Transfinite diameter on the graph of a polynomial mapping and the DeMarco–Rumely formula

We study Chebyshev constants and transfinite diameter on the graph of a polynomial mapping f:C2→C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:{{\mathbb {C}}}^2\rightarrow {{\mathbb {C}}}^2$$\end{document}. We show that two transfinite diameters of a compact subset of the graph (i.e., defined with respect to two different collections of monomials) are equal when the set has a certain symmetry. As a consequence, we give a new proof in C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {C}}}^2$$\end{document} of a pullback formula for transfinite diameter due to DeMarco and Rumely that involves a homogeneous resultant.


Introduction and summary
In this paper we study transfinite diameter on a graph, i.e., an affine algebraic variety where f : C 2 → C 2 is a polynomial mapping.The main goal of the paper is to give a direct proof of the pullback formula of DeMarco and Rumely for transfinite diameter in C 2 .Their main result in [8] is the following: Theorem.Let f : C N → C N be a regular polynomial map of degree d (where d ∈ N). * For any compact set K ⊂ C N , we have the formula where f denotes the degree d homogeneous part of f .
The original proof utilized deep results in complex dynamics and arithmetic intersection theory; specifically, a formula of Bassanelli and Berteloot relating the resultant of a regular homogeneous polynomial mapping on C N to the Green current of the induced mapping on P N −1 [1]; a formula of Rumely relating the transfinite diameter of K to its Robin function; and a pullback formula for a global sectional capacity developed by Rumely and his collaborators [9].Using these ingredients, the theorem was first proved in the special case where K is the Julia set of a polynomial mapping, then extended to general compact sets by some clever approximation arguments.
In Theorem 6.2 we prove the case N = 2 using only basic notions of pluripotential theory and Zaharjuta's techniques for estimating Vandermonde determinants, combined with computational algebraic geometry and linear algebra.The resultant term in the pullback formula appears naturally as a result of eliminating variables, and connects the DeMarco-Rumely formula (at least the C 2 version of it) more directly to the ideas of elimination theory.A general proof (in C N ) using the methods of this paper seems too unwieldly as the resultant becomes more and more complicated in higher dimensions.
We now give a summary of the paper.Let V be the graph of a polynomial mapping f as in (1.1).A polynomial in the coordinate ring C[V ] may be identified with a polynomial in C[z] by elimination.It may also be identified with a normal form in C[w, z] V , defined in Section 2.
In Section 3 we study directional Chebyshev constants on V defined by a limiting process using normal forms.The methods are similar to [7], but simpler.A technical algebraic condition (property (⋆)) is useful to establish convergence.
In Section 4 we study the symmetry of sets of the form where K ⊂ C 2 is compact.Our main theorem of this section (Theorem 4.6) uses this symmetry to show that two different types of directional Chebyshev constants are equivalent.
In Section 5 we study transfinite diameters d (1) (K), d (2) (K) and d (3) (K) of a set K ⊂ V using various collections of monomials.The main theorem (Theorem 5.4) gives integral formulas for each of d (j) (K) in terms of directional Chebyshev constants.Actually, a proof is only sketched for d (2) (K), since d (1) (K), d (3) (K) may be identified with the classical (Fekete-Leja) transfinite diameters of the projections to the z and w axes respectively, whose formulas were established by Zaharjuta [13].When L is as in (1.2), these formulas together with Theorem 4.6 yield d (2) (L) = d (3) (L) as a consequence.
In Section 6 we continue with V being the graph of a regular polynomial mapping and L as in Section 4. We use w = f (z) to substitute for the w in terms of z in a basis C of monomials on V , thereby relating d (1) (L) to d (2) (L).Powers of the resultant Res( f ) (where f is the leading homogeneous part of f ) arise in the process.When L is as in (1.2) this gives a relation between d (1) (L) and d (3) (L).Identifying these last two quantities with the Fekete-Leja transfinite diameters of the projections to z and w respectively, we recover the DeMarco-Rumely formula in C 2 .

