一元多项式的重根通过子结式的封闭公式
Closed formulae for multiple roots of univariate polynomials through subresultants
摘要：
实代数平面曲线拓扑的计算，如果在每条垂直线上不超过一个临界点：一般位置条件，则大为简化。当这个条件不满足时，则坐标的有限次改变将使初始曲线移动到一般位置。我们将展示许多情况下，所考虑的曲线的拓扑结构周围的临界点是非常容易计算，即使曲线不是在一般的位置。这将是通过引入一个新的家庭的公式描述，在许多情况下，通过subresult，重根的一元多项式作为有理函数的考虑多项式涉及最多一个平方根。这种新的方法将被用来表明，三次，四次和五次的拓扑结构可以很容易地计算，即使曲线是不是在一般的位置和那些高次曲线，这种方法可以使用。我们也将应用这种技术来确定两个二次曲面的相交曲线，并研究如何确定两个椭球相交时所产生的曲线的类型。
英文摘要：
The computation of the topology of a real algebraic plane curve is greatly simplified if there are no more than one critical point in each vertical line: the general position condition. When this condition is not satisfied, then a finite number of changes of coordinates will move the initial curve to one in general position. We will show many cases where the topology of the considered curve around a critical point is very easy to compute even if the curve is not in general position. This will be achieved by introducing a new family of formulae describing, in many cases and through subresultants, the multiple roots of a univariate polynomial as rational functions of the considered polynomial involving at most one square root. This new approach will be used to show that the topology of cubics, quartics and quintics can be computed easily even if the curve is not in general position and to characterise those higher degree curves where this approach can be used. We will apply also this technique to determine the intersection curve of two quadrics and to study how to characterise the type of the curve arising when intersecting two ellipsoids.
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