We prove that the roots of a smooth monic polynomial with complex-valued coefficients defined on a bounded Lipschitz domain \(\Omega \) in \(\mathbb {R}^m\) admit a parameterization by functions of bounded variation uniformly with respect to the coefficients. This result is best possible in the sense that discontinuities of the roots are in general unavoidable due to monodromy. We show that the discontinuity set can be chosen to be a finite union of smooth hypersurfaces. On its complement the parameterization of the roots is of optimal Sobolev class \(W^{1,p}\) for all \(1 \le p < \frac{n}{n-1}\), where n is the degree of the polynomial. All discontinuities are jump discontinuities. For all this we require the coefficients to be of class \(C^{k-1,1}(\overline{\Omega })\), where k is a positive integer depending only on n and m. The order of differentiability k is not optimal. However, in the case of radicals, i.e., for the solutions of the equation \(Z^r = f\), where f is a complex-valued function and \(r\in \mathbb {R}_{>0}\), we obtain optimal uniform bounds.

中文翻译：

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光滑多项式根的有界变化选择

我们证明，在\（\ mathbb {R} ^ m \）的有界Lipschitz域\（\ Omega \）上定义的具有复数值系数的光滑单项多项式的根均接受有界变化函数关于系数。从根源上讲，根系的不连续通常是不可避免的，从这个意义上说，这种结果是最好的。我们表明，不连续集可以选择为光滑超曲面的有限并集。作为补充，对于所有\（1 \ le p <\ frac {n} {n-1} \），根的参数化是最佳Sobolev类\（W ^ {1，p } \），其中n是多项式的次数。所有不连续点都是跳跃不连续点。对于所有这些，我们要求系数属于\（C ^ {k-1,1}（\ overline {\ Omega}）\）类，其中k是仅取决于n和m的正整数。可微性的次序k不是最佳的。但是，对于部首而言，即对于方程\（Z ^ r = f \）的解，其中f是复数值函数，\（r \ in \ mathbb {R} _ {> 0} \），我们可以获得最佳的统一边界。