Normal Form
We consider an affine variety given by a graph over C 2 of polynomials of degree be irreducible polynomials and let Also, define the polynomial map f = (f 1 , f 2 ).Then V is the graph of w = f (z).Let C[V ] denote the coordinate ring of V , i.e., the polynomials restricted to V .We now construct a basis B of monomials on V using a grevlex ordering.Order the monomials in Then order monomials with the same total degree lexicographically, according to The basis B is then constructed by going through the monomials listed in increasing grevlex order, and removing any monomial that is linearly dependent with respect to earlier monomials in the list.The normal form of a polynomial p ∈ C[V ] is its representation as a linear combination of basis elements.Lemma 2.2.Every monomial in C[w] is an element of B.Moreover, there is a finite collection of monomials I ⊂ C[z] (which includes 1) such that any monomial in B is of the form z β w α for some z β ∈ I. Hence a normal form is an element of Proof.In the linear case (d 1 = d 2 = 1) any monomial containing z 1 , z 2 is removed, and a normal form is simply an element of C[w].
When d 1 ≥ d 2 > 1, then within each fixed total degree, all monomials of the form w α are not removed since they are linearly independent by the previous lemma and listed ahead of monomials containing z for that same degree.The monomials that are removed are monomials containing z that are leading monomials of elements of the ideal have the same leading terms in our grevlex ordering.In other words, leading monomials of elements of f 1 , f 2 as an ideal in C[z] are also leading monomials of elements of I.The number of monomials that are not leading monomials of elements of f 1 , f 2 is finite because the number of points in the common zero set of f 1 , f 2 in C 2 is finite (see e.g.[6], 5 §3).† This finite set of monomials, by definition, is I.

Chebyshev constants
Let V be defined as in (2.1), (2.2).Let I denote the finite collection of multi-indices β such that w α z β ∈ C[w, z] V for some multi-index α ∈ N 2 0 .We will assume that the basis of monomials B := {w α z β : α ∈ N 2 0 , β ∈ I} is ordered so that the following property holds: (⋆) For each β ∈ I there exists an index β and a constant for some γ ∈ N 2 0 .Here l.o.t.refers to terms of lower order according to our ordering.In the generic case of Example 2.3, (⋆) holds, and we can take z β to be a sufficiently large power of z 2 .We illustrate the case d = 2; here 2 are not basis monomials, and we have I = {1, z 1 , z 2 , z 2 1 }.The first few monomials listed according to the ordering are . . .To verify (⋆), take, say, the monomial z 2 1 ; then multiplying by z 2 2 , we obtain Before stating the next result we first recall some notation.Let Σ ⊂ R 2 denote the closed line segment joining (1, 0) to (0, 1), and let Σ • = Σ \ {(1, 0), (0, 1)}.Let h(s) denote the number of monomials of degree s, i.e., the number of elements in {α ∈ N 2 0 : |α| = s}.As a consequence of the limiting properties of submultiplicative functions (see [3]), we have the following result.
Proposition 3.2.The limit exists for each θ ∈ Σ • , and θ → T B (K, θ) defines a logarithmically convex function on Σ • .Moreover, we have the convergence where θ(t) = (t, 1 − t).Definition 3.3.We call the function θ → T B (K, θ) the Chebyshev transform of K with respect to the basis B. A fixed θ is called a direction and T B (K, θ) is a directional Chebyshev constant.
The above terminology is based on [12] and [4].
We can also replace the basis B with the subset of monomials in C[w] ordered by grevlex.Let This is also submultiplicative and the analogue of Proposition 3.2 holds, replacing Y K by Z K and T B (K, θ) by T C[w] (K, θ), the Chebyshev transform of K with respect to the monomials in C[w], defined by the same kind of limit.Lemma 3.4.
