continuation, a polynomial homotopy $F(x,t)=0$ defines solution paths $x(t)$ beginning at known regular solutions and ending at solutions of a target system. When the endpoint at $t=1$ is an isolated singular solution, the Taylor series of the path converges only logarithmically, which makes direct summation slow and makes extrapolation highly sensitive to the analytic structure near the disk of convergence (Verschelde et al., 2024).

The canonical model is

$$x^q=(1-t)^p,\qquad t\in[0,1],\quad p\ge 2,\ q\ge 1,$$

with solution path

$x(t)=(1-t)^{p/q}.$

If

$x(t)=a_0+a_1 t+a_2 t^2+\cdots,$

then the nonzero coefficient ratios satisfy $a_n/a_{n+1}\to 1$ very slowly. The paper identifies the decisive condition for acceleration: a lone nearby singularity induces a clean inverse-power asymptotic expansion

$E\subset\mathbb F$0

This is the local single-pole behavior, and it matches the structure targeted by Richardson extrapolation and related sequence transformations.

The analytic foundation is Fabry’s ratio theorem. For a Taylor series $E\subset\mathbb F$1, if the coefficient ratios converge to a limit $E\subset\mathbb F$2, then $E\subset\mathbb F$3 is a singular point on the boundary of the circle of convergence and the radius of convergence is $E\subset\mathbb F$4. After a coordinate change, the nearest singularity may be placed at $E\subset\mathbb F$5; if the radius is $E\subset\mathbb F$6, then asymptotically $E\subset\mathbb F$7. This makes the ratio sequence itself the object to be extrapolated.

The approximation becomes nontrivial when a second singularity is introduced, as in

$E\subset\mathbb F$8

where $E\subset\mathbb F$9 is a second singularity near the Taylor disk of convergence. The paper models the resulting coefficient sequence through a convolution

$\mu$0

showing that if one factor dominates, then $\mu$1 inherits the same $\mu$2-expansion as the dominant factor. If the two factors contribute comparably, the regular inverse-power structure is destroyed and extrapolation becomes unreliable.

The main approximation principle is therefore explicitly local. If one pole is sufficiently far away, then on a finite index range $\mu$3,

$\mu$4

with coefficients $\mu$5 valid on that range. This justifies local extrapolation: a finite window of coefficients behaves as though controlled by one dominant pole, even when the global analytic structure is more complicated.

Algorithmically, the paper tests the $\mu$6 algorithm on coefficient-derived sequences. In the ideal one-pole case, a small number of terms suffices to recover the pole accurately; when a second pole lies near the convergence disk, the algorithm can fail completely. As $\mu$7 moves farther from $\mu$8, the smallest error in the $\mu$9 table decreases substantially, consistent with the single-pole asymptotic theory.

3. Recursive one-pole reduction in multistage feedforward OTAs

In analog circuit theory, the closely related successive one-pole approximation (SOPA) is a staged stability-analysis method for multistage feedforward OTAs. Its central idea is that each feedforward stage creates a zero that can cancel the dominant pole of the preceding stage when the feedforward transconductance is sufficiently large. The preceding multistage block is then replaced by an equivalent single-pole subsystem, and the same reduction is applied recursively at the next stage (Lee, 2024).

For a two-stage feedforward OTA, the key condition is

$F(x,t)=0$0

which yields

$F(x,t)=0$1

Under this near-cancellation, the two-stage OTA can be approximated as a single-pole system. The paper states the corresponding simplified equivalent gain and pole explicitly: the gain is approximately $F(x,t)=0$2, and the pole is approximately $F(x,t)=0$3.

The three-stage case applies the same logic to the reduced two-stage block. The internal two-stage feedforward structure is first replaced by a single-pole system, and the third-stage feedforward path generates a new zero. If the second feedforward transconductance is large enough, the new zero approximately cancels the pole of the reduced two-stage block. The paper notes, however, that practical design often places the zero slightly above the pole rather than exactly on it, because of limits on how large $F(x,t)=0$4 can be and because the prior-stage gain is already large. This mismatch produces phase peaking.

