Cosine and Computation

TL;DR

This paper investigates the decidability of a class of Diophantine equations involving cosine functions and exponential terms, proposing a polynomial-time algorithm for certain cases without relying on transcendental number theory.

Contribution

It introduces a PTIME algorithm for solving the cosine Diophantine problem when the angle is not a rational multiple of pi, bypassing complex transcendental methods.

Findings

Decidability remains open for general cases involving cosine and exponential functions.

A polynomial-time algorithm is proposed for specific cases where heta is not a rational multiple of pi.

Reduction of the Skolem problem to real-variable equations is demonstrated.

Abstract

We are interested in solving decision problem $\exists? t \in \mathbb{N}, \cos t \theta = c$ where $\cos \theta$ and $c$ are algebraic numbers. We call this the $\cos t \theta$ problem. This is an exploration of Diophantine equations with analytic functions. Polynomial, exponential with real base and cosine function are closely related to this decision problem: $ \exists ? t \in \mathbb{N}, u^T M^t v = 0$ where $u, v \in \mathbb{Q}^n, M \in \mathbb{Q}^{n\times n}$. This problem is also known as "Skolem problem" and is useful in verification of linear systems. Its decidability remains unknown. Single variable Diophantine equations with exponential function with real algebraic base and $\cos t \theta$ function with $\theta$ a rational multiple of $\pi$ is decidable. This idea is central in proving the decidability of Skolem problem when the eigenvalues of $M$ are roots of real numbers. The main difficulty with the cases when eigenvalues are not roots of reals is that even for small order cases decidability requires application of trancendental number theory which does not scale for higher order cases. We provide a first attempt to overcome that by providing a $PTIME$ algorithm for $\cos t \theta$ when $\theta$ is not a rational multiple of $\pi$. We do so without using techniques from transcendental number theory. \par One of the main difficulty in Diophantine equations is being unable to use tools from calculus to solve this equation as the domain of variable is $\mathbb{N}$. We also provide an attempt to overcome that by providing reduction of Skolem problem to solving a one variable equation (which involves polynomials, exponentials with real bases and $\cos t \theta$ function with $t$ ranging over reals and $\theta \in [0, \pi]$) over reals.