(1) We have Z K (α) ≥ Y K (α) for each α ∈ N 2 0 , and hence (2) Let K w denote the orthogonal projection of K to the w-axis.Then Proof.Fix α ∈ N 2 0 and let p(w) = w α + l.o.t.be a polynomial in C[w] that satisfies p K = Z K (α).Since p ∈ C[w, z] V also, p K ≥ Y K (α) by definition.The inequality of Chebyshev transforms follows by taking |α|-th roots and a limit.This proves the first statement.
For the second statement, observe that the z-components of a point do not enter into computation when evaluating polynomials in C[w].
We return to B again.Given β ∈ I, let Using property (⋆), we show that Y K,β gives the same limit, independent of β.
Using (3.3), we also calculate that Taking the lim sup on both sides as s → ∞, and using Proposition 3.2, we obtain If z β K = 0, we may handle this by letting q n := (z β + 1)p n and replacing z β in the above arguments with z β + 1.
We now show that using a similar argument.Denote the lim inf by L.

Group invariance
Let V be defined as in (2.1), (2.2).The projection to w exhibits V as a finite branched cover over C 2 .Let D ⊆ C 2 be the complement of the branch locus, let Ω := {(w, z) ∈ V : w ∈ D}, and let In what follows, we will assume V to be irreducible, hence Ω, D and D z are connected ( [5], §5.3).Given paths α, β : [0, 1] → D with α(1) = β(0), denote by αβ the concatenated path, . We describe the monodromy action of the fundamental group π(D).Let (w, z) ∈ V and assume for the moment that w ∈ D. Consider a loop γ : Since the lifting is locally holomorphic, this formula is well-defined by the monodromy theorem in several complex variables (i.e., if η is another such loop at w with lift η, ).We get a group action on Ω. Write (w, z) := [γ] • (w, z).Projecting to z coordinates, put This map is locally holomorphic and bounded on D z for fixed γ.The set C 2 \ D z is locally a hypersurface.By a theorem of Tsuji on removable singularities of holomorphic functions in several variables (see [11], Theorem 3), z → [γ] • z extends holomorphically to all of C 2 .Hence the action of π(D) given by (4.2) extends to an action on C 2 .Adjoining w-coordinates, the action of π(D) on Ω given by (4.1) extends to an action on V .Before continuing we recall some general notions of invariance.
Definition 4.1.Suppose G is a finite group that acts on a complex manifold X, We will also call a set Now take the sup over all x ∈ E.
We now return to the π(D)-actions on C 2 and V .
Proof.By Lemma 4.3, the function (w, z) → p π(D) (w, z) on Ω is independent of z, so p π(D) (w, z) = ϕ(w) for some ϕ ∈ O(D).Since ϕ is locally bounded, it extends to an entire function on C 2 by Tsuji's removable singularity theorem [11].Since This follows from applying w α z β = o(|w| |α+β| ) (valid for any monomial considered locally as a function in w) to the monomials of p π(D) .By standard complex analysis arguments in C 2 , the coefficients of the power series in w for ϕ must be zero for all powers greater than deg(p).Hence it is a polynomial, i.e., an element of C[w] ⊆ C[w, z] V .By uniqueness of normal forms, this polynomial (of degree at most deg(p)) is precisely p π(D) .
We next look at invariant sets in V and C 2 under the action of π(D).Given a set K ⊂ V , denote by K w and K z the projections to w and z coordinates respectively.(In particular, Ω w = D.) Proposition 4.5.Let K ⊂ V .The following are equivalent.
(2) K z is invariant under the π(D)-action given by (4.2). (3) Proof.We first prove the proposition when K ⊆ Ω. (Hence K w ⊆ D and elements of π(D) may be obtained from loops based at points of K w .) (1) ⇔ (2): By adjoining and dropping w-coordinates, we get the equivalence of these two statements.(2) ⇒ (3): We have K z ⊆ f −1 (K w ) by definition, so we need to show that invariance implies , where we can take η to be a loop based at f (b).
