arxiv: 2604.20888 · v1 · submitted 2026-04-18 · 🧮 math.GM

Recognition: unknown

A Limit-Free Algebraic-Geometric Construction of the Derivative with a Foundational Model in the Class of Polynomial Functions

Davit Kapanadze

Authors on Pith no claims yet

Pith reviewed 2026-05-10 07:07 UTC · model grok-4.3

classification 🧮 math.GM

keywords derivativetangencydouble rootpolynomialsalgebraic constructionlinear approximationlimit emergence

0 comments

The pith

The derivative is defined for polynomials by the unique linear approximation whose difference has a double root at the point, with the limit form emerging later.

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

The paper establishes an algebraic-geometric definition of the derivative that begins inside the polynomials and avoids limits at the outset. Tangency is fixed by the requirement that the difference between the function and a linear function shares a double root at the given point; this condition determines a unique slope at each point. All standard differentiation rules then follow from polynomial algebra alone. The construction extends conceptually to elementary functions and is linked to linear decomposition of functions, from which the classical limit expression appears as a derived analytic form rather than an initial definition.

Core claim

Within the class of polynomial functions, the existence, uniqueness, and basic rules of differentiation follow directly from the algebraic condition that the difference between a polynomial and its linear approximation has a double root at the point of interest. This supplies a functional correspondence that assigns to each point the slope of the tangent line. The model extends conceptually to elementary functions and connects to the linear decomposition of functions, from which the familiar limit representation of the derivative is recovered as a consequence.

What carries the argument

The double-root condition on the difference between a function and a linear approximation, which selects the unique tangent slope at each point.

If this is right

The sum, product, and power rules of differentiation hold for all polynomials through direct algebraic manipulation of the double-root condition.
Every polynomial possesses a unique derivative that is itself a polynomial of lower degree.
The construction yields the classical limit formula as the explicit analytic expression arising from the linear decomposition of the original function.
The same tangency concept applies to elementary functions once the polynomial case is secured.

Where Pith is reading between the lines

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

The reversal of order, with algebraic definition preceding the limit, could permit introductory calculus to develop derivative rules before introducing limits.
The link between double-root multiplicity and tangency suggests possible extensions to algebraic curves or varieties where similar multiplicity conditions define contact order.
If the linear-decomposition step generalizes cleanly, other analytic operations such as integration or series expansion might be reachable through related algebraic identities.

Load-bearing premise

The double-root algebraic condition for tangency extends to elementary functions while preserving uniqueness and the differentiation rules without reintroducing limits or creating inconsistencies.

What would settle it

A concrete polynomial for which either no linear approximation produces a double root at a given point or more than one slope satisfies the double-root condition.

read the original abstract

This paper presents an algebraic-geometric construction of the derivative developed initially within the class of polynomial functions without introducing limits at the initial stage. Tangency is characterized by an algebraic condition: the difference between a function and a linear approximation has a double root at a given point. On this basis, the derivative is defined as a functional correspondence assigning to each point the slope of the tangent. Within the class of polynomials, the existence, uniqueness, and fundamental rules of differentiation are established purely algebraically. The constructed model is then extended conceptually to elementary functions and connected to the linear decomposition of functions, from which the classical limit representation of the derivative naturally emerges. Thus, the limit appears not as a starting point but as an analytic expression of an already constructed concept.

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 polynomial algebra is standard and the conceptual extension to elementary functions lacks the needed rigor.

read the letter

Hey, the main takeaway is that the algebraic construction for polynomials via the double-root condition is already standard, and the paper's sketch of an extension to other functions does not supply enough detail to support a genuinely limit-free foundation. What the paper does well is to lay out the polynomial case in a straightforward way. It defines the derivative by requiring that f(x) minus the linear approximation f(a) plus m times (x minus a) has a double root at a, which pins down m uniquely through polynomial division or coefficient matching. From there the sum, product, and chain rules follow as algebraic identities, all without limits. That section is clean and could serve as a useful exercise for students working in polynomial rings. The soft spots appear once the paper moves beyond polynomials. It states that the model is extended conceptually to elementary functions and that the classical limit representation then emerges naturally from the linear decomposition of the function. Yet no explicit definition is given for what a double root or tangency condition would mean for exp, sin, or log without either power series or some implicit limiting step. Without that definition the claim that limits appear only as a derived expression rather than a foundational ingredient cannot be verified. The circularity burden is low for the polynomial part, but the extension reintroduces the very issue it aims to avoid. This is aimed at readers interested in alternative foundations for introductory calculus. Someone looking for a reminder of the algebraic derivative on polynomials might find the first half useful, but the overall argument does not add a new framework. I would not bring it to a reading group, would not cite it, and would not send it to peer review; the authors would need a precise construction for the general case before the central claim can be assessed.

Referee Report

1 major / 0 minor

Summary. The paper proposes an algebraic-geometric construction of the derivative that begins with polynomial functions, defining tangency via the algebraic condition that f(x) − [f(a) + m(x−a)] has a double root at x = a. Within polynomials this yields existence, uniqueness, and the standard differentiation rules (product, chain) purely from coefficient matching and the division algorithm in k[x], without limits. The model is then extended conceptually to elementary functions and linked to linear decomposition, from which the classical limit form of the derivative is said to emerge naturally as a derived expression rather than a foundational primitive.

