🔬 Puiseux Series Solutions of Algebraic ODEs via $x^a = t^b$

A complete, mathematically rigorous method for detecting and computing Puiseux series solutions (fractional exponent power series) of algebraic ordinary differential equations (AODEs). Developed during a research internship at CRIStAL (Centre de Recherche en Informatique, Signal et Automatique de Lille) under the supervision of François Boulier.

Report: report/main.pdf — Full 15-page paper with proofs, algorithms, and examples.

Preprint: The theoretical results are being prepared for submission.

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📖 Overview

The Problem

Given a polynomial differential equation:

$$P(x, y, y', \dots, y^{(R)}) = 0$$

Does it admit a Puiseux series solution?

$$y(x) = \sum_{n=N}^{\infty} a_n , x^{n/s}, \qquad s \in \mathbb{N}^*$$

The Denef–Lipshitz algorithm (1984/1989) decides existence of formal power series ($s = 1$) solutions, recently implemented by Boulier, Lemaire & Vu (2025) in the DifferentialAlgebra package. But when the answer is "no FPS", we need to investigate fractional-exponent series — Puiseux series.

Our Approach: $x^a = t^b$

We apply a monomial change of independent variable:

$$x^a = t^b \quad \Longleftrightarrow \quad t = x^{a/b}, \qquad \gcd(a,b) = 1$$

This transforms the original polynomial $P$ into a new differential polynomial $Q(t, z, \dot{z}, \dots)$, then we multiply by a power of $t$ to clear negative exponents, obtaining a normalized polynomial $K_{a,b}$.

The core insight:

$P$ admits a Puiseux series solution $\Longleftrightarrow$ there exists a pair $(a,b)$ such that $K_{a,b}$ admits a formal power series (FPS) solution — which is decidable by Denef–Lipshitz.

$$\boxed{P(x, y, \dot{y}, \dots) ;\xrightarrow{x^a = t^b}; Q(t, z, \dot{z}, \dots) ;\xrightarrow{\text{normalize}}; K_{a,b}(t, z, \dot{z}, \dots) ;\xrightarrow{\text{Denef–Lipshitz}}; \bar{z}(t) ;\xrightarrow{t = x^{a/b}}; \bar{y}(x)}$$

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🎯 Key Results

1. Congruence Condition (fast filter)

Before calling the costly Denef–Lipshitz elimination, we derive an arithmetic necessary condition on $b$. For a monomial of the form $(z')^m / c - t^{f(b)}$, the condition is:

$$f(b) \equiv 0 \pmod{m}$$

This generalizes to multi-monomial $K$ via a generalized congruence framework that analyzes $K^{(n)}|_{t=0}$ at each differentiation order $n$.

2. $a = 1$ Always Suffices

We prove that $a = 1$ is always a valid choice (Proposition 1), so the search reduces to finding admissible $b$ coprime to $a = 1$ — i.e., all $b \geq 2$.

3. Effective Bound $b \leq \deg(P)$

Using the Newton differential polygon, we establish that $b$ need only be searched up to the total degree of $P$ (Theorem 4). This makes the decision procedure finite.

4. Completeness: Monomial Changes are Enough

Any Puiseux solution accessible via an arbitrary algebraic change of variable $t = f(x)$ is already accessible via a monomial change $x^a = t^b$. This follows from a factorization theorem: $f = \varphi \circ (x \mapsto x^{1/s})$ where $\varphi$ is analytic, and analytic reparameterizations preserve FPS existence.

5. Extension to Poles

Puiseux series with poles are handled via the transformation $y = x^{-\nu} w$, reducing to the pole-free case (see docs/poles_puiseux.tex).

