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 $\Omega$ 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.

Appendix A: Sobolev regularity of continuous roots

The next theorem is a refinement of [33, Theorem 2] in the case that $\Omega$ is a Lipschitz domain.

Theorem A.1

Let $\Omega \subseteq \mathbb {R}^m$ be a bounded Lipschitz domain. Let $P_a$ be a monic polynomial (1.1) with coefficients $a_j \in C^{n-1,1}({\overline{\Omega }})$, $j = 1,\ldots ,n$. Let $\lambda \in C^0(V)$ be a root of $P_a$ on an open subset $V \subseteq \Omega$. Then $\lambda$ belongs to the Sobolev space $W^{1,p}(V)$ for every $1 \le p < \frac{n}{n-1}$. The distributional gradient $\nabla \lambda$ satisfies

$$\begin{aligned} \Vert \nabla \lambda \Vert _{L^p(V)} \le C(m,n,p,\Omega ) \max _{1 \le j \le n} \Vert a_j\Vert ^{1/j}_{C^{n-1,1}({\overline{\Omega }})}. \end{aligned}$$

(A.1)

Proof

By [33, Theorem 1], $\lambda$ is absolutely continuous along affine lines parallel to the coordinate axes (restricted to V). So $\lambda$ possesses the partial derivatives $\partial _i \lambda$, $i=1,\ldots ,m$, which are defined almost everywhere and are measurable.

Set $x=(t,y)$, where $t=x_1$, $y=(x_2,\ldots ,x_m)$, and let $V_1$ be the orthogonal projection of V on the hyperplane $\{x_1=0\}$. For each $y \in V_1$ we denote by $V^y := \{t \in \mathbb {R}{:}\,(t,y) \in V\}$ the corresponding section of V.

Let $\lambda ^y_j$, $j=1,\ldots ,n$, be a continuous system of the roots of $P_a(\cdot ,y)$ on $V^y$ such that $\lambda (\cdot ,y) = \lambda ^y_1$; it exists since $\lambda (\cdot ,y)$ can be completed to a continuous system of the roots of $P_a(\cdot ,y)$ on each connected component of $V^y$ by [38, Lemma 6.17]. Our goal is to bound

$$\begin{aligned} \Vert \partial _t\lambda (\cdot ,y)\Vert _{L^p(V^y)} = \Vert (\lambda ^y_1)'\Vert _{L^p(V^y)} \end{aligned}$$

uniformly with respect to $y \in V_1$.

Let $R= I_1 \times \cdots \times I_m \subseteq \mathbb {R}^m$ be an open box containing $\Omega$ and such that $|I_i| \le {\text {diam}}(\Omega )$ for all $i =1,\ldots ,m$. By Whitney’s extension theorem (cf. Sect. 3.2), the coefficients $a_j$ of $P_a$ admit a $C^{n-1,1}$-extension ${\hat{a}}_j$ to $\mathbb {R}^m$ such that

$$\begin{aligned} \max _{1 \le j \le n} \Vert {\hat{a}}_j\Vert ^{1/j}_{C^{n-1,1}({\overline{R}})} \le C(m,n,\Omega )\, \max _{1 \le j \le n} \Vert a_j\Vert ^{1/j}_{C^{n-1,1}(\overline{\Omega })}. \end{aligned}$$

(A.2)

Let $\mathcal {C}^y$ denote the set of connected components J of the open subset $V^y \subseteq \mathbb {R}$. For each $J \in \mathcal {C}^y$ we extend the system of roots $\lambda ^y_j|_J$, $j=1,\ldots ,n$, continuously to $I_1$, i.e., we choose continuous functions $\lambda ^{y,J}_j$, $j = 1,\ldots , n$, on $I_1$ such that $\lambda ^{y,J}_j|_J = \lambda ^y_j|_J$ for all j and

$$\begin{aligned} P_{{\hat{a}}}(t,y)(Z) = \prod _{j=1}^n (Z-\lambda ^{y,J}_j(t)), \quad t \in I_1. \end{aligned}$$

This is possible since $\lambda ^y_j|_J$ has a continuous extension to the endpoints of the (bounded) interval J, by [26, Lemma 4.3], and can then be extended on the left and on the right of J by a continuous system of the roots of $P_{{\hat{a}}}(\cdot ,y)$ after suitable permutations.

