  Coupled cluster theory produced arguably the most widely used high-accuracy
computational quantum chemistry methods. Despite the approach's overall great
computational success, its mathematical understanding is so far limited to
results within the realm of functional analysis. The coupled cluster
amplitudes, which are the targeted objects in coupled cluster theory,
correspond to solutions to the coupled cluster equations, which is a system of
polynomial equations of at most degree four. The high dimensionality of the
electronic Schr\"odinger equation and the non-linearity of the coupled cluster
ansatz have so far stalled a formal analysis of this polynomial system. In this
article, we present algebraic investigations that shed light on the coupled
cluster equations and the root structure of this ansatz. This is of importance
for the a posteriori evaluation of coupled cluster calculations. To that end,
we investigate the root structure by means of Newton polytopes. We derive a
general v-description, which is subsequently turned into an h-description for
explicit examples. This perspective reveals an apparent connection between
Pauli's exclusion principle and the geometrical structure of the Newton
polytopes. We also propose an alternative characterization of the coupled
cluster equations projected onto singles and doubles as cubic polynomials on an
algebraic variety with certain sparsity patterns. Moreover, we provide
numerical simulations of two computationally tractable systems, namely, the two
electrons in four spin-orbitals system and the three electrons in six
spin-orbitals system. These simulations provide novel insight into the root
structure of the coupled cluster solutions when the coupled cluster ansatz is
truncated.
