TY - JOUR
T1 - Integer-valued polynomials on valuation rings of global fields with prescribed lengths of factorizations
AU - Fadinger-Held, Victor
AU - Frisch, Sophie
AU - Windisch, Daniel
N1 - Publisher Copyright:
© 2023, The Author(s).
PY - 2023/12
Y1 - 2023/12
N2 - Let V be a valuation ring of a global field K. We show that for all positive integers k and 1 < n1≤ ⋯ ≤ nk there exists an integer-valued polynomial on V, that is, an element of Int(V)={f∈K[X]∣f(V)⊆V} , which has precisely k essentially different factorizations into irreducible elements of Int(V) whose lengths are exactly n1, … , nk . In fact, we show more, namely that the same result holds true for every discrete valuation domain V with finite residue field such that the quotient field of V admits a valuation ring independent of V whose maximal ideal is principal or whose residue field is finite. If the quotient field of V is a purely transcendental extension of an arbitrary field, this property is satisfied. This solves an open problem proposed by Cahen, Fontana, Frisch and Glaz in these cases.
AB - Let V be a valuation ring of a global field K. We show that for all positive integers k and 1 < n1≤ ⋯ ≤ nk there exists an integer-valued polynomial on V, that is, an element of Int(V)={f∈K[X]∣f(V)⊆V} , which has precisely k essentially different factorizations into irreducible elements of Int(V) whose lengths are exactly n1, … , nk . In fact, we show more, namely that the same result holds true for every discrete valuation domain V with finite residue field such that the quotient field of V admits a valuation ring independent of V whose maximal ideal is principal or whose residue field is finite. If the quotient field of V is a purely transcendental extension of an arbitrary field, this property is satisfied. This solves an open problem proposed by Cahen, Fontana, Frisch and Glaz in these cases.
KW - Discrete valuations domains
KW - Factorizations
KW - Global fields
KW - Integer-valued polynomials
KW - Irreducible polynomials
UR - http://www.scopus.com/inward/record.url?scp=85169816827&partnerID=8YFLogxK
U2 - 10.1007/s00605-023-01895-2
DO - 10.1007/s00605-023-01895-2
M3 - Article
AN - SCOPUS:85169816827
SN - 0026-9255
VL - 202
SP - 773
EP - 789
JO - Monatshefte fur Mathematik
JF - Monatshefte fur Mathematik
IS - 4
ER -