The four-stage analysis repeats the reduction one level higher. The intended stability mechanism is successive cancellation: $F(x,t)=0$5 leaving a single dominant pole. The paper emphasizes that this requires progressively larger feedforward transconductances,

$F(x,t)=0$6

$F(x,t)=0$7

with the design trend

$F(x,t)=0$8

The approximation rests on several explicit assumptions: the feedforward transconductance is sufficiently large, the preceding stage can be modeled by a single dominant pole, $F(x,t)=0$9 is neglected in the derivation and later reintroduced for low-frequency gain isolation, and Miller capacitors are initially neglected for analytic simplicity. The paper contrasts feedforward compensation with Miller compensation: feedforward compensation improves phase margin through zero-pole cancellation but requires strong feedforward devices, whereas Miller compensation improves phase margin via pole splitting and often sacrifices bandwidth.

Within this literature, “local” refers less to spatial locality than to recursive locality in the circuit hierarchy: a local pole-cancellation event is used to compress one subnetwork before the next stage is analyzed.

4. Local Fano reduction in two-giant-atom waveguide QED

In waveguide QED, local single-pole approximation is used to simplify a multi-resonant single-photon frequency-conversion problem into a one-resonance-plus-background description near the target inelastic peak. The full inelastic transmission spectrum is governed by three complex resonance poles because, in the single-excitation sector, the effective atomic response is a $x(t)$0 linear problem with cubic denominator. The approximation is therefore explicitly local in frequency: it is not a global fit to the entire spectrum (Xiang et al., 8 Jul 2026).

The system consists of a two-level giant atom and a $x(t)$1-type three-level giant atom coupled to a common one-dimensional waveguide at two separated points each. Multiple coherent propagation paths arise because photons acquire phase factors between coupling points. Under the equal-spacing and symmetric-coupling conditions

$x(t)$2

the exact inelastic transmission amplitude $x(t)$3 is written with a common cubic denominator, and the target conversion window is centered near

$x(t)$4

Defining

$x(t)$5

the first-order local expansion of the inelastic transmission amplitude is

$x(t)$6

This is then rewritten as

$x(t)$7

with

$x(t)$8

The resulting local lineshape is a Fano form: a coherent superposition of a smooth background channel and a single resonant pole.

The paper identifies a background-suppression condition by defining

$x(t)$9

where $t=1$0 and $t=1$1. The background magnitude is

$t=1$2

so the suppression condition is

$t=1$3

When this holds, the background becomes negligible and the local Fano form reduces to an approximately Lorentzian single-pole line,

$t=1$4

To quantify pole dominance, the paper defines the single-pole resonance weight

$t=1$5

This leads to a phase-selection criterion: $t=1$6 Large $t=1$7 indicates strong resonant weight, weak effective background, and high expected inelastic transmission in the target window.

The local approximation also clarifies the role of propagation phase. The bright region of large $t=1$8 lies near $t=1$9, and the background-suppression condition yields the approximate curve

$$x^q=(1-t)^p,\qquad t\in[0,1],\quad p\ge 2,\ q\ge 1,$$0

Conversely, near $$x^q=(1-t)^p,\qquad t\in[0,1],\quad p\ge 2,\ q\ge 1,$$1 or $$x^q=(1-t)^p,\qquad t\in[0,1],\quad p\ge 2,\ q\ge 1,$$2, one of the coherent envelopes vanishes and conversion is strongly suppressed. Compared with both the small-atom model and the single $$x^q=(1-t)^p,\qquad t\in[0,1],\quad p\ge 2,\ q\ge 1,$$3-type giant-atom model, the two-giant-atom scheme achieves substantially enhanced inelastic transmission over a broader frequency-conversion range.

5. Related simple-pole frameworks in control and rational spectral theory

A broader mathematical family of ideas uses simple poles locally without adopting the exact phrase “local single-pole approximation.” In system level synthesis, “Approximation by Simple Poles -- Part II: System Level Synthesis Beyond Finite Impulse Response” replaces FIR truncation by rational approximants built from simple-pole basis functions in the open unit disk. The design variables are the closed-loop transfer matrices $$x^q=(1-t)^p,\qquad t\in[0,1],\quad p\ge 2,\ q\ge 1,$$4 and $$x^q=(1-t)^p,\qquad t\in[0,1],\quad p\ge 2,\ q\ge 1,$$5, and the approximation takes the form

$$x^q=(1-t)^p,\qquad t\in[0,1],\quad p\ge 2,\ q\ge 1,$$6

$$x^q=(1-t)^p,\qquad t\in[0,1],\quad p\ge 2,\ q\ge 1,$$7

The paper characterizes this as a strict generalization of the FIR-based SLS formulation of Anderson et al.; because poles are not forced to the origin, the resulting closed-loop response is not FIR and therefore does not produce deadbeat control (Fisher et al., 2022).

