  For $d \geq 2$ and $n \in \mathbb{N}$ even, let $p_n = p_n(d)$ denote the
number of length $n$ self-avoiding polygons in $\mathbb{Z}^d$ up to
translation. The polygon cardinality grows exponentially, and the growth rate
$\lim_{n \in 2\mathbb{N}} p_n^{1/n} \in (0,\infty)$ is called the connective
constant and denoted by $\mu$. Madras [J. Statist. Phys. 78 (1995) no. 3--4,
681--699] has shown that $p_n \mu^{-n} \leq C n^{-1/2}$ in dimension $d=2$.
Here we establish that $p_n \mu^{-n} \leq n^{-3/2 + o(1)}$ for a set of even
$n$ of full density when $d=2$. We also consider a certain variant of
self-avoiding walk and argue that, when $d \geq 3$, an upper bound of $n^{-2 +
d^{-1} + o(1)}$ holds on a full density set for the counterpart in this variant
model of this normalized polygon cardinality.
