TY - JOUR
T1 - Tropical algebraic sets, ideals and an algebraic nullstellensatz
AU - Izhakian, Zur
N1 - Funding Information:
The author would like to thank Professor Eugenii Shustin for his invaluable help. The author is deeply grateful to him for his support and the fertile discussions we had. The author has been supported by the grant of the High Council for Scientific and Technological Cooperation between Israel and France.
PY - 2008/9
Y1 - 2008/9
N2 - 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.
AB - 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.
KW - Algebraic sets and com-sets
KW - Idempotent semiring
KW - Max-plus algebra
KW - Nullstellensatz
KW - Polynomial ideals
KW - Tropical algebraic geometry
UR - http://www.scopus.com/inward/record.url?scp=52949149021&partnerID=8YFLogxK
U2 - 10.1142/S0218196708004779
DO - 10.1142/S0218196708004779
M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???
AN - SCOPUS:52949149021
SN - 0218-1967
VL - 18
SP - 1067
EP - 1098
JO - International Journal of Algebra and Computation
JF - International Journal of Algebra and Computation
IS - 6
ER -