By [33, Theorem 1], for each $y \in V_1$, $J \in \mathcal {C}^y$, and $j=1,\ldots ,n$, the function $\lambda ^{y,J}_j$ is absolutely continuous on $I_1$ and $(\lambda ^{y,J}_j)' \in L^p(I_1)$, for $1 \le p < n/(n-1)$, with

$$\begin{aligned} \Vert (\lambda ^{y,J}_j)'\Vert _{L^p(I_1)} \le C(n,p,|I_1|) \, \max _{1 \le i \le n} \Vert {\hat{a}}_i\Vert ^{1/i}_{C^{n-1,1}({\overline{R}})}. \end{aligned}$$

(A.3)

Let $J,J_0 \in \mathcal {C}^y$ be arbitrary. By [32, Lemma 3.6], $(\lambda ^y_j)'$ as well as $(\lambda ^{y,J_0}_j)'$ belong to $L^p(J)$ and we have

$$\begin{aligned} \sum _{j=1}^n \Vert (\lambda ^y_j)'\Vert _{L^p(J)}^p = \sum _{j=1}^n \Vert (\lambda ^{y,J}_j)'\Vert _{L^p(J)}^p = \sum _{j=1}^n \Vert (\lambda ^{y,J_0}_j)'\Vert _{L^p(J)}^p. \end{aligned}$$

Thus,

$$\begin{aligned} \sum _{j=1}^n \Vert (\lambda ^y_j)'\Vert _{L^p(V^y)}^p&= \sum _{J \in \mathcal {C}^y} \sum _{j=1}^n \Vert (\lambda ^y_j)'\Vert _{L^p(J)}^p = \sum _{J \in \mathcal {C}^y} \sum _{j=1}^n \Vert (\lambda ^{y,J_0}_j)'\Vert _{L^p(J)}^p \\&= \sum _{j=1}^n \Vert (\lambda ^{y,J_0}_j)'\Vert _{L^p(V^y)}^p \le \sum _{j=1}^n \Vert (\lambda ^{y,J_0}_j)'\Vert _{L^p(I_1)}^p. \end{aligned}$$

In particular, by (A.3),

$$\begin{aligned} \Vert \partial _t\lambda (\cdot ,y)\Vert _{L^p(V^y)} = \Vert (\lambda ^y_1)'\Vert _{L^p(V^y)} \le C(n,p,|I_1|) \, \max _{1 \le i \le n} \Vert {\hat{a}}_i\Vert ^{1/i}_{C^{n-1,1}({\overline{R}})}, \end{aligned}$$

and so, by Fubini’s theorem,

$$\begin{aligned} \int _{V} |\partial _1 \lambda (x)|^p\, dx&= \int _{V_1} \int _{V^y} |\partial _1 \lambda (t,y)|^p\, dt\, dy\\&\le \Big (C(n,p,|I_1|) \, \max _{1 \le i \le n} \Vert {\hat{a}}_i\Vert ^{1/i}_{C^{n-1,1}({\overline{R}})} \Big )^p \int _{V_1} \, dy. \end{aligned}$$

Thus, thanks to $|I_1| \le {\text {diam}}(\Omega )$,

$$\begin{aligned} \Vert \partial _1 \lambda \Vert _{L^p(V)} \le C(n,p,{\text {diam}}(\Omega )) \, \max _{1 \le i \le n} \Vert {\hat{a}}_i\Vert ^{1/i}_{C^{n-1,1}({\overline{R}})}. \end{aligned}$$

In view of (A.2) this implies (A.1), since the other partial derivatives $\partial _i \lambda$, $i \ge 2$, are treated analogously. $\square$