Harmonic Properties of the Logarithmic Potential and the Computability of Elliptic Fekete Points

Abstract

We investigate the properties of the function sending each N-tuple of points to minus the logarithm of the product of their mutual distances. We prove that, as a function defined on the product of N spheres, this function is subharmonic, and indeed its (Riemannian) Laplacian is constant. We also prove a mean value equality and an upper bound on the derivative of the function. We use these results to get sharp upper bounds for the precision needed to describe an approximation to elliptic Fekete points (in the sense demanded by Smale’s 7th problem). We also conclude that Smale’s 7th problem has solutions given by rational spherical points of bounded (small) bit length, proving that there exists an exponential running time algorithm which solves it on the Turing machine model.

Appendix: Topics from Riemannian Geometry and Harmonic Analysis in Manifolds

To facilitate the reading of this manuscript, we include two short sections with some results from Riemannian geometry and harmonic analysis that have been used in the paper. The contents of this appendix are mainly taken from [8, 14].

1.1 A.1 Riemannian Geometry

By a “Riemannian manifold” $\mathcal{M}$, we mean here a smooth (C ^∞) differentiable manifold with a smooth Riemannian structure, that is, a smooth section $\langle\cdot,\cdot\rangle:T\mathcal{M}\times T\mathcal{M}\,{\rightarrow}\,\mathbb{R}$, where $T\mathcal{M}$ is the tangent bundle of $\mathcal{M}$ and for each $p\in \mathcal{M}$, $\langle \cdot,\cdot\rangle_{p}:T_{p}\mathcal{M}\times T_{p}\mathcal{M}\,{\rightarrow}\,\mathbb{R}$ is a definite positive, symmetric bilinear map. We denote by n the dimension of $\mathcal{M}$. Associated with 〈⋅,⋅〉 _p , we consider the norm $\|v\|_{p}=\langle v,v\rangle_{p}^{1/2}$ for $v\in T_{p}\mathcal{M}$. We denote by C ²(U) the set of C ² functions defined in some open set $U\subseteq \mathcal{M}$.

Given a collection $\mathcal{M}^{(1)},\ldots,\mathcal{M}^{(r)}$ of Riemannian manifolds, we can define in $\mathcal{M}=\mathcal{M}^{(1)}\times\cdots\times \mathcal{M}^{(r)}$ a product Riemannian structure as follows: Let $p=(p^{(1)},\ldots,p^{(r)})\in \mathcal{M}$, and let $v^{(i)},w^{(i)}\in T_{p^{(i)}}\mathcal{M}^{(i)}$, 1≤i≤r, be a collection of tangent vectors. Then define

$$\bigl\langle\bigl(v^{(1)},\ldots,v^{(r)}\bigr),\bigl(w^{(1)},\ldots,w^{(r)}\bigr)\bigr\rangle_p=\bigl\langle v^{(1)},w^{(1)}\bigr\rangle_{p^{(1)}}+\cdots+\bigl\langle v^{(r)},w^{(r)}\bigr\rangle_{p^{(r)}}.$$

The length of a piecewise C ¹ curve $\gamma:[a,b]\rightarrow \mathcal{M}$ with tangent vector $\dot{\gamma}$ (i.e., $\dot{\gamma}(t)\in T_{\gamma (t)}\mathcal{M},t\in[a,b]$) is defined as

$$L(\gamma)=\int_a^b\|\dot{\gamma}\|\,dt.$$

The distance between two points $p,q\in \mathcal{M}$ is then defined as the infimum of the lengths of piecewise C ¹ curves with extremes p,q. This gives $\mathcal{M}$ a structure of metric space and allows us to define open and closed balls as usual.

Given a coordinate chart $\phi:U\subseteq \mathcal{M}\,{\rightarrow}\, V\subseteq \mathbb{R}^{n}$, the Riemannian product is represented by a positive definite, symmetric matrix

$$G(x)=\bigl(g_{ij}(x)\bigr),\quad x\in V,$$

such that given $v,w\in T_{p} \mathcal{M}$ with p∈U, v=(v ₁,…,v _n ) and w=(w ₁,…,w _n ) in coordinates, we have

$$\langle v,w\rangle_x=w^TG(x)v,$$

where w ^T is the transpose of w. A function f:U → ℝ is called (Lebesgue-)measurable or integrable if f(ϕ ⁻¹(x))|det(DG(x))|^1/2 is (Lebesgue-)measurable or integrable as a function of x∈V. In that case, the integral of f in U is defined as

$$\int_Uf(p)\,dp=\int_Vf\bigl(\phi^{-1}(x)\bigr)\bigl|\det\bigl(DG(x)\bigr)\bigr|^{1/2}\,dx.$$

