Selections of bounded variation for roots of smooth polynomials

Abstract

We prove that the roots of a smooth monic polynomial with complex-valued coefficients defined on a bounded Lipschitz domain Ω in 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 ≤ 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 C k−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 Z r = f , where f is a complex-valued function and r ∈ R >0 , we obtain optimal uniform bounds.

Sobolev Lifting over Invariants

“…When continuous lifting is impossible, we expect that a general BV -lifting result is true analogous to the existence of BV -roots for smooth polynomials proved in [17]. We shall not pursue that question in this paper.…”

“…This paper arose from our wish to understand and extend the principles behind our proof of the optimal Sobolev regularity of roots of smooth families of polynomials [13,15,16,17]. Here we look at this problem from a representation theoretic view point.…”

Quantitative tame properties of differentiable functions with controlled derivatives

“…The right-hand side of (25) depends uniformly on f (see Remark 4.10). Now it is easy to conclude Note that the Sobolev regularity in the results of [15,16] used above is optimal. Remark 4.9.…”

Sobolev Lifting over Invariants

“…Motivation and introduction to the problem. This paper arose from our wish to understand and extend the principles behind our proof of the optimal Sobolev regularity of roots of smooth families of polynomials [13,15,16,17]. Here we look at this problem from a representation theoretic view point.…”

“…for all 1 ≤ p < d/(d − 1), where C is a constant which depends only on the representation G V , on Ω, m, and p. When continuous lifting is impossible, we expect that a general BV -lifting result is true analogous to the existence of BV -roots for smooth polynomials proved in [17]. We shall not pursue that question in this paper.…”