An Algebraic Proof of Barlet's Join Theorem

In this article we give an algebraic proof of Barlet's join theorem, using an improved version of an algorithm for computing the join that was presented in (J. Dalbec and B. Sturmfels,in“Invariant Methods in Discrete and Computational Geometry, Curaçao, 1994,” pp. 37–58, Kluwer Academic, Dordrecht, 1995). The improvements in the algorithm consist of proving the correctness of the algorithm in large prime characteristics and giving explicit formulas for the functionsp(q,L) that were left unspecified in the original. The first three sections are largely expository; the fourth presents the modified algorithm and the proof of the main theorem. The last section gives an example of the algorithm; unfortunately, the complexity of the process makes it difficult to compute anything without resorting to tricky shortcuts, as we do in the example.