Discrete maximum principle for polynomials

The polynomially convex hull of a compact set {K\subset \mathbb C} is defined as the set of all points {z\in \mathbb C} such that the inequality {|p(z)|\le \sup_K |p|} (a form of the maximum principle) holds for every polynomial {p}. For example, the polynomially convex hull of a simple closed curve is the union of that curve with its interior region. In general, this process fills up the holes in the set {K}, resulting in the complement of the unbounded connected component of {\mathbb C\setminus K}.

We can recover the usual convex hull from this construction by restricting {p} to the polynomials of first degree. Indeed, when {p} is linear, the set {|p|\le M} is a closed disk, and we end up with the intersection of all closed disks that contain {K}. This is precisely the convex hull of {K}.

What if we restrict {p} to the polynomials of degree at most {n}? Let’s call the resulting set the degree-{n} convex hull of {K}, denoted {K_n}. By definition, it is contained in the convex hull and contains the polynomially convex hull. To exactly compute {K_n} for general {K} appears to be difficult even when {n=2}.

Consider finite sets. When {K} has at most {n} points, we have {K_n=K} because there is a polynomial of degree {n} whose zero set is precisely {K}. So, the first nontrivial case is of {K} having {n+1} points. Let us write {K=\{z_0, \dots, z_n\}}.

Depending on the location of the points, {K_n} may be strictly larger than {K}. For example, if {K} consists of the vertices of a regular {(n+1)}-gon, then {K_n} also contains its center. Here is why. By a linear transformation, we can make sure that {K=\{\zeta^k\colon k=0, \dots, n\}} where {\zeta = \exp(2\pi i/(n+1))}. For {j=1, \dots, n} we have {\sum_{k=0}^n \zeta^{kj} = (\zeta^{(n+1)j}-1)/(\zeta^j - 1) = 0}. Hence, for any polynomial {p} of degree at most {n}, the sum {\sum_{k=0}^n p(\zeta^k)} is equal to {(n+1)p(0)}. This implies {|p(0)|\le \max_{0\le k\le n}|p(\zeta^k)|}, a kind of a discrete maximum principle.

A more systematic approach is to use the Lagrange basis polynomials, that is {L_j(z) = \prod_{k\ne j} (z-z_k)/(z_j-z_k)}, which satisfy {L_j(z_k) = \delta_{jk}}. Since {p = \sum_j p(z_j)L_j} for any polynomial of degree at most {n}, it follows that {z\in K_n} if and only if {\left|\sum_j c_j L_j(z)\right|\le \max |c_j| } holds for every choice of scalars {c_0, \dots, c_n}. The latter is equivalent to the inequality {\sum_j |L_j(z)|\le 1}.

This leads us to consider the function {S=\sum_j |L_j|}, the sum of the absolute values of the Lagrange basis polynomials. (Remark: S is called a Lebesgue function for this interpolation problem.) Since {\sum_j L_j\equiv 1}, it follows that {S\ge 1} everywhere. By construction, {S=1} on {K}. At a point {z\notin K}, the equality {S(z)=1} holds if and only if {\arg L_j(z)} is the same for all {j}.

In the trivial case {K=\{z_0, z_1\}}, the function {S(z) = (|z-z_0|+|z-z_1|)/|z_0-z_1|} is equal to {1} precisely on the linear segment with endpoints {z_0, z_1}. Of course, this only repeats what we already knew: the degree-1 convex hull is the ordinary convex hull.

If {K=\{x_0, \dots, x_n\}} with {x_0<\cdots<x_n} real and {n\ge 2}, then {K_n=K}. Indeed, if {x\in K_n\setminus K}, then {x} lies in the convex hull of {K}, and therefore {x_{j-1}<x<x_j} for some {j}. The basis polynomial {L_j} is positive at {x}, since it is equal to {1} at {x_j} and does not vanish outside of {K}. Since a polynomial changes its sign at every simple zero, it follows that {L_{j+1}(x) < 0}. Well, there is no {L_{j+1}} if {j=n}, but in that case, the same reasoning applies to {L_{j-2}(x)<0}. In any case, the conclusion is that {\arg L_k(x)} cannot be the same for all {k}.

At this point one might guess that the vertex set of a regular polygon is the only kind of finite sets that admit a nontrivial discrete maximum principle for polynomials. But this is not so: the vertices of a rhombus work as well. Indeed, if {K=\{a, -a, ib, -ib\}} with {a, b>0}, then {L_j(0)>0} for all {j}, hence {S(0)=1}.

The vertices of a non-square rectangle do not work: if {K} is the set of these vertices, the associated function {S} is strictly greater than 1 on the complement of {K}.

Are there any other finite sets that support a discrete maximum principle for polynomials?

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