  We calculate the helicity trace index $B_{14}$ for $\mathcal{N} = 8$ pure
D-brane black holes using various techniques of computational algebraic
geometry and find perfect agreement with the existing results in the
literature. For these black holes, microstate counting is equivalent to finding
the number of supersymmetric vacua of a multi-variable supersymmetric quantum
mechanics which in turn is equivalent to solving a set of multi-variable
polynomial equations with syzygies. We explore four different techniques to
solve a set of polynomial equations, namely Newton Polytopes, Homotopy
continuation, Monodromy and Hilbert series. Unlike other schemes, these methods
are almost exact with a very high success rate for finding all solutions. A
gauge invariant analysis is also possible using the symbolic Hilbert series.
Furthermore, exploiting various exchange symmetries, we show the non-existence
of a quartic and(or) higher order terms in the potential which if present would
have spoiled the counting. Incorporating recent developments in mathematics,
algorithms and multi-threading, we have extended the scope of one of the
author's previous works to other charges and presented a new perspective for
the microstate counting problem. This further establishes the pure D-brane
system as a consistent model, bringing us a step closer to $\mathcal{N} = 2$
black hole counting.
