Lengths of factorizations of integer-valued polynomials on Krull domains with prime elements

Area of Science:

• Abstract Algebra
• Commutative Algebra
• Number Theory

Background:

• Krull domains are fundamental structures in abstract algebra, generalizing unique factorization domains.
• Integer-valued polynomials play a crucial role in studying the arithmetic of integral domains.
• Understanding factorization properties of these polynomials is key to characterizing algebraic structures.

Purpose of the Study:

• To demonstrate the existence of integer-valued polynomials with a prescribed number of distinct factorizations into irreducibles.
• To characterize the lengths of these factorizations within specific types of Krull domains.
• To solve an open problem posed by Cahen, Fontana, Frisch, and Glaz regarding factorization lengths.

Main Methods:

• Construction of integer-valued polynomials within the ring Int(D) for a Krull domain D.
• Analysis of the number and lengths of factorizations into irreducible elements of Int(D).
• Characterization of factorization lengths in unique factorization domains and discrete valuation domains.

Main Results:

• For any Krull domain D with a prime element and finite residue field, integer-valued polynomials exist with k distinct factorizations.
• The lengths of these factorizations can be precisely controlled to match any given sequence n1, ..., nk.
• Factorization lengths are characterized for unique factorization domains and discrete valuation domains.

Conclusions:

• The study provides a comprehensive understanding of factorization properties of integer-valued polynomials in Krull domains.
• The results confirm and extend previous conjectures in the area of abstract algebra.
• This work resolves a significant open problem, offering new insights into the structure of Int(D).