There exists a feasible procedure to decide whether or not an arbitrary polynomial belongs to a given ideal in Z[x] if the ideal's minimal basis is known. However, when this is not the case there is no feasible procedure to decide whether or not an arbitrary polynomial belongs to a given ideal. There already exists an effective procedure to find an ideal's minimal basis, but it depends on solving the membership problem for the ideal (i.e. the problem of deciding whether an arbitrary polynomial belongs to the ideal). Therefore, we develop a modification of the existing algorithm to find an ideal's minimal basis so that there is no need to solve the membership problem to carry it out, and then we use this minimal basis to solve the membership problem for this ideal.
Professor Luis F. Cáceres, Professor of Mathematics,University of Puerto Rico-Mayagüezlcaceresd@gmail.com
Arreche, Carlos E.
"The Membership Problem for Ideals in Z[X],"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 6
, Article 2.
Available at: http://scholar.rose-hulman.edu/rhumj/vol6/iss2/2