Membership tests for images of algebraic sets by linear projections

Given a witness set for an irreducible variety V and a linear map π, we describe membership tests for both the constructible algebraic set π(V) and the algebraic set π(V)¯. We also provide applications and examples of these new tests including computing the codimension one components of π(V)¯⧹π(V)¯. Additionally, we also describe computing the geometric genus of a curve section of an irreducible component of the solution set of a polynomial system and a test for deciding whether a plane quartic curve is a Lüroth quartic.