TY - JOUR
T1 - Polynomial products modulo primes and applications
AU - Klurman, Oleksiy
AU - Munsch, Marc
PY - 2020/3/1
Y1 - 2020/3/1
N2 - For any polynomial P(x) ∈ Z[x] , we study arithmetic dynamical systems generated by FP(n)=∏k≤nP(k)(modp),n≥ 1. We apply this to improve the lower bound on the number of distinct quadratic fields of the form Q(FP(n)) in short intervals M≤ n≤ M+ H previously due to Cilleruelo, Luca, Quirós and Shparlinski. As a second application, we estimate the average number of missing values of FP(n)(modp) for special families of polynomials, generalizing previous work of Banks, Garaev, Luca, Schinzel, Shparlinski and others.
AB - For any polynomial P(x) ∈ Z[x] , we study arithmetic dynamical systems generated by FP(n)=∏k≤nP(k)(modp),n≥ 1. We apply this to improve the lower bound on the number of distinct quadratic fields of the form Q(FP(n)) in short intervals M≤ n≤ M+ H previously due to Cilleruelo, Luca, Quirós and Shparlinski. As a second application, we estimate the average number of missing values of FP(n)(modp) for special families of polynomials, generalizing previous work of Banks, Garaev, Luca, Schinzel, Shparlinski and others.
KW - Diophantine equations
KW - Distribution of sequences modulo p
KW - Dynamical system modulo p
KW - Perfect powers
KW - Polynomials
KW - Prime ideals of number fields
UR - http://www.scopus.com/inward/record.url?scp=85077625522&partnerID=8YFLogxK
U2 - 10.1007/s00605-019-01359-6
DO - 10.1007/s00605-019-01359-6
M3 - Article
AN - SCOPUS:85077625522
SN - 0026-9255
VL - 191
SP - 577
EP - 593
JO - Monatshefte fur Mathematik
JF - Monatshefte fur Mathematik
IS - 3
ER -