  In this first of four articles, we study a homogeneous system of $2N+3$
linear partial differential equations (PDEs) in $2N$ variables that arises in
conformal field theory (CFT) and multiple Schramm-Lowner evolution (SLE). In
CFT, these are null-state equations and conformal Ward identities. They govern
partition functions for the continuum limit of a statistical cluster or loop
model, such as percolation, or more generally the Potts models and O$(n)$
models, at the statistical mechanical critical point. (SLE partition functions
also satisfy these equations.) For such a lattice model in a polygon
$\mathcal{P}$ with its $2N$ sides exhibiting a free/fixed side-alternating
boundary condition, this partition function is proportional to the CFT
correlation function
$\langle\psi_1^c(w_1)\psi_1^c(w_2)\dotsm\psi_1^c(w_{2N-1})\psi_1^c(w_{2N})\rangle^{\mathcal{P}}$
where the $w_i$ are the vertices of $\mathcal{P}$ and where $\psi_1^c$ is a
one-leg corner operator. When conformally mapped onto the upper half-plane,
methods of CFT show that this correlation function satisfies the system of PDEs
that we consider.
  This article is the first of four that completely and rigorously characterize
the space of all solutions for this system of PDEs that grow no faster than a
power law. In this first article, we use methods of analysis to prove that the
dimension of this solution space is no more than $C_N$, the $N$th Catalan
number. This proof is contained entirely within this article, except for the
proof of lemma 14, which constitutes the second article ("part II"). In the
third article ("part III"), we use the results of this article to prove that
the solution space of this system of PDEs has dimension $C_N$ and is spanned by
solutions constructed with the CFT Coulomb gas (contour integral) formalism. In
the fourth article ("part IV"), we prove further CFT-related properties about
these solutions.
