Key Positivity Problem Overview

• The Key Positivity Problem is a decision problem determining if all elements in sequences, matrices, or operators remain nonnegative across various indices and time steps.
• It spans diverse fields such as analytic number theory, dynamical systems, quantum information, and computational mathematics, offering insights into stability and threshold behavior.
• The problem presents significant complexity and decidability challenges that intersect with Diophantine approximation, model checking, and numerical method robustness.

The Key Positivity Problem refers to the decision problem of determining whether all terms of a given sequence, process, or operator remain nonnegative (or positive) for all valid indices or time steps. This problem appears in a wide variety of mathematical disciplines—analytic number theory (through linear recurrence sequences), dynamical systems (matrix powers and Markov chains), combinatorics (Schubert structure constants, rational series coefficients), quantum information (positivity of key rates and integral operators), and computational mathematics (positivity-preserving numerical methods). Despite its apparent simplicity, the Key Positivity Problem is a gateway to deep unresolved questions and exhibits a complexity-theoretic boundary that intersects with fields such as Diophantine approximation, real algebraic geometry, and model checking.

1. Formal Definitions Across Domains

1.1 Linear Recurrence Sequences (LRS)

Let $k \geq 2$ and $$\alpha_1, \dots, \alpha_k, \beta_0, \dots, \beta_{k-1} \in \mathbb{Q}$$. The sequence $(u_n)_{n \geq 0}$ is the LRS of order $k$ defined by

$$u_0 = \beta_0,\,\, u_1 = \beta_1,\,\, ...,\,\, u_{k-1} = \beta_{k-1},\,\,\,\,\, u_{n+k} = \alpha_1 u_{n+k-1} + \alpha_2 u_{n+k-2} + \cdots + \alpha_k u_n.$$

The Positivity Problem asks whether $u_n \geq 0$ for all $n \geq 0$ (Piribauer et al., 2023).

1.2 Matrix Powers and Weighted Sums

Given matrices $A_1, ..., A_m \in \mathbb{Q}^{d \times d}$ and weights $w_i \in \mathbb{Q}$, define

$S(n) = \sum_{i=1}^m w_i A_i^n.$

The Key Positivity Problem asks whether, for all large $n$, all entries of $S(n)$ are nonnegative (eventual nonnegativity), or strictly positive (eventual positivity) (Akshay et al., 2022).

1.3 Generalized Moment Problem

Given $A \in \mathbb{Q}^{d \times d}$, $v, w \in \mathbb{Q}^d$, and a cone $\mathcal{P}$, decide whether $w^T A^n v \geq 0$ for all $n$ (generalized moment membership) (Coves et al., 2024).

1.4 Markov Chains and Markov Decision Processes

Given a finite-state Markov chain or MDP, initial and goal states, and a threshold $r$, decide if the probability (or expected cost, accumulated weight, etc.) at step $n$ remains above the threshold for all $n$, or crosses it (Vahanwala, 2023, Piribauer et al., 2023).

1.5 Polynomial Recurrences and Differential Equations

For recurrences with polynomial coefficients (P-finite/Poincaré type), or positive-graph-Laplacian ODEs, decide if all terms or solutions remain nonnegative (Ibrahim, 18 Mar 2025, Blanes et al., 2021).

1.6 Quantum Operators

Given a polynomial-Gaussian integral kernel $\kappa_{PG}(x,y) = P(x,y)\kappa_G(x,y)$, decide whether the corresponding operator is positive semidefinite (Balka et al., 2024).

2. Known Decidability Frontiers and Complexity Barriers

The classical Key Positivity Problem for rational LRS is undecidable or open in general beyond low order. For order $k \leq 3$ (and some order 4, special cases), decidability is known by deep Diophantine arguments; in general, it is open and directly tied to the Skolem Problem. The problem is coNP-hard and in PSPACE but essentially driven by number-theoretic, rather than traditional complexity, barriers (Piribauer et al., 2023, Coves et al., 2024).