The approximation is “local” in the sense that the pole atoms are anchored near the unknown optimal poles and the quality is geometry-dependent. The main suboptimality result is

$$x^q=(1-t)^p,\qquad t\in[0,1],\quad p\ge 2,\ q\ge 1,$$8

where $$x^q=(1-t)^p,\qquad t\in[0,1],\quad p\ge 2,\ q\ge 1,$$9 is the worst-case geometric approximation error from the optimal pole set to the chosen poles. For Archimedes spiral pole placement, the paper states

$x(t)=(1-t)^{p/q}.$0

for sufficiently large $x(t)=(1-t)^{p/q}.$1, with $x(t)=(1-t)^{p/q}.$2. In this setting, a simple-pole basis serves as a rational alternative to horizon-truncated impulse-response models.

In rational-matrix theory, the notion of locality is formalized through local linearizations of rational matrices in subsets $x(t)=(1-t)^{p/q}.$3. A linear polynomial system matrix $x(t)=(1-t)^{p/q}.$4 is a linearization of a rational matrix $x(t)=(1-t)^{p/q}.$5 at $x(t)=(1-t)^{p/q}.$6 when it is minimal at $x(t)=(1-t)^{p/q}.$7 and there exist rational matrices $x(t)=(1-t)^{p/q}.$8, regular at $x(t)=(1-t)^{p/q}.$9, such that

$x(t)=a_0+a_1 t+a_2 t^2+\cdots,$0

The paper proves that such local linearizations preserve pole and zero elementary divisors in the target set, and it explicitly motivates this framework by rational approximations of nonlinear eigenvalue problems, where only a chosen spectral region matters (Dopico et al., 2019).

These works do not define a named local single-pole approximation. Nonetheless, they show that the local-use-of-poles viewpoint extends beyond asymptotic extrapolation and resonance modeling to convex control synthesis and rational spectral computation.

6. Boundary cases, limitations, and non-equivalent usages

Recent literature also marks out what local single-pole approximation is not. In the uniform electron gas, the paper on screened second-order exchange introduces a reduction-compatible single-pole (RC-SP) screened interaction,

$x(t)=a_0+a_1 t+a_2 t^2+\cdots,$1

and proves that exact reduction to a one-variable kernel is possible if and only if the one-pole screened interaction has a momentum-independent pole scale $x(t)=a_0+a_1 t+a_2 t^2+\cdots,$2 (Imoto, 24 Mar 2026).

The paper is explicit that RC-SP is not a local single-pole approximation in the usual physical sense of a locally varying plasmon-pole model. It is an exactly reducible reference model, not a faithful model of actual plasmon dispersions in real materials. Its significance lies elsewhere: it yields an analytically controlled benchmark, theorem-level asymptotics for the SOSEX energy, and basis elements for more general finite-rank separable or rational expansions.

Taken together, these literatures support several general conclusions. First, one-pole locality is always conditional. In homotopy extrapolation it requires one nearby singularity to dominate the Taylor coefficients; in OTA analysis it requires feedforward zeros to nearly cancel preceding poles; in waveguide QED it requires one resonance to dominate inside a chosen scattering window. Second, the approximation is usually fragile with respect to competing singular structures. Multiple nearby poles of comparable influence can destroy the inverse-power ratio expansion of Taylor coefficients, incomplete pole-zero cancellation can produce phase peaking, and non-negligible background channels turn Lorentzian reduction back into a Fano problem. Third, locality may refer to very different mathematical objects: coefficient index windows, recursive subsystem reductions, frequency intervals, target spectral sets, or exactly reducible reference kernels.

A common misconception is to treat “single-pole approximation” as a universally global statement about an entire model. The cited work points in the opposite direction. The approximation is typically local both in validity and in purpose: it is a controlled reduction designed for a narrowly defined computational or physical regime, and its success is governed by whether one pole is genuinely dominant there.