Height bounds, nullstellensatz and primality

ABSTRACT

In this study, we find height bounds in the polynomial ring over the field of algebraic numbers to test the primality of an ideal. We also obtain height bounds in the arithmetic Nullstellensatz. We apply nonstandard analysis and hence our constants will be ineffective.

KEYWORDS:

• Height
• model theory
• nonstandard analysis
• primality

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 11G50, 03C98, 13L05

Acknowledgments

This work is partially based on the author’s Ph.D thesis [10]. The author thanks his Ph.D supervisors Amador Martin-Pizarro and Frank Wagner for many motivating discussions. The author also would like to thank the anonymous referee for the invaluable suggestions, which immensely improved the quality and presentation of the paper. Also Theorems 2, 3 and their proofs, and the other proof of Theorem 1 in Remark 3.1 were suggested by the referee.