Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Cosine and Computation   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.

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The categories of this paper are: comp-gas
5
The categories of this paper are: gr-qc
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-ex
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The categories of this paper are: nucl-th
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The categories of this paper are: dg-ga
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The categories of this paper are: acc-phys
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The categories of this paper are: bayes-an
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The categories of this paper are: nucl-ex
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The categories of this paper are: ao-sci
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The categories of this paper are: quant-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: econ
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The categories of this paper are: physics
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The categories of this paper are: hep-ph
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The categories of this paper are: q-fin