On the residues vectors of a rational class of complex functions. Application to autoregressive processes

Guillermo Daniel Scheidereiter; Omar Roberto Faure

arxiv: 1907.05949 · v1 · pith:MZB4LFHInew · submitted 2019-07-12 · 💰 econ.EM

On the residues vectors of a rational class of complex functions. Application to autoregressive processes

Guillermo Daniel Scheidereiter , Omar Roberto Faure This is my paper

Pith reviewed 2026-05-24 21:48 UTC · model grok-4.3

classification 💰 econ.EM

keywords rational functionsresidues vectorp-norm boundautoregressive processescomplex analysiseconometricstime series

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The pith

Rational functions of a given class obey a lower bound on the p-norm of their residues vector.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper focuses on a particular class of rational functions of a complex variable. From this class it derives two basic properties about the residues at the poles. It then establishes a lower bound for the p-norm of the vector of these residues. The result is applied to autoregressive processes with illustrations from electric generation history and econometric time series. A sympathetic reader would see value in having an explicit bound that constrains model behavior in these applied domains.

Core claim

This work begins with a particular class of rational functions of a complex variable; over this is deduced two elemental properties concerning the residues and is proposed one result which establishes one lower bound for the p-norm of the residues vector. Applications to the autoregressive processes are presented and the exemplifications are indicated in historical data of electric generation and econometric series.

What carries the argument

the residues vector and the lower bound on its p-norm for the specified class of rational functions

If this is right

The bound applies directly to autoregressive process models.
The derived properties constrain the residues without extra pole restrictions.
Examples can be computed from electric generation historical records.
The same bound can be checked on econometric series.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This bound could be used to validate fitted AR models by comparing computed residues norms.
Similar algebraic arguments might yield bounds for other function classes or norms.
The connection to autoregressive processes suggests utility in forecasting or stability checks in economics.

Load-bearing premise

The specific class of rational functions possesses the algebraic structure that permits the two elemental residue properties and the subsequent p-norm bound to hold without additional restrictions on the poles or coefficients.

What would settle it

Observing a function in the class whose residues vector has a p-norm smaller than the claimed lower bound would falsify the result.

read the original abstract

Complex functions have multiple uses in various fields of study, so analyze their characteristics it is of extensive interest to other sciences. This work begins with a particular class of rational functions of a complex variable; over this is deduced two elementals properties concerning the residues and is proposed one results which establishes one lower bound for the p-norm of the residues vector. Applications to the autoregressive processes are presented and the exemplifications are indicated in historical data of electric generation and econometric series.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper claims a lower bound on the p-norm of residue vectors for one class of rational complex functions plus an AR application, but the abstract supplies no equations or proofs so the claim cannot be checked.

read the letter

The central claim is a lower bound on the p-norm of the residues vector for a stated class of rational functions, built from two basic residue properties, followed by an application to autoregressive processes and two real-data examples from electric generation and econometric series. That is the whole contribution as described. The paper does connect residue calculus to AR modeling in a direct way and includes concrete data illustrations, which is better than a purely formal note. Those are the only clear positives. The soft spots are substantial and central. No definition of the rational class appears, no derivation of the two properties or the bound is shown, and there is no comparison to existing inequalities from complex analysis or time-series literature. Without those steps it is impossible to tell whether the bound is new, whether it holds under the stated conditions, or whether it adds anything useful beyond standard AR checks. The application section would also need to demonstrate that the bound improves diagnostics or forecasts to carry weight. This work is for a narrow group of readers who already use residue vectors in econometric or signal-processing contexts and might want an analytic inequality for quick checks. Most AR researchers would not engage. The paper deserves peer review because the claim is specific enough that a referee could verify the math if the full derivations are present; a desk reject would be premature if the proofs turn out to be correct and non-trivial.

Referee Report

1 major / 1 minor

Summary. The paper considers a particular class of rational functions of a complex variable, deduces two elemental properties of the residues, and proposes a lower bound on the p-norm of the residues vector. Applications to autoregressive processes are presented, with examples on historical electric generation data and econometric series.

