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An implicit function theorem for the stream calculus

Michele Boreale, Luisa Collodi, Daniele Gorla

TL;DR

This work develops an implicit function theorem for the stream calculus, providing conditions under which a polynomial system $\mathcal{E}$ has a unique stream solution $\boldsymbol{\sigma}$ and an accompanying polynomial SDE system to compute it. It introduces a syntactic stream derivative and a chain rule for streams, and shows that the stream solution corresponds to the Taylor-coefficient stream of the analytic function solving $\mathcal{E}(x,f(x))=0$, i.e., $\boldsymbol{\sigma} = \mathcal{T}[f]$, while preserving a direct link to the classical IFT via the same Jacobian invertibility condition. The relationship is tight: the classical solution's Taylor series matches the stream solution, enabling two computational pathways, with the stream-based approach offering significant efficiency advantages for coefficient generation. The three-coloured-trees example illustrates algebraicity and demonstrates the practical computation of coefficients, outperforming the traditional ODE-based method and highlighting the method's applicability to combinatorial generating functions.

Abstract

In the context of the stream calculus, we present an Implicit Function Theorem (IFT) for polynomial systems, and discuss its relations with the classical IFT from calculus. In particular, we demonstrate the advantages of the stream IFT from a computational point of view, and provide a few example applications where its use turns out to be valuable.

An implicit function theorem for the stream calculus

TL;DR

This work develops an implicit function theorem for the stream calculus, providing conditions under which a polynomial system has a unique stream solution and an accompanying polynomial SDE system to compute it. It introduces a syntactic stream derivative and a chain rule for streams, and shows that the stream solution corresponds to the Taylor-coefficient stream of the analytic function solving , i.e., , while preserving a direct link to the classical IFT via the same Jacobian invertibility condition. The relationship is tight: the classical solution's Taylor series matches the stream solution, enabling two computational pathways, with the stream-based approach offering significant efficiency advantages for coefficient generation. The three-coloured-trees example illustrates algebraicity and demonstrates the practical computation of coefficients, outperforming the traditional ODE-based method and highlighting the method's applicability to combinatorial generating functions.

Abstract

In the context of the stream calculus, we present an Implicit Function Theorem (IFT) for polynomial systems, and discuss its relations with the classical IFT from calculus. In particular, we demonstrate the advantages of the stream IFT from a computational point of view, and provide a few example applications where its use turns out to be valuable.
Paper Structure (10 sections, 10 theorems, 41 equations, 1 figure)

This paper contains 10 sections, 10 theorems, 41 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Streams
  4. Polynomial stream differential equations
  5. An implicit function theorem for the stream calculus
  6. Relations with the classical IFT
  7. An extended example: three-coloured trees
  8. Classical vs. stream IFT: computational aspects
  9. Conclusion
  10. Three-coloured trees example: details

Key Result

Theorem 2.2

Every polynomial SDE initial value problem of the form eq:ivp has a unique solution.

Figures (1)

  • Figure 1: Execution time as a function of the number of computed coefficients for the stream solution of system \ref{['eq:trees']} (left) and of system \ref{['eq:nonzerod']} (right). The orange (lower) curve is the recurrence relation \ref{['eq:reconv']} computed via Lemma \ref{['lemma:algo']} (authors' Python code); the blue (upper) curve is the power series solution of \ref{['eq:jacobODE']} (Maple's dsolve).

Theorems & Definitions (25)

  • Definition 2.1: SDE Rut03
  • Theorem 2.2: existence and uniqueness of solutions
  • Remark 2.3: stream coefficients computation
  • Definition 3.1: syntactic stream derivative
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Lemma 3.4: from rational to polynomial SDE s
  • Lemma 3.5
  • proof
  • ...and 15 more