The Membership Problem for Ideals in Z[X]

Authors

Carlos E. Arreche, Princeton UniversityFollow

Abstract

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.

Author Bio

I am currently an undergraduate Mathematics major at Princeton University. I completed this paper under the guidance of Professor Luis F. C攼㸱ceres-Duque as a senior year project for my Mathematical Research class in the University Gardens High School in San Juan, Puerto Rico. This paper was eventually presented at Intel's 2004 International Science and Engineering Fair held in Portland, Oregon, where it obtained a Third Place Award in Mathematics. I would like to obtain a Ph.D. in Mathematics and become a mathematician.

Faculty Sponsor

Luis F. Cáceres

Recommended Citation

Arreche, Carlos E. (2005) "The Membership Problem for Ideals in Z[X]," Rose-Hulman Undergraduate Mathematics Journal: Vol. 6: Iss. 2, Article 2.
Available at: https://scholar.rose-hulman.edu/rhumj/vol6/iss2/2