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Positivity Proofs for Linear Recurrences through Contracted Cones

Alaa Ibrahim, Bruno Salvy

TL;DR

This work develops a geometric, cone-based method to decide positivity for a broad class of linear recurrences with polynomial coefficients (P-finite sequences). By reformulating recurrences as matrix recurrences U_{n+1}=A(n)U_n and constructing cones K contracted by the limit A, the authors prove that, for generic initial conditions, the trajectory enters K and positivity can be certified via cone membership. They build and compare multiple contracted-cone constructions (Vandergraft-type, polyhedral, and approximate), derive algorithms to compute stability indices, and demonstrate the approach on P-finite and GRZ families, often outperforming previous techniques. The results establish decidability of positivity in a wide setting and offer practical tools for automatic positivity proofs with potential extensions to broader recurrences and coefficient fields.

Abstract

Deciding the positivity of a sequence defined by a linear recurrence with polynomial coefficients and initial condition is difficult in general. Even in the case of recurrences with constant coefficients, it is known to be decidable only for order up to~5. We consider a large class of linear recurrences of arbitrary order, with polynomial coefficients, for which an algorithm decides positivity for initial conditions outside of a hyperplane. The underlying algorithm constructs a cone, contracted by the recurrence operator, that allows a proof of positivity by induction. The existence and construction of such cones relies on the extension of the classical Perron-Frobenius theory to matrices leaving a cone invariant.

Positivity Proofs for Linear Recurrences through Contracted Cones

TL;DR

This work develops a geometric, cone-based method to decide positivity for a broad class of linear recurrences with polynomial coefficients (P-finite sequences). By reformulating recurrences as matrix recurrences U_{n+1}=A(n)U_n and constructing cones K contracted by the limit A, the authors prove that, for generic initial conditions, the trajectory enters K and positivity can be certified via cone membership. They build and compare multiple contracted-cone constructions (Vandergraft-type, polyhedral, and approximate), derive algorithms to compute stability indices, and demonstrate the approach on P-finite and GRZ families, often outperforming previous techniques. The results establish decidability of positivity in a wide setting and offer practical tools for automatic positivity proofs with potential extensions to broader recurrences and coefficient fields.

Abstract

Deciding the positivity of a sequence defined by a linear recurrence with polynomial coefficients and initial condition is difficult in general. Even in the case of recurrences with constant coefficients, it is known to be decidable only for order up to~5. We consider a large class of linear recurrences of arbitrary order, with polynomial coefficients, for which an algorithm decides positivity for initial conditions outside of a hyperplane. The underlying algorithm constructs a cone, contracted by the recurrence operator, that allows a proof of positivity by induction. The existence and construction of such cones relies on the extension of the classical Perron-Frobenius theory to matrices leaving a cone invariant.

Paper Structure

This paper contains 33 sections, 11 theorems, 68 equations, 4 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Matrix Recurrences
  3. Scalars to Vectors
  4. Constant Matrices and the Power Method
  5. Recurrences of Poincaré Type
  6. Cones
  7. Definitions
  8. Properties
  9. Proof of the Algorithm
  10. Cone construction
  11. Stability index
  12. Convergence into the Cone
  13. Proof of \ref{['theorem:main']} (termination of the algorithm)
  14. Proof of \ref{['thm:decidability']}
  15. Constructions of Contracted Cones
  16. ...and 18 more sections

Key Result

Theorem 1

For all linear recurrences of the form given in rec, of order $d$ and of Poincaré type, having a unique simple dominant eigenvalue and such that $0\not\in p_0p_d(\mathbb N)$, the positivity of the solution $(u_n)_n$ is decidable for any $U_0=(u_0,u_1,\dots,u_{d-1})$ outside a hyperplane in $\mathbb{

Figures (4)

  • Figure 1: The first values of $U_n=(u_n,u_{n+1},u_{n+2})$ (in blue) in \ref{['example:3']}, together with the corresponding cone $K$ (in red).
  • Figure 2: Cone from \ref{['example:ConeVandergraft']} for the recurrence of \ref{['example:3']}.
  • Figure 3: The first values of $U_n=(u_n,u_{n+1},u_{n+2})$ (in blue) in \ref{['example:3']}, together with the corresponding polyhedral cone (in red) with its extremal vectors (in black)
  • Figure 4: Polyhedral cone (red) with its image by the matrix $A$ (blue)

Theorems & Definitions (26)

  • Definition 1: Positivity problem for P-finite sequences
  • Example 1
  • Theorem 1
  • Definition 2: Dominant eigenvalues
  • Lemma 1
  • Definition 3
  • Example 2
  • Proposition 1: Vandergraft Vandergraft1968
  • Theorem 2
  • Proposition 2: Cone with fixed signs
  • ...and 16 more