This says that K z is invariant.Now suppose each point of K is a limit point of the set K ∩ Ω.By continuity of the extended action, statement 1 holds for K ∩ Ω if and only if it holds for K; similarly, statement 2 holds for K z ∩ Ω z if and only if it holds for K z .Also, K z ∩ Ω z = f −1 (K w ∩ D) by continuity of f .Hence the proposition holds for K because it holds for K ∩ Ω by the first part of the proof.
For general K, write K = ∞ j=1 K j where the sets are decreasing in j: Each K j satisfies the conditions of the previous paragraph, so the proposition holds for these sets.Moreover, using continuity of the π(D)-action and continuity of f , it can be shown by standard analysis that • K z is invariant if and only if (K j ) z is invariant for sufficiently large j; and for sufficiently large j.Hence the proposition holds for K by taking a limit as j → ∞.
Our main theorem of this section is the following.Theorem 4.6.Let K ⊂ V be a compact π(D)-invariant set.Then Let q(w) := p(w) − w α denote the polynomial given by the lower order terms.By Proposition 4.4, p π(D) , q π(D) ∈ C[w] and

By the definition of Z
which was to be shown.
We will return to π(D)-invariant sets at the end of the next section (see equation (5.15)) and throughout Section 6.

Transfinite diameter
We recall the construction of transfinite diameter.Let B denote a countable collection of monomials that span a subspace A ⊂ C[w, z].(Later, A will be one of We will assume that B = {e j } j∈N lists the elements of B with respect to some ordering ≺, i.e., e j ≺ e k whenever j ≤ k.
Let X ⊂ C 2 × C 2 be an algebraic variety.We will suppose that there is a filtration • is an increasing sequence of finite-dimensional linear subspaces with the property that A k A l ⊆ A k+l .This means that given p 1 ∈ A k and p 2 ∈ A l , there exists p 12 ∈ A k+l such that Denote the filtration (equivalently, the sequence {A k } ∞ k=0 ) by A. We require A to be compatible with the ordering in the sense that if p ∈ A k and q ∈ A \ A k then p ≺ q.For each k, let Given a finite set of points {ζ j } n j=1 ⊂ X, define (5.1) and given a compact set K ⊂ X and n ∈ N, define Let m k denote the number of elements in B k (equivalently, the dimension of A k ), and let K) is the Fekete-Leja transfinite diameter in C 2 , which we will denote simply by d(K).
Example 5.2.We will consider the following cases of transfinite diameter on Hence the restriction of the map (5.3) to C[w, z] V is an isomorphism, which we use to define the the elements of A = {A k }:  The relation among all these transfinite diameters will be important later.
Example 5.3.We give a concrete computation of A in case 2 of the above example.Consider w = f (z) given by This can be verified by a straightforward induction.
Theorem 5.4.Let K ⊂ V be compact, and let θ(t) = (t, 1 − t).We have where B in (5.5) denotes the monomial basis of C[w, z] V .Sketch of Proof.The derivation of these formulas uses the standard method of bounding ratios of Vandermonde determinants above and below.Formulas (5.4) and (5.6) are the classical formula of Zaharjuta for the sets K z and K w in C 2 .We sketch the proof of (5.5), giving the main ideas but omitting a number of technical details.
First, we have the inequality and we have the lower bound in (5.7).
The upper bound is obtained by expanding the determinant in a different way; we omit the details.Now (assuming all quantities are nonzero, see also the next remark) we use (5.7) to estimate the telescoping product Van B,mn (K) Van B,mn−1 (K) and obtain (5.9) Regrouping and relabelling terms, we rewrite the products on each side of (5.9) as (5.10) where Then s β,n → ∞ as n → ∞, and by Proposition 3.5, Fix ǫ > 0; then (5.11) implies (5.12) for sufficiently large n, where s n = β∈I s β,n .
