Complete solution to the TP2 completion problem

For any pattern of the specified entries, an explicit finite list of polynomial inequalities in the specified entries is given that characterizes the completability of a partial positive matrix to a TP matrix. Each polynomial happens to be a monomial, and an algorithm, whose input is the pattern, is given for finding the polynomials. The method uses some new partial orders on matrices and the logarithmic method to reduce the question to determining the generators of a certain finitely generated, pointed cone.