POTENTIAL AT INFINITY OF A

POLYNOMIAL IMAGE OF THE DISK

S.Yu.Orevkov

Let D be a domain in C which is the image q(Δ) of the unit disk Δ under the mapping t ↦ q(t) where

q(t) = a_{0}t + a_{1}t^{2} + ... + a_{n}t^{n+1},  a_{0} ∈ R, |a_{0}| > 0  (1)

is a polynomial, univalent in D.

Let p(z) = π^{−1} ∫_{D}(z − ζ)^{−1}dμ(ζ) be the potential of D (dμ(x + iy) = dx dy). The inverse problem of potential theory is the problem of reconstructing D given the germ of p at ∞ (see [1] and references therein). For |z| ≫ 1 one has

p(z) = ∑_{k≥0} \frac{c_{k}}{z^{k+1}} ,  where c_{k} = \frac{1}{π} ∫_{D} ζ^{k}dμ(ζ) = \frac{1}{π} ∫_{Δ} q(t)^{k}q′(t)\overline{q′(t)} dμ(t). (2)

(c_{k} are the moments of D). Using the right hand side of (2) one can define p(z) for any polynomial q(t) of the form (1), not necessarily univalent. Since

\frac{1}{π} ∫_{Δ} t^{k}t̄^{m} dμ(t) = \{ \begin{matrix}1/(k + 1), & if k = m \\ 0, & if k ≠ m, \end{matrix}  (3)

(2) allows us to express c_{k} as polynomials in a_{0}, a_{1}, ..., a_{n}, ā_{1}, ..., ā_{n}, with rational coefficients. It follows from (2), (3) that c_{k} = 0 for k > n. Hence,

p(z) = c_{0}z^{−1} + c_{1}z^{−2} + ... + c_{n}z^{−(n+1)},  c_{0} ∈ R, |c_{0}| > 0  (4)

Thus, we obtain a polynomial mapping η : V^{+} → W^{+} where V (resp. W) is the vector space over R, isomorphic to R × C^{n}, with the coordinates (a_{0}, ..., a_{n}) (resp. (c_{0}, ..., c_{n})), V^{+} (resp. W^{+}) is the half-spaces a_{0} > 0, (resp. c_{0} > 0) and η is defined by η(q) = p. We identify points of V^{+} (resp. W^{+}) with polynomials q of the form (1) (resp. of the form (4)).

Denote by j(η) the jacobian of η with respect to the volume forms da_{n} ∧ ... ∧ da_{1} ∧ da_{0} ∧ dā_{1} ∧ ... ∧ dā_{n} and dc_{n} ∧ ... ∧ dc_{1} ∧ dc_{0} ∧ dc̄_{1} ∧ ... ∧ dc̄_{n}. Given q ∈ V, let us define t_{1}, ..., t_{n} by q′(t) = ∏_{i=1}^{n}(1 − t_{i}t).

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Partially supported by Grants RFFI-96-01-01218 and DGICYT SAB95-0502

Theorem. j(η) = 2a_{0}^{n^{2}+n+1} Res_{t}(t^{n}q̄′(t^{−1}), q′(t)) = 2a_{0}^{n^{2}+3n+1} ∏_{i=1}^{n} ∏_{j=1}^{n}(1 − t_{i}t̄_{j}).

Remark. (P. Etingof) A similar formula appears in the Conformal Field Theory as a formula for the cocycle defining the central extension of the complexification of the group Diff(S^{1}).

Let I_{n} = {q ∈ V^{+} | q′ has no root in Δ}.

Corollary 1. (see [1]) The restriction of η onto Int I_{n} is an immersion.

Corollary 2. The closure of the fold of η is the (real) hypersurface F = {q ∈ V^{+} | ∃t_{i}, q′(t_{i}) = 0 and |t_{i}| = 1}.

Corollary 3. The closure of the branching set of η is the codimension 2 subvariety B = {q ∈ V^{+} | ∃t_{i} ≠ t_{j}, t_{i}t̄_{j} = 1}

Proof of Theorem. Denote ∂/∂a_{m} by ∂_{m} and ∂/∂ā_{m} by ¯∂_{m}, m = 0, ..., n. (∂_{0} = ¯∂_{0}). Put

J = ( \begin{matrix}¯∂_{n}c_{n} … ¯∂_{1}c_{n} ∂_{0}c_{n} ∂_{1}c_{n} … ∂_{n}c_{n} \\ · · · · · · · · · \\ ¯∂_{n}c_{1} … ¯∂_{1}c_{1} ∂_{0}c_{1} ∂_{1}c_{1} … ∂_{n}c_{1} \\ ¯∂_{n}c_{0} … ¯∂_{1}c_{0} ∂_{0}c_{0} ∂_{1}c_{0} … ∂_{n}c_{0} \\ ¯∂_{n}c̄_{1} … ¯∂_{1}c̄_{1} ∂_{0}c̄_{1} ∂_{1}c̄_{1} … ∂_{n}c̄_{1} \\ · · · · · · · · · \\ ¯∂_{n}c̄_{n} … ¯∂_{1}c̄_{n} ∂_{0}c̄_{n} ∂_{1}c̄_{n} … ∂_{n}c̄_{n} \end{matrix} ) ,

