  Polynomial neural networks have been implemented in a range of applications
and present an advantageous framework for theoretical machine learning. A
polynomial neural network of fixed architecture and activation degree gives an
algebraic map from the network's weights to a set of polynomials. The image of
this map is the space of functions representable by the network. Its Zariski
closure is an affine variety known as a neurovariety. The dimension of a
polynomial neural network's neurovariety provides a measure of its
expressivity. In this work, we introduce the notion of the activation threshold
of a network architecture which expresses when the dimension of a neurovariety
achieves its theoretical maximum. In addition, we prove expressiveness results
for polynomial neural networks with equi-width~architectures.