Significance. A non-trivial, verifiable lower bound on the p-norm of residues for a class relevant to AR characteristic polynomials could supply a useful analytic tool in time-series econometrics. The mention of real-data illustrations is positive, but no reproducible code, machine-checked proofs, or explicit falsifiable predictions are referenced.

major comments (1)

Abstract: the central claim (a lower bound on the p-norm of the residues vector) is never stated explicitly, nor is the class of rational functions defined; without these the result cannot be verified or assessed for load-bearing assumptions on poles or coefficients.

minor comments (1)

Abstract: grammatical and phrasing issues ('so analyze their characteristics it is of extensive interest', 'two elementals properties', 'one results which establishes one lower bound', 'exemplifications are indicated') impair readability and should be revised.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comment. We address the point raised below.

read point-by-point responses

Referee: Abstract: the central claim (a lower bound on the p-norm of the residues vector) is never stated explicitly, nor is the class of rational functions defined; without these the result cannot be verified or assessed for load-bearing assumptions on poles or coefficients.

Authors: We agree that the abstract is insufficiently precise. It refers only to 'a particular class' and 'one results which establishes one lower bound' without stating the bound or defining the class. The body of the paper (Section 2) defines the class via the rational function form with prescribed pole locations and derives the bound in Theorem 1, but the abstract must be self-contained. We will revise the abstract to state the class explicitly (rational functions with simple poles inside/outside the unit circle, as relevant to AR characteristic polynomials) and to formulate the lower bound on the p-norm of the residue vector. Assumptions on pole moduli will be stated. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The abstract describes a mathematical derivation establishing elemental residue properties and a lower bound on the p-norm of the residues vector for a defined class of rational complex functions, followed by applications to autoregressive processes. No equations, parameter fitting, self-citations, or ansatzes are referenced in the available text. The claim is presented as deduced from the algebraic structure of the function class without any indication that the bound or properties reduce to the inputs by construction. As a pure mathematical result with no visible fitting or renaming steps, the derivation is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No full text available; cannot enumerate free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5600 in / 1073 out tokens · 18543 ms · 2026-05-24T21:48:50.480507+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

∈ ℝ %& ' = 1, … , *; (II) The poles , , … , of are simple poles that satisfy ,

Choice of Theme Consider the class of complex rational functions: : ⊂ ℂ → ℂ such that = 1 1 − − − ⋯ − 1 that meet the following conditions: (I) ≠ 0 and | |+ | |+ ⋯ + | | < 1 with " ∈ ℝ %& ' = 1, … , *; (II) The poles , , … , of are simple poles that satisfy , ", > 1 for ' = 1, … , *. Be the subclass of functions of (1): = 1 1 − 2 Here, | | < 1 and ≠ 0. Th...

work page
[2]

These functions will have as many poles as indicates the degree the polynomial of the denominator and, therefore, a series of Laurent associated to each pole

Definition of the problem The object of study is the residuals ; < " for ' = 1, … , * of the type of complex functions of the form defined in (1) that satisfy t he conditions (I) and (II). These functions will have as many poles as indicates the degree the polynomial of the denominator and, therefore, a series of Laurent associated to each pole. Starting ...

work page
[3]

g , ", 11 ‖_‖h = iK,

Conceptual theoretical framework An open set iii Δ in the complex plane is connected if in any division of the same in two non-empty subsets, without common elements, ∆ and ∆ , at least one of these sets contains a point of accumulation of the other set (Markushevich, 1970). Also, if a set is open and connected, it is a domain (Churchill & Brown, 1992). I...

work page 1970
[4]

, > for ' = 1,2, then max

Justification of the study Taking into account function (1) with the condition s (I) and (II), the subclass of functions where the denominator is of degree * = 2 satisfy the following result: If = 1 1 − − , 14 then ,;< " , > 1 2 for ' = 1,2. 15 In effect, the function has two poles that depend on the coefficients and : , = ± o + 4 −2 . In this case, 234 5...

work page
[5]

6 , ,;<

Limitations Suppose we consider the function: = 1 1 − 2 + 3 , which satisfies (14) as a particular form of (1). I n such a case, the residue vector is: v<w = x− √2 4 C,√2 4 Cy × Calculating the norms it is observed that they are not fulfilled (17), (18) and (19): ‖v<w‖ = 1 2 < 1 √2 ‖v<w‖ = √2 2 < 1 ‖v<w‖L = max "6 , ,;< " , = √2 4 < 1 2 . It is clear that...