Consider the upper bound in (5.9).Using (5.12) and (5.10), (5.13) Van B,mn for sufficiently large n, say n ≥ N. Now apply (5.13) to the telescoping product to obtain the estimate (5.14) where We take l n -th roots on both sides of (5.14) and apply the following limits (calculations omitted) as n → ∞: This yields lim sup and since ǫ > 0 was arbitrary, lim sup n→∞ (Van B,mn (K)) 1/ln ≤ exp By a similar argument as above, this time using the lower bound in (5.9), These last two inequalities yield (5.5).Then L w = K and L z = f −1 (K).
Corollary 5.6.Let K ⊂ C 2 be compact, and let L ⊂ V be defined as in (5.15).Then Proof.We have K = L w and L z = f −1 (L w ).By Proposition 4.5, L is invariant under the group action of π(D) described in Section 4. Hence by Theorem 4.6, ) for all t ∈ (0, 1), where θ(t) = (t, 1 − t).Finally, by formulas (5.5) and (5.6) of the previous theorem, d (2) The relation with d (1) (L) will be studied in the next section.

Pullback formula
Let d ∈ N and (w 1 , w 2 ) = f (z 1 , z 2 ) where f = (f 1 , f 2 ) is a regular polynomial mapping of degree d.This means that f −1 (0, 0) = {(0, 0)}, where f = ( f 1 , f 2 ) denotes the leading homogeneous part of f .Its components are of the form (6.1) Remark 6.1.Note that f 1 , f 2 are products of linear factors, whose zero sets are lines through the origin.When f is not regular, f −1 (0, 0) must contain at least one complex line through the origin which is the zero set of a common factor.
We will prove the following special case (in dimension 2) of the main result of [8].
Theorem 6.2.Let d ∈ N and let f : C 2 → C 2 be a regular polynomial map of degree d.For any compact set K ⊂ C 2 , we have the formula On both sides, d denotes the Fekete-Leja transfinite diameter.The term Res( f ) on the right-hand side is the resultant of f , which is the 2d × 2d determinant , where a j , b j are defined as in (6.1) and the triangular regions above the a 0 , b 0 diagonals and below the a d , b d diagonals are filled with zeros.We relate the transfinite diameters d(f −1 (K)), d(K) to transfinite diameters on the graph V = {(w, z) : Also, write V = {(w, z) : w = f (z)}.Let K ⊂ C 2 be compact, and let L ⊂ V be defined as in (5.15).Then where we use Corollary 5.6.Hence we need to relate d (1) (L) to d (2) (L).We consider a filtration of the monomials in C[w, z] V .Similar to (5.3), let We also let F m,n := F n \ F m when m < n; similarly, define B m,n where B is the monomial basis of C[w, z] V .Hence The following lemma is a straightforward consequence of the definitions.
Remark 6.5.We will assume in what follows that f is as in Example 2.3, where the powers of z in B are given by monomials not in z 1 z d−1 2 , z d 2 , i.e., the powers of z 2 are as small as possible.Pre-and post-composing with generic rotations R 1 , R 2 will transform f into a mapping g := R 2 • f • R 1 as in that example.This changes the set f −1 (K) in Theorem 6.2 to g −1 (K), and the resultant to Res ) by invariance of the Fekete-Leja transfinite diameter under a rotation, and by standard properties of resultants and the fact that det(R 1 ) = det(R 2 ) = 1.Hence the formula for f follows from the formula for g, so we can reduce to this case.
Then we inductively define Following the pattern in the above lemma, define C k := G dk for each k ∈ N and C := k C k .Observe that (parts of) the bases can be constructed inductively.For example, when n is sufficiently large, where Gn−1,n = G n−1,n \ {z 1 z n−j 2 : j = 0, . . ., r − 1}.(Note that in both cases there is lots of overlap in the two pieces of the union.)A specific illustration is given a bit later in Example 6.7.
The elements of G n−1,n , constructed at each stage, form the correct number for a basis.The issue is whether they are linearly independent.We have the following lemma which relies on the fact that f is regular.Lemma 6.6.The collection of monomials G 2d−1 is linearly independent.