Given $f:\mathcal{M}\,{\rightarrow}\,\mathbb{R}$, we define

$$\int_\mathcal{M}f(p)\,dp=\sum_{\alpha}\int_{U_\alpha}f(p)\rho_\alpha(p)\,dp,$$

where {ρ _α } is a partition of the unity subordinate to some open cover of $\mathcal{M}$ by coordinate charts {(U _α ,ϕ _α )} (f is called measurable or integrable if it is so in each U _α ). The volume of some measurable subset $U\subseteq \mathcal{M}$ is

$$\operatorname {Vol}{(U)}=\int_\mathcal{M}\chi_U(p)\,dp,\quad \chi_U(p)=\begin{cases}1&p\in U,\\0&\mathrm{otherwise}.\end{cases}$$

If $\operatorname {Vol}{(U)}<\infty$, we define the expected value of f:U → ℝ in U as

$$\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle-}{\int}$}\vcenter{\hbox{$\textstyle-$}}\kern-.5\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle-}{\int}$}\vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$}\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$}\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}\!\int _U f(p)\,dp=\frac{1}{\operatorname {Vol}{(U)}}\int_Uf(p)\,dp.$$

We denote by g ^ij the components of the inverse matrix G(x)⁻¹. The Chirstoffel symbols associated with ϕ are then

$$\varGamma _{jk}^i=\frac{1}{2}\sum _{l=1}^ng^{il} \biggl(\frac{\partial g_{jl}}{\partial x_k}+\frac{\partial g_{kl}}{\partial x_j}-\frac{\partial g_{jk}}{\partial x_l} \biggr).$$

A smooth curve $\gamma:[a,b]\rightarrow \mathcal{M}$ is a geodesic if it is a critical point of the energy functional $\int_{a}^{b}\|\dot{\gamma}\|^{2}\,dt$. In coordinates, denoting x(t)=(x ₁(t),…,x _n (t))=ϕ(γ(t)), for i∈{1,…n},

$$\ddot{x}_i(t)=-\dot{x}(t)^T\varGamma ^i\bigl(x(t)\bigr)\dot{x}(t),\quad \varGamma ^i=\bigl(\varGamma ^i_{jk}\bigr)_{j,k=1\ldots n}.$$

Given $p\in \mathcal{M}$ and $v\in T_{p}\mathcal{M}$, there exists ε>0 and a unique geodesic $\gamma:[0,\varepsilon]\,{\rightarrow}\,\mathcal{M}$ such that γ(0)=p and $\dot{\gamma}(0)=v$. From the geodesic equation above, we can easily see that if $\mathcal{M}^{(1)}\times\cdots\times \mathcal{M}^{(r)}$ has the product structure, then a curve $\gamma=(\gamma^{(1)},\ldots,\gamma ^{(r)}):[a,b]\,{\rightarrow}\,\mathcal{M}$ is a geodesic in $\mathcal{M}$ if and only if $\gamma ^{(i)}:[a,b]\,{\rightarrow}\,\mathcal{M}^{(i)}$ is a geodesic in $\mathcal{M}^{(i)}$ for i∈{1,…,r}. If $\mathcal{M}\subseteq \mathbb{R}^{k}$ is a smooth submanifold of ℝ^k with the Riemannian structure inherited from ℝ^k (that is, 〈v,w〉 _p =〈v,w〉 the usual inner product in ℝ^k), the geodesic equation reads

$$\bigl\|\dot{\gamma}(t)\bigr\|=\text{constant},\qquad\ddot{\gamma}(t)\perp T_{\gamma(t)}\mathcal{M}\subseteq \mathbb{R}^k,\quad\text{for all }t.$$

We are most interested in the cases $\mathcal{M}=\mathbb{S}$ with the Riemannian structure inherited from ℝ³ and $\mathcal{M}=\mathbb{S}^{N}=\mathbb{S}\times\cdots \times \mathbb{S}$ (N times) with the product Riemannian structure. Of course, this product structure is also the Riemannian structure of $\mathbb{S}^{N}$ as a submanifold of ℝ^3N. Note that $\mathbb{S}$ has dimension 2 and $\mathbb{S}^{N}$ has dimension 2N. Geodesics in $\mathbb{S}$ are great circles parametrized with constant speed. More exactly, the geodesic γ(t) such that $\gamma(0)=p\in \mathbb{S}$ and $\dot{\gamma}(0)=v\in T_{p}\mathbb{S}$, where ∥v∥=1, is given by

$$ \gamma_{p,v}(t)=p\cos(2t)+\frac{v}{2}\sin(2t)+\frac{1}{2}\bigl(0,0,1-\cos(2t)\bigr)^T.$$

(A.1)

Indeed, the reader may check that $\gamma(0)=p,\dot{\gamma}(0)=v,\dot{\gamma}(t)\perp T_{\gamma(t)}\mathbb{S},\|\dot{\gamma}(t)\|=1$ for all t.

