Hensel引理的一个结果
A resultant for Hensel's Lemma
摘要：
设R是完备离散赋值环，其极大理想由π生成.设f（X）是R[X]中的一个首一多项式，其判别式Delta（f）为非零.设s >= v_pi（Delta（f））+ 1。假设给定f（X）在（R/pi^s R）[X]中的因子分解为几个因子，不一定在（R/pi R）[X]中互质。我们将它唯一地提升为f（X）在R[X]中的因子分解。这推广了Hensel-Rychlik引理，它涵盖了两个因子的情况。我们直接使用分解为几个因子的提升，避免了将分解分解为两个因子。为了这个目的，我们定义一个结果为几个多项式在一个变量作为行列式的广义西尔维斯特矩阵。
英文摘要：
Let R be a complete discrete valuation ring with maximal ideal generated by pi. Let f(X) in R[X] be a monic polynomial with nonzero discriminant Delta(f). Let s >= v_pi(Delta(f)) + 1. Suppose given a factorisation of f(X) in (R/pi^s R)[X] into several factors, not necessarily coprime in (R/pi R)[X]. We lift it uniquely to a factorisation of f(X) in R[X]. This generalises the Hensel-Rychlik Lemma, which covers the case of two factors. We work directly with lifts of factorisations into several factors and avoid having to iterate factorisations into two factors. For this purpose we define a resultant for several polynomials in one variable as determinant of a generalised Sylvester matrix.
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