关于函数域中相交多项式的一个注记

以Z表示有理整数环。设L为一个特征为p的域, f(x)=∑ⁿ_j=0a_jx^j∈L［x］,L［x］表示L上的多项式环。假定在L的某个代数闭包上, f(x)=a∏^r_i=1(x-η_i)^e_i。此处,a∈L,一切η_i是两两不同的,r,e₁,e₂,…,e_r是正整数,且r≥2, n=∑^r_j=1e_j。f的半判别式Δ(f)被定义为Δ(f)=a^2n-1∏_{1≤i,j≤ri≠j}(η_i-η_j)^{e_i e_j}。证明了下面的结果: 如果n1,e₂,…,e_r)有关的正整数m与G∈Z［x₀,x₁,…,x_n］,使得Δ(f)=1/mG(a₀,a₁,…,a_n)且m|n!。此外,当L为有限域时,还应用此结果研究了与环L［x］上相交多项式有关的一个问题。

A note on intersective polynomials in function fields

Let Z denote the ring of rational integers. Let L be a field, and let p be its characteristic. Let f(x)=∑ⁿ_j=0a_jx^j∈L［x］, the polynomial ring over L. Suppose that f(x)=a∏^r_i=1(x-η_i)^ei over some algebraic closure of L, where a∈L, all the η_i are distinct, and r,e1,e2,…,e_r are positive integers with r≥2 and n=∑^r_j=1e_j. The semidiscriminant Δ(f) of f is defined by Δ(f)=a2n-1∏1≤i,j≤r_i≠j(η_i-η_j)^{e_i e_j}. It is proved that if n, then there exist a positive integer m with m|n! and a polynomial G∈Z［x₀,x₁,…,x_n］, which depend only on the vector (e1,e2,…,e_r), such that Δ(f)=1/mG(a0,a1,…,a_n). This result is applied to investigate a question on intersective polynomials over the ring L［x］, where L is a finite field.