关键词: 函数域, 半判别式, 相交多项式

Abstract: 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.