  An Ulam sequence U(1,n) is defined as the sequence starting with integers 1,n
such that n > 1, and such that every subsequent term is the smallest integer
that can be written as the sum of distinct previous terms in exactly one way.
This family of sequences is notable for being the subject of several remarkable
rigidity conjectures. We introduce an analogous notion of an Ulam sequence
inside the polynomial ring Z[X], and use it both to give new, constructive
proofs of old results as well as producing a new conjecture that implies many
of the other existing conjectures.