1.2 A.2 Harmonic Analysis in Manifolds

The Hessian of a function f∈C ²(U) is a bilinear form defined as

$$\operatorname{Hess}({\mathcal{E}})(p) (v,w)=X\bigl(Y(f)\bigr)-(\nabla_XY)(f),\quad p\in U,v,w\in T_p\mathcal{M},$$

where ∇ is the Levi-Civita connection and X,Y are vector fields such that X(p)=v, Y(p)=w. In coordinates,

$$\operatorname{Hess}({f})\bigl(p(x)\bigr) (v,w)=w^t\bigl (h_{ij}(x)\bigr)v,\quad h_{ij}(x)=\frac{\partial^2f}{\partial x_i\partial x_j}-\sum_{k=1}^n\frac{\partial f}{\partial x_k}\varGamma _{ij}^k.$$

If γ(t) is the unique geodesic such that γ(0)=p and $\dot{\gamma}(0)=v$, then

$$\operatorname{Hess}({f})(p) (v,v)=\frac{d^2}{dt^2}\bigg {|}_{t=0}f\bigl(\gamma (t)\bigr).$$

The Laplacian³ $\Delta {f}=\operatorname {div}(\operatorname{grad}(\mathcal{E}))$ of f is the trace of $\operatorname{Hess}({f})$ as a bilinear operator, that is, the trace of (h _ij ), or in coordinates,

$$\Delta {f}=\sum_{i=1}^nh_{ii}(x)=\sum_{i=1}^n \biggl(\frac{\partial^2f}{\partial x_i\partial x_i}\biggr)-\sum_{k=1}^n\frac{\partial f}{\partial x_k}\Biggl(\sum_{i=1}^n\varGamma _{ii}^k\Biggr).$$

A function f∈C ²(U), $U\subseteq \mathcal{M}$ open, is called harmonic (subharmonic) if Δf=0 (Δf≥0). Note that Δ is a uniformly elliptic operator and satisfies the hypotheses of [13, Sect. 24]. Thus, the classical strong maximum principle applies. Namely:

Theorem 6.1

(Strong maximum principle)

Let f∈C ²(U) be a nonconstant subharmonic function. Then for any open set Ω⊆U such that $\overline{\varOmega }\subseteq U$, we have

$$x\in \varOmega \quad\Rightarrow\quad f(x)<\sup_{x\in \varOmega }f.$$

In particular, the supremum is reached only in the boundary of Ω.

The trace of (h _ij ) is equal to the trace of Q ^∗(h _ij )Q for any orthogonal matrix Q. Hence, we also have

$$\Delta {f}=\sum_{{v^{(1)},\ldots,v^{(2N)}}\text{\ an orthonormal basis of}\ T_x\mathbb{S}^N}\operatorname{Hess}({f})(x) \bigl(v^{(i)},v^{(i)} \bigr).$$

The classical mean value theorem for harmonic functions in ℝⁿ (see for example [11, Sect. 2.2]) claims that, if f:U⊆ℝⁿ → ℝ is harmonic (U an open set), then $f(x)= \mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle-}{\int}$}\vcenter{\hbox{$\textstyle-$}}\kern-.5\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle-}{\int}$}\vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$}\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$}\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}\!\int _{B(x,\varepsilon)}f(y)\,dy$ for each ball B(x,ε)⊆U. Unfortunately, no similar equality is valid in general for harmonic functions on Riemannian manifolds, but there is a class of manifolds for which it holds. These are called locally harmonic manifolds, and the corresponding mean value equality was first proved by Willmore [22] (see also [5, Chap. 6]). Following the proof of [5, Prop. 6.21], if $\mathcal{M}$ is a locally harmonic manifold and $f:\mathcal{M}\,{\rightarrow}\,\mathbb{R}$ is such that Δf=C is constant, then for small enough t>0, we have

$$ \frac{d}{dt} \mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle-}{\int}$}\vcenter{\hbox{$\textstyle-$}}\kern-.5\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle-}{\int}$}\vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$}\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$}\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}\!\int _{S(x,t)}f=C \frac{\mathrm {Vol}(B(x,t))}{\mathrm {Vol}(S(x,t))}.$$

(A.2)

The sphere $\mathbb{S}\subseteq \mathbb{R}^{3}$ is locally harmonic, and moreover, (A.2) can be stated for every t>0 such that f is defined in B(x,t). Thus, the classical mean value theorem is valid in the case of $\mathbb{S}=\mathcal{M}$:

Theorem 6.2

Let f:U → ℝ ($U\subseteq \mathbb{S}$ an open set) be such that Δf=C. Let p∈U. Then, for every ε>0 such that B _p (ε)⊆U, we have

$$\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle-}{\int}$}\vcenter{\hbox{$\textstyle-$}}\kern-.5\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle-}{\int}$}\vcenter{\hbox{$\scriptstyle-$}}\kern-.5\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle-}{\int}$}\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle-}{\int}$}\vcenter{\hbox{$\scriptscriptstyle-$}}\kern-.5\wd0}}\!\int _{S(p,\varepsilon)}f=f(p)+C\int_0^\varepsilon \frac{\pi\sin^2t}{\pi\sin(2t)}=f(p)-\frac{C}{2}\log(\cos \varepsilon).$$