Proof.When n < 2d − 1, G n = F n , i.e., the monomials are normal forms in C[w, z] V which are automatically linearly independent.Hence to study the linear independence of G 2d−1 , we only need to study the linear independence of We need to verify that any linear combination of monomials does not reduce to a linear combination of monomials in F 2d−2 .Substituting for w 1 , w 2 in the above elements, we obtain the collection of polynomials We need to show that no linear combination of these polynomials has degree strictly less than 2d − 1.This is equivalent to showing that if for all z, then c j = 0 for all j = 1, . . ., 2d.If not, then the left-hand side of (6.3) is a nonzero homogeneous polynomial, given by a product of linear factors at least one of which is nonzero.Clearly we cannot factor out either z 1 or z 2 from this sum, so the nonzero linear factor must be a common factor of f 1 and f 2 .This contradicts the fact that f is regular.
For larger values of n, we can use the lemma to verify that the monomials in G n are linearly independent by an inductive argument based on the step by step construction described above.We omit the details.Example 6.7.We illustrate these bases using a simple example.Let d = 2 and suppose I = {1, z 1 , z 2 , z 2 1 }.(This is the generic case, recall Example 2.3.)Then 2 }.The last four monomials (which are in F 3,4 and G 3,4 respectively) are slightly different.At the next level, 6  2 } where G3,4 = G 3,4 \ {z 4  2 }.More generally, one can show inductively that , one can use the inductive formula to match up monomials w α z β ∈ G n,n+1 with monomials w α ′ z β ′ ∈ F n,n+1 so that the differences |α − α ′ | and |β − β ′ | are uniformly bounded above by the maximum difference in powers between pairs of monomials in F 3,4 ∪G 3,4 .Example 6.8.For a slightly more general example, let d ≥ 2 and consider n = 2d (i.e., k = 1 and r = 1 in (6.2)).We have In both cases there are 2d + 1 monomials.The restriction of The elements of F 2d−1,2d may be obtained from the elements of the basis {z j 1 z 2d−j 2 } 2d j=0 of C[z] =2d by computing normal forms in C[w, z] V and reading off the monomials.Similar to the previous example, we have for k > 1 that where G2d−1,2d = G 2d−1,2d \{z 2d 2 }.Thus we can create a one-to-one mapping between all but one pair of monomials in F (k+1)d−1,(k+1)d and G (k+1)d−1,(k+1)d so that the difference in powers is bounded above by the maximum difference in powers between pairs of monomials in F 2d−1,2d and G 2d−1,2d .
We have the following calculations (proof omitted).
Lemma 6.9.Let m n be the number of elements in C n and let Define Van C,n (ζ 1 , . . ., ζ n ), Van C,n (K) similar to (5.1), (5.2), using the basis C. We need a lemma whose detailed proof is quite technical and tedious.We sketch a proof and write out the details for a particular example.The basic idea behind the proof is that the monomials in B and C are the same except for a few factors that become negligible as n → ∞.
In the next lemma, recall the set L defined in (5.15).For each j = 2d − 2, . . ., n, all but at most d − 1 elements of G j−1,j have total degree in z uniformly bounded above by 2d.(For G n−1,n these are the last r elements.)All elements of F j−1,j have total degree in z uniformly bounded above by d − 1.
Consider replacing rows of the Vandermonde matrix for Van C (ζ 1 , . . ., ζ mn ) with rows of the matrix for Van this changes the determinant by a factor of mn j=1 w α−α z β−β (ζ j ).We may compare, one by one, rows of Van C corresponding to monomials w α z β ∈ G j−1,j with rows of Van B corresponding to monomials w αz β ∈ F j−1,j for which the differences α − α and β − β are uniformly bounded, as in Examples 6.7 and 6.8 .We do this for all j = 2d − 2, . . ., n.If L avoids the coordinate axes in z and w one can find constants c, C such that . ., ζ mn )| using powers of c and C by applying (6.4) to each row that we compare.This estimate works for all rows except those which contain no powers of w; altogether, fewer than m n = o(l n ) rows.
