Engineering / Mathematics — Fields, Series and Factorisation
Reversed Formal Laurent Series
Reversed formal Laurent series, the valuation by degree, and their role in rational function reconstruction.
Executive summary
Reversing the exponent convention makes degree play the role of absolute value, so a polynomial of higher degree is larger rather than smaller.
This is the setting in which rational function reconstruction and the continued fraction expansion of a rational function are naturally expressed.
Learning objectives
- Define the reversed convention and its valuation.
- Relate it to polynomial degree.
- Connect it to reconstruction algorithms.
01The reversed convention
Reversed formal Laurent series
Series in X^{−1} of the form Σ_{i ≤ k} aᵢX^i — finitely many positive-exponent terms, infinitely many negative ones.
The valuation is −deg, so the leading term is the one of highest degree.
Under this convention a polynomial has a finite expansion and its size is its degree. Rational functions expand as infinite series in X^{−1}, exactly as rational numbers expand as decimals.
| Integers | Reversed Laurent setting |
|---|---|
| Absolute value | Degree |
| Integer part of a rational | Polynomial part of a rational function |
| Decimal expansion | Expansion in powers of X⁻¹ |
| Continued fraction | Continued fraction of a rational function |
| Convergents | Pade approximants |
02Why this convention
The Euclidean algorithm on polynomials reduces degree at each step, so degree is the natural size function. The reversed convention aligns the series valuation with that size function.
With the convention fixed, the theory of continued fractions for real numbers transfers to rational functions essentially unchanged, and the best rational approximation of bounded degree is a convergent of the continued fraction expansion.
03Rational function reconstruction
Approximation by convergents
Given the first N coefficients of the expansion of a rational function, the convergents of the continued fraction expansion — equivalently, the intermediate results of extended Euclid — give the best rational approximations of each bounded degree.
Take a truncated expansion
The first N coefficients, or equivalently a polynomial residue modulo X^N.
Run extended Euclid
On the pair consisting of X^N and the truncated series.
Stop at the degree bound
The first remainder below the target numerator degree gives the answer.
Read off numerator and denominator
The remainder and the accumulated coefficient.
This is the same procedure as integer rational reconstruction with degree replacing magnitude. The unified view is the reason both appear in this collection and why one implementation pattern serves both.
04Frequently asked questions
Is the reversed convention standard?
It is standard in the algorithmic literature on rational reconstruction and coding theory, precisely because it aligns with the Euclidean reduction. Analytic treatments usually keep the ordinary convention.
What is a Pade approximant?
A rational function matching a given series to as many terms as its degrees permit. It is the polynomial analogue of a continued fraction convergent, and it is what rational function reconstruction computes.
Why does extended Euclid produce the convergents?
Because the quotient sequence in the Euclidean algorithm is exactly the continued fraction expansion, and the accumulated coefficients are the convergent numerators and denominators. The two algorithms are the same computation.
Sources and method
Structural reference: Victor Shoup, A Computational Introduction to Number Theory and Algebra, Version 1, Cambridge University Press, 2005 — book pages 382-383.
This page carries the durable method layer only: definitions, constructions, algorithms, complexity results and selection criteria, authored originally for KEVOS. No text is transcribed or paraphrased from the source, and no numeric tables or benchmark data are reproduced — these are routed to live authoritative sources instead.
Author: Kevin Jogin. Last reviewed 2026-08-07.