Key results:

• LRS order ≤5: Positivity decidable in the Counting Hierarchy/coNP^[PP^PP^PP], Ultimate Positivity in P (Ouaknine et al., 2013).
• Simple LRS order ≤9: Decidable, in Counting Hierarchy (Ouaknine et al., 2013).
• Order 6 barrier: Decidability at order 6 would yield major breakthroughs in Diophantine approximation (e.g., explicit computation of Lagrange constants for transcendental numbers) (Ouaknine et al., 2013).
• Weighted matrix powers (eventual positivity): coNP-hard for $m \geq 2$, reduces to Ultimate Positivity for LRS (Akshay et al., 2022).
• Polynomial and non-commutative rings: Positivity is undecidable once the base ring is $\mathbb{Z}[x_1, ..., x_d]$ or $\mathbb{Z}\langle z_1, ..., z_d \rangle$ for suitable dimension (Coves et al., 2024).
• Robustness setting: "Robust" positivity (positivity over a ball/neighborhood of the initial vector) is as hard as the classical problem. The Positivity Problem for an explicit neighborhood remains Diophantine-hard at order 6; existential variants are PSPACE-decidable for bounded order (Akshay et al., 2022, Vahanwala, 2023).

3. Methodological Reductions and Hardness Transfers

A landmark result by Piribauer & Baier establishes polynomial-time reductions from the LRS Positivity Problem to threshold questions for a broad family of objectives in Markov decision processes, including:

• Termination/energy objectives.
• Expected termination time.
• Quantile/cost or CVaR for accumulated weights.
• Stochastic shortest path (partial and conditional).
• Two-sided variants (Piribauer et al., 2023).

The reductions use modular gadget constructions:

• Initial gadget creates start weights and converts the sign question for $u_w$ to an action selection in the MDP.
• Recurrence gadget ensures the difference in value functions follows the recurrence relation for LRS.
• Initial-value gadget tunes initial states to match LRS initials.

Consequently, any algorithmic advance on these MDP threshold problems would entail a breakthrough for the general LRS Positivity Problem.

Analogously, Markov chain reachability (especially for ergodic chains) is shown reducible to the LRS Positivity and Skolem Problems, with chain size nearly matching LRS order (Vahanwala, 2023).

Matrix-weighted power sum problems and positivity in trace/moment sequences also reduce to ultimate positivity for LRS, sharing their hardness and decidability frontiers (Akshay et al., 2022, Coves et al., 2024).

4. Decidable and Efficient Subcases

Complete decision procedures exist in certain constrained settings:

• Low-order LRS, simple spectrum: Quantifier elimination and exponential-polynomial analysis yield explicit decision algorithms (PTIME for ultimate positivity order ≤5, Counting Hierarchy for positivity) (Ouaknine et al., 2013, Ouaknine et al., 2013).
• Orthogonal/unitary matrices and real spectrum: Positivity of trace/moment sequences (e.g., tr$A^n$) decidable for fixed dimension via real quantifier elimination and group structure (Coves et al., 2024).
• Three-term polynomial recurrences: Explicit necessary and sufficient positivity criteria based on discriminants, continued fractions, and initial value tests. Algorithmic procedures are provided for such systems (Pei et al., 2023).
• BB84 QKD key rates: Jain & Adhikari derive analytic bounds guaranteeing positive secret key rate, resulting in an explicit protocol abort condition (Jain et al., 23 Apr 2025).
• Numerical methods for ODEs with positivity constraints: For systems with graph-Laplacian structure, splitting and Magnus expansion schemes yield unconditional, positivity- and mass-preserving methods of order 2 (Blanes et al., 2021).

5. Geometric and Cone-Based Approaches for Recurrences

The geometric method of constructing invariant cones provides decidability and proofs of positivity for recurrences with a unique simple dominant eigenvalue. Ibrahim extends this to polynomial-coefficient recurrences of Poincaré type, via sequences of cones contracted under sequential application of the difference operator. This method proves positivity inductively for families with several simple dominant eigenvalues, under spectral separation and genericity hypotheses (Ibrahim, 18 Mar 2025).

