Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Positivity Proofs for Linear Recurrences with Several Dominant Eigenvalues

Alaa Ibrahim

TL;DR

This work extends the positivity problem for $P$-finite sequences by generalizing the cone-based method to Poincaré-type recurrences with multiple simple dominant eigenvalues. It replaces a single contracted cone by a sequence of cones $(K_n)$ that are mapped forward by the varying operator $A(n)$ under asymptotic eigenvalue separation. The authors provide asymptotic conditions on the dominant eigenvalues guaranteeing decidability of positivity for recurrences with several simple dominant eigenvalues, and illustrate the method with concrete examples and practical approximations. The approach combines vector form, Vandergraft style cone construction, and inductive arguments to extend decisiveness results beyond the unique dominant eigenvalue setting, offering a new algebraic tool for positivity verification with potential applications in analysis and combinatorics.

Abstract

Deciding the positivity of a sequence defined by a linear recurrence and initial conditions is, in general, a hard problem. When the coefficients of the recurrences are constants, decidability has only been proven up to order 5. The difficulty arises when the characteristic polynomial of the recurrence has several roots of maximal modulus, called dominant roots of the recurrence. We study the positivity problem for recurrences with polynomial coefficients, focusing on sequences of Poincaré type, which are perturbations of constant-coefficient recurrences. The dominant eigenvalues of a recurrence in this class are the dominant roots of the associated constant-coefficient recurrence. Previously, we have proved the decidability of positivity for recurrences having a unique, simple, dominant eigenvalue, under a genericity assumption. The associated algorithm proves positivity by constructing a positive cone contracted by the recurrence operator. We extend this cone-based approach to a larger class of recurrences, where a contracted cone may no longer exist. The main idea is to construct a sequence of cones. Each cone in this sequence is mapped by the recurrence operator to the next. This construction can be applied to prove positivity by induction. For recurrences with several simple dominant eigenvalues, we provide a condition that ensures that these successive inclusions hold. Additionally, we demonstrate the applicability of our method through examples, including recurrences with a double dominant eigenvalue.

Positivity Proofs for Linear Recurrences with Several Dominant Eigenvalues

TL;DR

This work extends the positivity problem for -finite sequences by generalizing the cone-based method to Poincaré-type recurrences with multiple simple dominant eigenvalues. It replaces a single contracted cone by a sequence of cones that are mapped forward by the varying operator under asymptotic eigenvalue separation. The authors provide asymptotic conditions on the dominant eigenvalues guaranteeing decidability of positivity for recurrences with several simple dominant eigenvalues, and illustrate the method with concrete examples and practical approximations. The approach combines vector form, Vandergraft style cone construction, and inductive arguments to extend decisiveness results beyond the unique dominant eigenvalue setting, offering a new algebraic tool for positivity verification with potential applications in analysis and combinatorics.

Abstract

Deciding the positivity of a sequence defined by a linear recurrence and initial conditions is, in general, a hard problem. When the coefficients of the recurrences are constants, decidability has only been proven up to order 5. The difficulty arises when the characteristic polynomial of the recurrence has several roots of maximal modulus, called dominant roots of the recurrence. We study the positivity problem for recurrences with polynomial coefficients, focusing on sequences of Poincaré type, which are perturbations of constant-coefficient recurrences. The dominant eigenvalues of a recurrence in this class are the dominant roots of the associated constant-coefficient recurrence. Previously, we have proved the decidability of positivity for recurrences having a unique, simple, dominant eigenvalue, under a genericity assumption. The associated algorithm proves positivity by constructing a positive cone contracted by the recurrence operator. We extend this cone-based approach to a larger class of recurrences, where a contracted cone may no longer exist. The main idea is to construct a sequence of cones. Each cone in this sequence is mapped by the recurrence operator to the next. This construction can be applied to prove positivity by induction. For recurrences with several simple dominant eigenvalues, we provide a condition that ensures that these successive inclusions hold. Additionally, we demonstrate the applicability of our method through examples, including recurrences with a double dominant eigenvalue.

Paper Structure

This paper contains 17 sections, 7 theorems, 81 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contracted Cones associated to recurrences
  3. Conversion to Vector Form
  4. Poincaré type
  5. Cones
  6. Contracted Cones
  7. Application to Positivity
  8. Sequences of contracted cones
  9. Eigenvalues ordering
  10. Sequence of cones
  11. Vandergraft's Construction
  12. Basis Construction
  13. Construction of a sequence of contracted cones
  14. Several Simple Dominant Eigenvalues
  15. Examples
  16. ...and 2 more sections

Key Result

Theorem 1

Vandergraft1968 Let $A$ be a real matrix. There exists a proper cone $K$ contracted by $A$ if and only if $A$ has a unique dominant eigenvalue $\lambda_{1}>0$, and $\lambda_1$ is simple.

Figures (1)

  • Figure 1: Cones $K_n$ for $n = 1$ to $4$, where the color of $K_n$ becomes lighter for increasing $n$. The red vector is the limit of the cones, and the blue vectors represent $U_0, \dots, U_4$.

Theorems & Definitions (18)

  • Definition 1: Poincaré type
  • Remark 1
  • Definition 2: Dominant eigenvalues
  • Theorem 1: Vandergraft
  • Theorem 2
  • Remark 2
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • ...and 8 more