2017/05/31 by Adam Parusiński, Adam Parusinski, Armin Rainer
Computer Science · Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Bounded function #Bounded variation #Mathematical Analysis and Transform Methods #Mathematical analysis #Mathematics #Physics #Pure mathematics #Variation (astronomy) #math.CA
paper · pdf · doi:10.1007/s00029-020-0538-z
published as Sel. Math. New Ser. 26, 13 (2020) · 33 pages. This version covers the general case, while the previous one only treated the case of radicals. Minor changes. Accepted for publication in Selecta Mathematica
arxiv created 2020/01/14 · openalex publication_date 2020/01/28 · arxiv updated 2021/04/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
We prove that the roots of a smooth monic polynomial with complex-valued coefficients defined on a bounded Lipschitz domain Ω in \mathbb Rm 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 W1,p for all 1 ≤ p < (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 Ck-1,1( Ω), 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 Zr = f, where f is a complex-valued function and r∈ \mathbb R>0, we obtain optimal uniform bounds.