work page
[6]

21 The equation (20) can be written (Shumway & Stoffer, 2011): D| − = } D|< − + } D|< − + ⋯ + }~ D|<~ − + {|

Scope of Work Consider an autoregressive model of order 4, {2 4 , with equation: D | = + } D|< + } D|< + ⋯ + }~D|<~ + {| 20 8 where {| is a white noise process with zero mean and consta nt variance, {|~CCP 0, , the parameters } , } , … , }~ are the coefficients of the model and is a constant such that the mean of the model is: = 1 − } − } − ⋯ − }~ ...

work page 2011
[7]

Specifically, it is shown that the remainder of subclass (2) is such that ,; < , > 1 while the residues vector of subclass (14) verify ( 17), (18) and (19)

Objectives As shown in (3) and (15), subclasses (2) and (14) o f the general form (1), which satisfy (I) and (II), have residues with vect or properties of interest. Specifically, it is shown that the remainder of subclass (2) is such that ,; < , > 1 while the residues vector of subclass (14) verify ( 17), (18) and (19). We are looking, therefore, for a g...

work page
[8]

for ' = 1, … , * are the residuals at the poles

Hypothesis In the previous paragraphs it is shown that the com plex functions considered here have associated residues vectors whose norms s atisfy certain conditions for two particular cases. It is of interest, theref ore, to formulate a conjecture that considers general orders for the residues vector norm. From the properties that meet the residuals of ...

work page
[9]

Material and methods As previously indicated, the study focuses on the p roperties that verify the vector norms whose elements are the residues of complex functions that satisfy the definition (1) with conditions (I) and (II). Th e hypothesis will be demonstrated using the inductive method in mathematics, according to which if a propositional function is ...

work page
[10]

for ' = 1, … , * are the residues of in the poles,

Results Let be defined in (1) with conditions (I) and (II). Suppose that Ω is a simply connected domain xi and that @ is a simple closed contour in Ω that encloses the poles of and the unit circle. If is analytic in Ω except for the zeros of 1 − − − ⋯ − , then ‖v<w‖h > 91 *: h< h 25 where v<w = ;< , ;< , … , ;< and ;< " for ' = 1, … , * are the residues o...

work page
[11]

A specific example is t he series of daily electric generation records (MWh) of the Salto Grande Hydroelectric Dam, in the period from September 2018 to January 2019

Application The complex functions of form (1) appear in time series when they are modeled by autoregressive processes. A specific example is t he series of daily electric generation records (MWh) of the Salto Grande Hydroelectric Dam, in the period from September 2018 to January 2019. The logarithm o f the observed data and the estimation of the model wer...

work page 2018
[12]

Conclusions The first properties for the residuals of the comple x functions (1) with the conditions (I) and (II), show for the subclass (2) that they are outside a circle of radius 1, while for (14) they remain outside a c ircle of radius . In addition, the results (25) and (30), establish that the norms of the vector of residues of these complex rationa...

work page 2013

[1] [1]

∈ ℝ %& ' = 1, … , *; (II) The poles , , … , of are simple poles that satisfy ,

Choice of Theme Consider the class of complex rational functions: : ⊂ ℂ → ℂ such that = 1 1 − − − ⋯ − 1 that meet the following conditions: (I) ≠ 0 and | |+ | |+ ⋯ + | | < 1 with " ∈ ℝ %& ' = 1, … , *; (II) The poles , , … , of are simple poles that satisfy , ", > 1 for ' = 1, … , *. Be the subclass of functions of (1): = 1 1 − 2 Here, | | < 1 and ≠ 0. Th...

work page

[2] [2]

These functions will have as many poles as indicates the degree the polynomial of the denominator and, therefore, a series of Laurent associated to each pole

Definition of the problem The object of study is the residuals ; < " for ' = 1, … , * of the type of complex functions of the form defined in (1) that satisfy t he conditions (I) and (II). These functions will have as many poles as indicates the degree the polynomial of the denominator and, therefore, a series of Laurent associated to each pole. Starting ...

