Explicit factors of some iterated resultants and discriminants
Explicit factors of some iterated resultants and discriminants
复制标题
一些迭代结果和判别式的显式因子
作者：
Laurent Busé;B. Mourrain
本文分析了将迭代单变量合成构造应用于多元多项式的结果。我们将输入多项式视为给定次数的通用多项式，并显示出对几种结构的不可约因子的显式分解，这些结构涉及输入多项式系数的整数通用环上的两次迭代单变量结果和判别式。处理涉及两到四个通用多项式及其变量之一的结果或判别式的情况。我们得到的不可约因子的分解是通过利用单变量合式和判别式的基本性质以及多项式次数的归纳来获得的。因此，每个不可约因子都可以根据某个多元结果单独且显式地计算。通过这种方法，我们还获得了柯林斯和麦卡勒姆推测的一些结果作为直接推论，这些结果对应于多项式的情况，该多项式的系数本身就是其他变量中的通用多项式。最后，详细介绍了单个多项式的迭代判别式的代数因式分解的几何解释。
In this paper, the result of applying iterative univariate resultant constructions to multivariate polynomials is analyzed. We consider the input polynomials as generic polynomials of a given degree and exhibit explicit decompositions into irreducible factors of several constructions involving two times iterated univariate resultants and discriminants over the integer universal ring of coefficients of the entry polynomials. Cases involving from two to four generic polynomials and resultants or discriminants in one of their variables are treated. The decompositions into irreducible factors we get are obtained by exploiting fundamental properties of the univariate resultants and discriminants and induction on the degree of the polynomials. As a consequence, each irreducible factor can be separately and explicitly computed in terms of a certain multivariate resultant. With this approach, we also obtain as direct corollaries some results conjectured by Collins and McCallum which correspond to the case of polynomials whose coefficients are themselves generic polynomials in other variables. Finally, a geometric interpretation of the algebraic factorization of the iterated discriminant of a single polynomial is detailled.