A note on intersective polynomials in function fields
- QIAN Kun, LIU Bao-qing, LI Guo-quan
JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(12):
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Let Z denote the ring of rational integers. Let L be a field, and let p be its characteristic. Let f(x)=∑nj=0ajxj∈L［x］, the polynomial ring over L. Suppose that f(x)=a∏ri=1(x-ηi)ei over some algebraic closure of L, where a∈L, all the ηi are distinct, and r,e1,e2,…,er are positive integers with r≥2 and n=∑rj=1ej. The semidiscriminant Δ(f) of f is defined by Δ(f)=a2n-1∏1≤i,j≤ri≠j(ηi-ηj)ei ej. It is proved that if n<p, then there exist a positive integer m with m|n! and a polynomial G∈Z［x0,x1,…,xn］, which depend only on the vector (e1,e2,…,er), such that Δ(f)=1/mG(a0,a1,…,an). This result is applied to investigate a question on intersective polynomials over the ring L［x］, where L is a finite field.