Decidability of the Threshold Problem for Hypergeometric Sequences with Quadratic Parameters

Core Concepts

The Threshold Problem for hypergeometric sequences with monic polynomial coefficients that split over an imaginary quadratic field is decidable. The Threshold Problem for hypergeometric sequences with monic polynomial coefficients possessing Property S is decidable subject to the truth of Schanuel's conjecture.

Abstract

The paper considers the Threshold Problem for hypergeometric sequences, which asks to determine whether every term in a recursively defined sequence is bounded from below by a given threshold value.
The key insights and highlights are:

For hypergeometric sequences with monic polynomial coefficients that split over an imaginary quadratic field, the Threshold Problem is decidable. The proof leverages properties of the gamma function and algebraic independence results.

For hypergeometric sequences with monic polynomial coefficients possessing Property S, the Threshold Problem is decidable subject to the truth of Schanuel's conjecture. Property S is a condition on the roots of the polynomial coefficients that allows for a reduction to an equality testing problem involving the gamma function.

The classes of polynomials with Property S include even polynomials, irreducible monic quadratic polynomials, and polynomials with roots having rational real parts. This leads to conditional decidability results for hypergeometric sequences with parameters drawn from the integers of quadratic number fields, as well as sequences with unnested radical and cyclotomic parameters.

The decidability of the Membership Problem for hypergeometric sequences is shown to Turing-reduce to the decidability of the Threshold Problem, thus extending previous results in the literature.

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Key Insights Distilled From

The Threshold Problem for Hypergeometric Sequences with Quadratic Parameters

by George Kenis... at arxiv.org 04-25-2024

https://arxiv.org/pdf/2211.02447.pdf

Deeper Inquiries

What other classes of hypergeometric sequences, beyond those considered in the paper, could exhibit decidability of the Threshold Problem

In addition to the classes of hypergeometric sequences discussed in the paper, there are several other classes that could potentially exhibit decidability of the Threshold Problem. One such class is hypergeometric sequences with parameters that are related to algebraic numbers with specific properties, such as having rational real parts or being roots of certain types of polynomials. These sequences may exhibit symmetries or structural characteristics that allow for the reduction of the Threshold Problem to more manageable equality testing problems. Additionally, hypergeometric sequences with parameters derived from special functions or specific mathematical structures, such as Bessel functions or elliptic integrals, could also potentially demonstrate decidability of the Threshold Problem. By analyzing the properties of the parameters and the recurrence relations of these sequences, it may be possible to identify additional classes that exhibit decidability.

Can the conditional decidability results be strengthened to unconditional decidability by further restricting the structure of the polynomial coefficients

The conditional decidability results presented in the paper could potentially be strengthened to unconditional decidability by further restricting the structure of the polynomial coefficients in hypergeometric sequences. For example, imposing additional constraints on the roots of the polynomials, such as requiring them to belong to specific number fields or have certain algebraic properties, could lead to unconditional decidability of the Threshold Problem. By carefully examining the algebraic and transcendental properties of the parameters in hypergeometric sequences and their associated polynomials, it may be possible to identify more specific conditions under which the Threshold Problem is unconditionally decidable. Additionally, exploring the relationships between the parameters and the convergence properties of the sequences could provide insights into how to strengthen the conditional decidability results.

How might the techniques developed in this paper apply to the study of limits and identities involving hypergeometric sequences and the gamma function in other areas of mathematics and computer science

The techniques developed in the paper for studying the Threshold Problem for hypergeometric sequences with quadratic parameters and Property S could have applications in various areas of mathematics and computer science where hypergeometric sequences and the gamma function play a significant role. For example, in numerical analysis and computational mathematics, these techniques could be used to analyze the convergence properties of hypergeometric series and their associated sequences, leading to improved algorithms for numerical computations involving hypergeometric functions. In symbolic computation and algebraic algorithms, the methods developed in the paper could be applied to study identities involving hypergeometric sequences and the gamma function, potentially leading to new results in symbolic manipulation and algebraic simplification. Furthermore, in the field of automated verification and formal languages, the techniques for decidability of the Threshold Problem could be extended to other types of sequences and recurrence relations, providing a framework for analyzing the convergence and boundedness properties of a wide range of mathematical sequences.