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Abstract
A conjecture by R. Varley states that the Thom-Boardman invariant for polynomial multiplication maps can be computed by using the Euclidean algorithm on the degrees of the polynomials. This thesis provides some history of the problem, its connection with secant maps and Gauss maps, proofs of classes of cases, and it develops a theory which gives an upper bound on the invariant that agrees with the conjectured invariant. We con-struct monomial ideals which have the conjectured invariant and discuss the generaliza-tion of this construction to polynomial ideals. There is also a discussion on the symmetric product of a smooth curve, along with some basic deformation theory, that is used in the proof of a proposition concerning versality of families of hyperplane sections and transversality to a stratification of the symmetric product.