6. Extensions, Robustness, and Open Problems

Robustness under Initialization

Explicit robust positivity asks whether all recurrences with initial vector in a specified neighborhood remain positive. Hardness remains Diophantine at order 6, matching the classical initialized case (Akshay et al., 2022, Vahanwala, 2023). Existential robustness (existence of some positive-radius neighborhood where positivity holds) is PSPACE-decidable for fixed order, as are open-ball variants (Akshay et al., 2022).

Extension to Nearly Linear Recurrences

Nearly Linear Recurrence Sequences (NLRS), with bounded errors in the recursive update, admit a Positivity Problem that is algorithmically solvable for order ≤3, all characteristic roots of modulus ≤1, using a transcendence argument to ensure nonvanishing limit behaviors (Pouly et al., 31 Jul 2025).

Quantum and Operator Positivity

For polynomial-Gaussian integral operators, positivity is governed solely by the underlying Gaussian: if the pure Gaussian fails positivity, no polynomial prefactor can restore it. Odd-degree polynomials always yield non-positive operators. There is a preorder structure on Gaussian kernels, allowing systematic inference of positivity across operator families (Balka et al., 2024). For trace-class Weyl operators, the KLM conditions are made finitely checkable via Gabor frames (Cordero et al., 2017).

Combinatorial and Algebraic Positivity

Positivity of Schubert structure constants $c_{u,v}^w$ in the Schubert polynomial basis is a major open problem. Under GRH and strengthened derandomization hypotheses, positivity is in NP (i.e., admits a certificate), but combinatorial, unconditional rules are conjectural (Pak et al., 2024).

7. Summary Table: Decidability and Hardness Across Key Problems

Domain/Problem	General Status	Decidable Orders/Cases	Key Results/Barriers
LRS Positivity	Open (hard)	≤5 (all), ≤9 (simple)	Order-6 implies Diophantine breakthroughs (Ouaknine et al., 2013)
Matrix powers (eventual nonneg.)	coNP-hard for m≥2	m=1: Perron–Frobenius	Reduces to ULT-NONNEG for LRS (Akshay et al., 2022)
Generalized moment (trace positivity)	Decidable in O/unitary/real	Fixed d: quantifier elimination	Undecidable for polynomial rings (Coves et al., 2024)
Thresholded MDPs/Markov chains	Positivity-hard	Small orders, special spectral cases	Any algorithm would break LRS Positivity (Piribauer et al., 2023, Vahanwala, 2023)
Polynomial recurrences (Poincaré)	Partially open	Several simple dominants, generic	Sequence-of-cones extension (Ibrahim, 18 Mar 2025)
Robustness to initial perturbation	Diophantine-hard (explicit)	Existential open-ball, fixed order	PSPACE algorithms for existential variants (Akshay et al., 2022, Vahanwala, 2023)
Quantum polynomial-Gaussian operators	Positive iff Gaussian is	n/a	Preorder, criterion via Gaussian core (Balka et al., 2024)
Schubert coefficients	Open; in NP under GRH+MVA	Some small n, special types	NP-witness via Nullstellensatz (Pak et al., 2024)

8. Open Directions and Significance

The Key Positivity Problem serves as a mathematical "frontier" with broad implications. Any progress in its general algorithmic decidability would yield breakthroughs in analytic number theory, particularly Diophantine approximation. Conversely, any unconditional hardness or undecidability proof at higher orders would necessarily circumvent known barriers in LRS theory.

Research continues in extending decidable subclasses, refining robustness analyses, and unifying the problem across operator, combinatorial, and dynamical settings. The centrality of the LRS Positivity Problem in these reductions establishes it as a touchstone for complexity in discrete mathematics, control, spectral theory, formal methods, and quantum computation.