The remaining rows of G n contain no powers of w; for each j = 2d − 2, . . ., n this is at most d − 1 rows of G j−1,j ; so in total, O((d − 1)n) rows.For each of these rows we can use an estimate of the form c O(n) , C O(n) (perhaps replacing c, C as appropriate).The sum of all the exponents is O((d − 1)n 2 ) = o(l n ) by Lemma 6.9.
Putting the estimates in the previous two paragraphs together, we obtain where c > 0 is smaller than the minimum and C is larger than the maximum of  As shown in Example 6.7, F 2n,2n+1 = G 2n,2n+1 and no estimates are needed.Putting everything together, let k be a large integer; then k ∈ {2n, 2n + 1} for some n ∈ N and So the l k -th roots of these quantities go to 1 as k → ∞.
We will need the following elementary lemma which follows immediately from properties of determinants under row operations.Lemma 6.12.Let t ∈ N. Consider polynomials q j (z) = f j (z) + r j (z) for each j = 1 . . ., t.Let s > t and consider square matrices given by evaluating polynomials at a set of points {ζ 1 , . . ., ζ s }, where R(z) is a vector of s − t monomials.If every monomial of r j (z) is an entry of R(z) (for all j) then det F = det G.
In what follows let A := {z α } denote the monomial basis of C[z] under a graded ordering, so that in the proposition below, the rows of the Vandermonde matrix for Van A (ζ 1 , . . ., ζ mn ) are given by monomials in C[z] ≤dn , for each n ∈ N. (For convenience, local computations will use grevlex with z 1 ≺ z 2 , but reordering monomials will not affect the result.)Proof.We study the transformation of Vandermonde matrix columns.Consider the j-th column of the matrix for Van C (ζ 1 , . . ., ζ mn ); its entries are of the form w α z β (ζ j ).We transform these entries to linear combinations of expressions of the form z γ (ζ j ), where |γ| ≤ dn, in a step by step process.In the calculations that follow we assume that (6.5) For convenience in what follows, we suppress the dependence on ζ j .By using (6.5) and (6.1) we may transform a pair of factors w 1 , w 2 in the monomials contained in w 0 , w 1 , respectively, as follows: Adjoining w 2 , . . ., w ℓ to this calculation, we obtain Observe that R is the matrix of Res( f ) and det(R 1 ) = det(R) = Res( f ).We may transform, by the same method, another pair of factors w 1 , w 2 in a pair of respective blocks to monomials in z 1 , z 2 of degree d; for example, transform w 1 in the block w 2 and w 2 in the block w 3 .A similar calculation as above yields  where det(R 2 ) = Res( f ) also.
Suppose in what follows that ℓ is odd.Then there are an even number of blocks.By pairing up blocks we can eliminate one factor each of w 1 and w 2 in (ℓ + 1)/2 steps; overall, the total degree in w of each monomial is decreased by 1.We continue transforming powers of w to powers of z in (disjoint) block pairs.The transformation matrices R 2 , R 3 , . . ., etc. satisfy | det(R j )| = |Res( f )|.All powers of w are eliminated after doing these (ℓ + 1)/2 steps ℓ times (to reduce the total degree in w from ℓ to 0).The final expression on the right-hand side is . When ℓ is even, we first modify the basis monomials so that monomials containing w have degree ℓ − 1. § We account for this modification with estimates as in (6.4).We then do a similar calculation as in the odd case.Altogether, we obtain a change of variable matrix R k with the property that (6.7) .