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🧩 Two Algorithms

Algorithm	Purpose	Complexity	Calls Denef–Lipshitz?
fast_decision	Yes/No/Maybe answer	$O(N \cdot D^2)$	No
full_solver	Compute all Puiseux solutions	Dominated by DL	Only for passing $b$

fast_decision uses the congruence filter to classify equations into:

• YES — definitely has a Puiseux solution (polynomial in $z'$ factorizes over $\mathbb{Q}(t)$)
• NO — definitely has no Puiseux solution (congruence condition fails for all $b$)
• LIKELY — congruence passes but factorization fails; needs full_solver

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📁 Repository Structure

puiseux-differential-algebra/
├── README.md                       # This file
├── LICENSE                         # MIT License
├── report/
│   ├── main.pdf                    # Full research paper (15 pages)
│   ├── main.tex                    # LaTeX source
│   └── chapters/*.tex              # Individual chapters
├── src/
│   ├── __init__.py
│   ├── change_of_variable.py       # x^a = t^b transformation
│   ├── compute_a.py                # Determination of admissible a values
│   ├── utils.py                    # Symbolic utilities (min exponents, etc.)
│   ├── congruence_method.py        # Simple congruence: (z')^m / c - t^{f(b)}
│   ├── multi_monomial.py           # Generalized multi-monomial congruence
│   ├── theorem_with_z.py           # Extension: K contains z (not just z')
│   ├── two_algorithms.py           # fast_decision + full_solver
│   ├── puiseux_pipeline.py         # Complete pipeline (end-to-end)
│   ├── puiseux_solver.py           # Alternative solver (with z support)
│   ├── manual_denef_lipshitz.py    # Manual DL for degenerate cases
│   └── resolve_likely.py           # Factorization-based resolution
├── tests/
│   └── test_pipeline.py            # Test suite (5 test cases)
├── docs/
│   ├── equivalence_theorem.tex     # Proof that monomial changes are complete
│   ├── congruence_generalisee.tex  # Generalized congruence theory
│   ├── changements_exotiques.tex   # Non-monomial changes do not help
│   ├── borne_b_max.tex             # Proof of b ≤ deg(P) bound
│   └── poles_puiseux.tex           # Extension to Puiseux series with poles
└── examples/
    └── basic_examples.py           # Ready-to-run examples

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🚀 Quick Start

Prerequisites

• Python 3.11+
• SymPy — symbolic computation
• DifferentialAlgebra — the C extension by François Boulier implementing Rosenfeld-Gröbner and Denef–Lipshitz

Installation

# 1. Clone the repository
git clone https://github.com/AlexGit31/puiseux-differential-algebra.git
cd puiseux-differential-algebra

# 2. Install Python dependencies
pip install sympy

# 3. Install DifferentialAlgebra (C extension)
#    Clone from: https://codeberg.org/francois.boulier/DifferentialAlgebra
cd DifferentialAlgebra/bmi-install-sympy-c99/packaging
python3 setup.py build
pip install -e .

⚠️ Known limitation: The C implementation of DenefLipshitz can fail (bas_Yuple.c:348) when the separant vanishes at $t = 0$ (pure power cases). A Python fallback is under development in src/manual_denef_lipshitz.py.

Basic Usage

from sympy import var
from DifferentialAlgebra import indexedbase
from src.puiseux_pipeline import find_puiseux_solutions, filter_nontrivial

xi = var("xi")
x, y = indexedbase("x,y")

# Example 1: ẏ² - x = 0  →  y = C + (2/3)x^{3/2}
P1 = y[xi]**2 - x
solutions = find_puiseux_solutions(P1)
solutions = filter_nontrivial(solutions)
for s in solutions:
    print(f"(a,b) = ({s['a']}, {s['b']}), y(x) = {s['puiseux_solution']}")

# Example 2: 8ẏ³ - 24ẏ² - 243y + 32 = 0  →  y = 2x + 3x^{3/2} (class P_{(3/2,1)})
P2 = 8*y[xi]**3 - 24*y[xi]**2 - 243*y + 32
solutions = find_puiseux_solutions(P2)

# Example 3: Fast decision (no Denef–Lipshitz)
from src.two_algorithms import fast_decision

result, reason = fast_decision(P1, verbose=True)
print(f"Decision: {result} — {reason}")

See examples/basic_examples.py for more.