A = ( \begin{matrix}A_{n,n} & & 0 & 0 \\ ⋮ & ... & & ⋮ & & 0 \\ A_{1,n} & … & A_{1,1} & 0 \\ 0 & … & 0 & 1 & 0 & … & 0 \\ & & & 0 & Ā_{1,1} & … & Ā_{1,n} \\ & 0 & & ⋮ & & ... & ⋮ \\ & & & 0 & 0 & & Ā_{n,n} \end{matrix} ) , M = \frac{1}{2a_{0}} ( \begin{matrix}−(n+1)ā_{n} \\ 2a_{0}E  ⋮  0 \\ −2ā_{1} \\ 0 … 0 a_{0} 0 … 0 \\ 2a_{1} \\ 0  ⋮  2a_{0}E \\ (n+1)a_{n} \end{matrix} )

where A_{k,j}, j ≥ k > 1 are polynomials in a_{0}^{−1}, a_{n}, ..., a_{1}, a_{0}, ā_{1}, ..., ā_{n}, such that ∑_{j≥k} A_{k,j}q^{j} = t^{k}. Let us calculate AJ. Put A_{k,j} = 0 for j < k, A_{0,0} = 1, A_{0,j} = 0 for j > 0, and a_{m} = 0 unless 0 ≤ m ≤ n. Denote \frac{1}{π} ∫_{Δ} f(t)\overline{g(t)} dμ(t) by ⟨f, g⟩. Then (3) means ⟨t^{k}, t^{m}⟩ = δ_{km}/(k + 1). Note that ∂_{m}q = t^{m+1}, ∂_{m}q′ = (m + 1)t^{m} and

(∂_{m}q^{j}) · q′ = \frac{dq^{j}}{dq} · ∂_{m}q · q′ = \frac{dq^{j}}{dq} · \frac{dq}{dt} · ∂_{m}q = \frac{dq^{j}}{dt} · ∂_{m}q.

Hence, for any m ≥ 0 we have

∑_{j} A_{k,j}⟨∂_{m}(q^{j}q′), q′⟩ = ⟨(∂_{m}q) · \frac{d}{dt} ∑ A_{k,j}q^{j}, q′⟩ + ⟨(∂_{m}q′) · ∑ A_{k,j}q^{j}, q′⟩
= ⟨t^{m+1} · kt^{k−1}, q′⟩ + ⟨(m + 1)t^{m} · t^{k}, q′⟩
= (m + k + 1)⟨t^{m+k}, q′⟩ = (m + k + 1)ā_{m+k},
∑_{j} A_{k,j}⟨q^{j}q′, ∂_{m}q′⟩ = ⟨q′t^{k}, (m + 1)t^{m}⟩ = (m − k + 1)a_{m−k}

Therefore, the entries of the right upper quadrant of AJ (resp. the left upper quadrant, the upper part of the central column) are equal resp. to

∑ A_{k,j}∂_{m}c_{j} = ∑ A_{k,j}⟨∂_{m}(q^{j}q′), q′⟩ = (m + k + 1)ā_{m+k},  m > 0,
∑ A_{k,j} ¯∂_{m}c_{j} = ∑ A_{k,j}⟨q^{j}q′, ∂_{m}q′⟩ = (m − k + 1)a_{m−k},  m > 0,
∑ A_{k,j}∂_{0}c_{j} = ∑ A_{k,j}(⟨∂_{0}(q^{j}q′), q′⟩ + ⟨q^{j}q′, ∂_{0}q′⟩) = (k + 1)ā_{k} − (k − 1)a_{−k}.

Each entry of the lower part of AJ is conjugated to the centrally symmetric entry of the upper one. Thus,

AJ = ( \begin{matrix}a_{0} & & 0 & & (n+1)ā_{n} & & & 0 \\ 2a_{1} & a_{0} & & & nā_{n−1} & (n+1)ā_{n} \\ ⋮ & 2a_{1} & ... & & ⋮ & nā_{n−1} & ... \\ na_{n−1} & ⋮ & ... & a_{0} & 2ā_{1} & ⋮ & ... & (n+1)ā_{n} \\ (n+1)a_{n} & na_{n−1} & … & 2ā_{1} & 2a_{0} & 2ā_{1} & … & nā_{n−1} & (n+1)ā_{n} \\ & (n+1)a_{n} & ... & ⋮ & 2a_{1} & a_{0} & ... & ⋮ & nā_{n−1} \\ & & ... & nā_{n−1} & ⋮ & & ... & 2ā_{1} & ⋮ \\ & & & (n+1)ā_{n} & na_{n−1} & & & a_{0} & 2a_{1} \\ & 0 & & & (n+1)a_{n} & & 0 & & a_{0} \end{matrix} )

Multiplying this matrix by M from the right, we replace the central column with (0, ..., 0, a_{0}, 2a_{1}, ..., (n+1)a_{n})^{t} and obtain the matrix whose upper row is (a_{0}, 0, ..., 0) and the complementary minor of the a_{0} is the transposed Sylvester matrix for the resultant of a_{0} + 2a_{1}t + ... + (n + 1)a_{n}t^{n} and (n + 1)ā_{n} + nā_{n−1}t + ... + ā_{0}t^{n}. Thus,

det A det J det M = a_{0} Res_{t}(q′, t^{n}q̄′(t^{−1})).

Clearly that det M = ^{1}/_{2} and A_{k,k} = a_{0}^{−k} which implies det A = ∏ A_{k,k}^{2} = a_{0}^{−n(n+1)}. It remains to note that the reversing the order of ∂a_{m}’s in J changes the sign the same way as the swapping the arguments of the resultant. □

I am grateful to P. Etingof and N. Kruzhilin for usefull discussions.

References

1. A.I. Prilepko, Potential theory, inverse problems in, in: Mathematical Encyclopedia, vol. 4, pp. 266–269.

Steklov Math. Inst., Russ. Acad. Sci., Gubkina 8, Moscow, Russia