Significance. If the conceptual extension can be rigorized without reintroducing limits or series as hidden foundations, the work would supply a genuinely alternative starting point for differentiation that separates the algebraic notion of tangency from analytic limits. This could be pedagogically useful and contribute to foundational discussions in calculus. The polynomial portion is technically sound and self-contained; the significance therefore hinges on whether the extension preserves the claimed limit-free character for the broader class of functions.

major comments (1)

The central claim that the limit representation 'naturally emerges' after the model is extended to elementary functions rests on an imprecise step. The manuscript supplies a precise algebraic definition of the double-root tangency condition only for polynomials (via the division algorithm). For exp, sin, log, etc., the text offers only a high-level conceptual extension without an explicit algebraic definition of 'double root' or 'tangent line' that avoids power-series truncation or an implicit limiting argument. This gap is load-bearing: absent such a definition, it is not possible to verify that the construction remains foundational and limit-free when the limit form is later derived.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful and constructive review. The observation regarding the extension to elementary functions identifies a genuine point of imprecision in the current draft. We address it directly below and will revise the manuscript to supply the requested explicit definitions while preserving the algebraic character of the construction.

read point-by-point responses

Referee: The central claim that the limit representation 'naturally emerges' after the model is extended to elementary functions rests on an imprecise step. The manuscript supplies a precise algebraic definition of the double-root tangency condition only for polynomials (via the division algorithm). For exp, sin, log, etc., the text offers only a high-level conceptual extension without an explicit algebraic definition of 'double root' or 'tangent line' that avoids power-series truncation or an implicit limiting argument. This gap is load-bearing: absent such a definition, it is not possible to verify that the construction remains foundational and limit-free when the limit form is later derived.

Authors: We agree that the extension is currently stated at a conceptual level and that an explicit algebraic definition of the double-root condition for elementary functions is needed to substantiate the limit-free claim. In the revised manuscript we will add a dedicated subsection that defines the tangent line for the standard elementary functions by means of their characterizing functional equations and algebraic identities (addition formulas, recurrence relations, and the division algorithm in appropriate polynomial rings over the reals). For each function we will show that there exists a unique slope m such that the difference f(x) − [f(a) + m(x − a)] has a root of multiplicity at least two at x = a, established by direct coefficient comparison or substitution into the functional equation rather than by series truncation or limits. We will then derive the classical difference-quotient expression as a consequence of this already-constructed derivative. These additions will make the extension verifiable and will keep the foundational order intact. revision: yes

Circularity Check

0 steps flagged

No significant circularity; polynomial construction is algebraically self-contained

full rationale

The paper's core derivation for polynomials defines tangency via the algebraic condition that f(x) - [f(a) + m(x-a)] has a double root at x=a, which is checked by coefficient matching or the division algorithm in k[x] and yields the formal derivative without reference to limits or the target result. Existence, uniqueness, and rules (product, chain) follow as algebraic identities from this setup. The subsequent conceptual extension to elementary functions and emergence of the limit form are presented as later steps rather than load-bearing inputs to the polynomial case; no equation or step reduces the claimed result to a fit, self-definition, or self-citation chain. The derivation remains independent of the classical limit definition.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard properties of polynomial rings without introducing new free parameters or invented entities.

axioms (1)

standard math Polynomials over the reals form an integral domain in which the division algorithm holds and roots have well-defined multiplicity.
Invoked to guarantee existence and uniqueness of the linear approximation when the difference has a double root.

pith-pipeline@v0.9.0 · 5425 in / 1333 out tokens · 58335 ms · 2026-05-10T07:07:52.884757+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

9 extracted references

[1]

The Derivative Without Limits

Oved Shisha (1986). The Derivative Without Limits. Journal of Mathematical Analysis and Applications, 113(2), 280–287

1986
[2]

Morsi (1987)

Morris Falkowitz, Oved Shisha, & N. Morsi (1987). Note on Derivative without Limit. Journal of Mathematical Analysis and Applications, 127(2), 595–597

1987
[3]

An Elementary, Limit-Free Calculus for Polynomials

Christopher Sangwin (2010). An Elementary, Limit-Free Calculus for Polynomials. The Mathematical Gazette, 94(529), 67–83

2010
[4]

A Calculus Without Limits

Xiaoping Zhang & Yong Tong (2018). A Calculus Without Limits. arXiv

2018
[5]

J. C. Sparks (2004). Calculus Without Limits

2004
[6]

Using the Computer to Visualize Concepts in Calculus

David Tall (1986). Using the Computer to Visualize Concepts in Calculus

1986
[7]

Jerome Keisler (1986)

H. Jerome Keisler (1986). Elementary Calculus: An Infinitesimal Approach

1986
[8]

Non-standard Analysis

Abraham Robinson (1966). Non-standard Analysis

1966
[9]

Bell (2008)

John L. Bell (2008). A Primer of Infinitesimal Analysis (2nd ed.). Cambridge University Press

2008