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📊 Examples with Known Solutions

Equation	Puiseux Solution	$(a, b)$	Method
$\dot{y}^2 - x = 0$	$y = z_0 + \frac{2}{3}x^{3/2}$	$(1, 2)$	DL
$\dot{y}^3 - x^2 = 0$	$y = z_0 + \frac{3}{5}x^{5/3}$	$(1, 3)$	DL
$8\dot{y}^3 - 24\dot{y}^2 - 243y + 32 = 0$	$y = 2x + 3x^{3/2}$	$(1, 2)$	DL
$\ddot{y}^2 - y - x^2 - 1 = 0$	$y = -1 + z_1 x^{5/3} + \dots$	$(1, 3)$	Congruence
$81\dot{y}^4 - 216\dot{y}^3 + 162\dot{y}^2 - 256y - 27 = 0$	Multiple solutions	multiple	-

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📚 Theoretical Background

Denef–Lipshitz Algorithm

The Denef–Lipshitz algorithm decides the existence of formal power series solutions for systems of algebraic ordinary differential equations. The key steps are:

1. Differentiate the equations to reveal constraints on the series coefficients
2. Rename derivatives as new variables ($z^{(i)} \to z_i$)
3. Evaluate at $t = 0$ to obtain algebraic equations in the coefficients
4. Solve the resulting triangular system

The practical implementation relies on differential elimination (Rosenfeld-Gröbner) as a preprocessing step.

Congruence Framework

For a normalized polynomial of the form:

$$K(t, z, \dot{z}, \dots) = \sum_{i=1}^{N} c_i \cdot \prod_r \big(z^{(r)}\big)^{m_i^{(r)}} \cdot t^{e_i(b)}$$

the $n$-th derivative evaluated at $t = 0$ gives algebraic equations for the series coefficients $z_k$. The generalized congruence condition identifies orders $n$ where a constant term (from a $t$-only monomial with $e_i = n$) coincides with a pure $z$-power term (from a monomial with $m_j > 0$ and $n - e_j \equiv 0 \pmod{m_j}$). This yields $z_k \neq 0$ for $k = (n - e_j)/m_j + 1$, providing a non-trivial FPS branch.

Completeness Theorem

Any algebraic change of variable $t = f(x)$ factors as $\varphi \circ (x \mapsto x^{1/s})$ where $\varphi$ is an invertible FPS. Since analytic reparameterizations preserve the existence of FPS solutions, the search over $(a,b) \in \mathbb{N}^2$ with $\gcd(a,b) = 1$ is complete — no more exotic changes are needed.

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🔗 References

• Denef & Lipshitz (1984) — Power Series Solutions of Algebraic Differential Equations, Math. Ann.
• Denef & Lipshitz (1989) — Decision Problems for Differential Equations, J. Symbolic Logic
• Boulier, Lemaire, Vu (2025) — The Denef–Lipshitz Algorithm in Differential Algebra, HAL-05294349
• Grigoriev & Singer (1991) — Solving ODEs in Terms of Series with Real Exponents, Trans. AMS
• Cano, Falkensteiner, Sendra (2019) — Puiseux Series Solutions for Autonomous First-Order ODEs, arXiv:1908.09196
• Ayad (2007) — Complexity of Solving ODEs in Terms of Puiseux Series, arXiv:0705.2127
• Falkensteiner, Zhang, Vo (2018) — Formal Power Series Solutions of AODEs, arXiv:1803.09646

The DifferentialAlgebra package: codeberg.org/francois.boulier/DifferentialAlgebra

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👤 Author

Alexis Evaristo
Research intern, CRIStAL (UMR 9189), Université de Lille
Polytech Lille
📧 alexis.evaristo@polytech-lille.fr

Supervisor: François Boulier, Professor, Université de Lille

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📄 License

This project is licensed under the MIT License — see LICENSE for details.

The DifferentialAlgebra package is developed by François Boulier and collaborators (see its own license).

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If you use this code in your research, please cite the report and/or the forthcoming preprint.