  Finding the solutions to a system of multivariate polynomial equations is a
fundamental problem in mathematics and computer science. It involves evaluating
the polynomials at many points, often chosen from a grid. In most current
methods, such as subdivision, homotopy continuation, or marching cube
algorithms, polynomial evaluation is treated as a black box, repeating the
process for each point. We propose a new approach that partially evaluates the
polynomials, allowing us to efficiently reuse computations across multiple
points in a grid. Our method leverages the Compressed Sparse Fiber data
structure to efficiently store and process subsets of grid points. We
integrated our amortized evaluation scheme into a subdivision algorithm.
Experimental results show that our approach is efficient in practice. Notably,
our software \texttt{voxelize} can successfully enclose curves defined by two
trivariate polynomial equations of degree $100$, a problem that was previously
intractable.