work page

[3] [3]

g , ", 11 ‖_‖h = iK,

Conceptual theoretical framework An open set iii Δ in the complex plane is connected if in any division of the same in two non-empty subsets, without common elements, ∆ and ∆ , at least one of these sets contains a point of accumulation of the other set (Markushevich, 1970). Also, if a set is open and connected, it is a domain (Churchill & Brown, 1992). I...

work page 1970

[4] [4]

, > for ' = 1,2, then max

Justification of the study Taking into account function (1) with the condition s (I) and (II), the subclass of functions where the denominator is of degree * = 2 satisfy the following result: If = 1 1 − − , 14 then ,;< " , > 1 2 for ' = 1,2. 15 In effect, the function has two poles that depend on the coefficients and : , = ± o + 4 −2 . In this case, 234 5...

work page

[5] [5]

6 , ,;<

Limitations Suppose we consider the function: = 1 1 − 2 + 3 , which satisfies (14) as a particular form of (1). I n such a case, the residue vector is: v<w = x− √2 4 C,√2 4 Cy × Calculating the norms it is observed that they are not fulfilled (17), (18) and (19): ‖v<w‖ = 1 2 < 1 √2 ‖v<w‖ = √2 2 < 1 ‖v<w‖L = max "6 , ,;< " , = √2 4 < 1 2 . It is clear that...

work page

[6] [6]

21 The equation (20) can be written (Shumway & Stoffer, 2011): D| − = } D|< − + } D|< − + ⋯ + }~ D|<~ − + {|

Scope of Work Consider an autoregressive model of order 4, {2 4 , with equation: D | = + } D|< + } D|< + ⋯ + }~D|<~ + {| 20 8 where {| is a white noise process with zero mean and consta nt variance, {|~CCP 0, , the parameters } , } , … , }~ are the coefficients of the model and is a constant such that the mean of the model is: = 1 − } − } − ⋯ − }~ ...

work page 2011

[7] [7]

Specifically, it is shown that the remainder of subclass (2) is such that ,; < , > 1 while the residues vector of subclass (14) verify ( 17), (18) and (19)

Objectives As shown in (3) and (15), subclasses (2) and (14) o f the general form (1), which satisfy (I) and (II), have residues with vect or properties of interest. Specifically, it is shown that the remainder of subclass (2) is such that ,; < , > 1 while the residues vector of subclass (14) verify ( 17), (18) and (19). We are looking, therefore, for a g...

work page

[8] [8]

for ' = 1, … , * are the residuals at the poles

Hypothesis In the previous paragraphs it is shown that the com plex functions considered here have associated residues vectors whose norms s atisfy certain conditions for two particular cases. It is of interest, theref ore, to formulate a conjecture that considers general orders for the residues vector norm. From the properties that meet the residuals of ...

work page

[9] [9]

Material and methods As previously indicated, the study focuses on the p roperties that verify the vector norms whose elements are the residues of complex functions that satisfy the definition (1) with conditions (I) and (II). Th e hypothesis will be demonstrated using the inductive method in mathematics, according to which if a propositional function is ...

work page

[10] [10]

for ' = 1, … , * are the residues of in the poles,

Results Let be defined in (1) with conditions (I) and (II). Suppose that Ω is a simply connected domain xi and that @ is a simple closed contour in Ω that encloses the poles of and the unit circle. If is analytic in Ω except for the zeros of 1 − − − ⋯ − , then ‖v<w‖h > 91 *: h< h 25 where v<w = ;< , ;< , … , ;< and ;< " for ' = 1, … , * are the residues o...

work page

[11] [11]

A specific example is t he series of daily electric generation records (MWh) of the Salto Grande Hydroelectric Dam, in the period from September 2018 to January 2019

Application The complex functions of form (1) appear in time series when they are modeled by autoregressive processes. A specific example is t he series of daily electric generation records (MWh) of the Salto Grande Hydroelectric Dam, in the period from September 2018 to January 2019. The logarithm o f the observed data and the estimation of the model wer...

work page 2018

[12] [12]

Conclusions The first properties for the residuals of the comple x functions (1) with the conditions (I) and (II), show for the subclass (2) that they are outside a circle of radius 1, while for (14) they remain outside a c ircle of radius . In addition, the results (25) and (30), establish that the norms of the vector of residues of these complex rationa...

work page 2013