  This paper introduces the foundations of the polynomial algebra and basic
structures for algebraic geometry over the extended tropical semiring. Our
development, which includes the tropical version for the fundamental theorem of
algebra, leads to the reduced polynomial semiring -- a structure that provides
a basis for developing a tropical analogue to the classical theory of
commutative algebra. The use of the new notion of tropical algebraic com-sets,
built upon the complements of tropical algebraic sets, eventually yields the
tropical algebraic Nullstellensatz.
