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.

中文翻译：

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余弦和计算

我们有兴趣解决决策问题 $\exists? t \in \mathbb{N}, \cos t \theta = c$ 其中 $\cos \theta$ 和 $c$ 是代数数。我们称之为 $\cos t \theta$ 问题。这是对带有解析函数的丢番图方程的探索。多项式、实数指数和余弦函数与这个决策问题密切相关： $ \exists ? t \in \mathbb{N}, u^TM^tv = 0$ 其中 $u, v \in \mathbb{Q}^n, M \in \mathbb{Q}^{n\times n}$。此问题也称为“Skolem 问题”，可用于验证线性系统。它的可判定性仍然未知。具有实代数底指数函数的单变量丢番图方程和具有 $\theta$ 的 $\pi$ 有理倍数的 $\cos t \theta$ 函数是可判定的。当 $M$ 的特征值是实数根时，这个想法是证明 Skolem 问题的可判定性的核心。特征值不是实数根的情况的主要困难在于，即使对于小阶情况，可判定性也需要应用超越数理论，而该理论不适用于高阶情况。当$\theta$ 不是$\pi$ 的有理倍数时，我们通过为$\cos t \theta$ 提供$PTIME$ 算法来首次尝试克服这个问题。我们这样做没有使用超越数论的技术。\par 丢番图方程的主要困难之一是无法使用微积分中的工具来求解这个方程，因为变量域是 $\mathbb{N}$。