More details are given in an example which follows the proof.The above procedure is carried out on monomials in G k,k−1 for all k ∈ {d, . . ., dn} to eliminate powers of w.We obtain for the full Vandermonde matrix (6.8) where the entries of Z are z α (ζ j ) for |α| ≤ dn.For convenience, the monomials that give the components of W have been reordered so that all monomials containing only powers of z have been transferred to the top region (denoted by * ).¶ We now compute | det R| = nd j=d | det R j |.For each ℓ = 1, . . ., n − 1 we have 2 +o(ℓ 2 ) .The computation of each of the above R j s involves the same calculation, using the same value of ℓ, with r = 0, . . ., d − 1.This accounts for all but a few factors | det R j | whose number stays bounded as n → ∞.Hence (6.9) Finally, apply (6.8) to all columns of the Vandermonde matrix.We obtain (using the points ζ j to distinguish the columns).Now take the absolute value of the determinant on both sides and apply (6.9) and (6.10).The result follows as long as (6.5) holds.
In general, f may consist of f plus lower order terms.By calculations similar to the above, one can show that Given a row of the matrix on left-hand side whose leading term corresponds to some monomial, the monomials in l.o.t.correspond to rows further up the matrix.We can reduce the calculations in this case to those of the previous one by applying Lemma 6.12 and an inductive argument.Example 6.14.We illustrate how powers of the resultant appear as a result of the change of variable calculation when d = 2.For simplicity, we consider when f is homogeneous, i.e., We compute the details of a case in which ℓ is even, which needs a first step to modify it into a similar form as the odd case.Suppose k = 9, so that ℓ = 4 and r = 0 in (6.6).The 10 rows of the Vandermonde determinant for k = 9 correspond to G 8,9 = {w 4  1 z 1 , w 4 1 z 2 , w 3 1 w 2 z 1 , w We now carry out the change of variables from w to z for the first 8 monomials of H, similar to the proof.The first two steps yield Proof of Theorem 6.2.Suppose for each n ∈ N we take Fekete points ζ 1 , . . ., ζ mn in L (i.e., points that maximize the determinant).Then by the above Proposition, (Note that in the third line, a root of (nd) 3 + o(n) is the appropriate one for d (1) (L), where we take the sum of the degrees of all monomials in C[z] ≤nd .)The theorem follows from these limits by letting n → ∞ on both sides of (6.11).
Since the pullback formula depends only on the resultant of the leading homogeneous part of the polynomial mapping, we have the following immediate consequence, which is a special case of Theorem 5 in [2].Corollary 6.15.Let f : C 2 → C 2 be a regular polynomial mapping with leading homogeneous part f .Then d(f −1 (K)) = d( f −1 (K)).

Final Remarks
It seems straightforward to adapt the method of this paper to relate transfinite diameters on K and f −1 (K) in other situations, e.g., • when using more general notions of transfinite diameter, such as the so-called C-transfinite diameter defined using a convex body [10], or weighted transfinite diameter [4]; • when f = (f 1 , f 2 ) is not necessarily regular, or f 1 , f 2 are of different degrees.The idea is to lift K to the graph V and study projections to w and z.The z variables may be eliminated using symmetry and the w variables may be eliminated using substitution; patterns in the coefficients due to substitution then give rise to normalization factors.It would be interesting to see what other types of resultants arise in this way.

Lemma 2 . 1 .
The monomials w α = w α 1 1 w α 2 2 are linearly independent in C[V ].Proof.Consider a linear combination R(w) := α c α w α = 0. Let U ⊂ V be a connected open set on which the projection V ∋ (w, z) → w ∈ C 2 is holomorphic.Using this, identify R(w) with a holomorphic function on U w ⊂ C 2 which is the image of U under the projection.Since R(w) is identically zero on U w and U w is open, we must have c α = 0 for all α.

Lemma 4 . 3 .
Let f : V → C be a function that is locally the restriction of a holomorphic function in a neighborhood (in C 2 × C 2 ) of every point of V .Fix a ∈ D and let b (1) , . . ., b (d) be the corresponding points in D z , i.e., V ∩ {(w, z) : w = a} = {(a, b (1) ), . . ., (a, b (d) )}.Then for any j, k ∈ {1, . . ., d}, ) Let A = {C[z] ≤k } and B = {z α }.We obtain the Fekete-Leja transfinite diameter of the projection to z, d A,B (K) = d(K z ).Elements of C[z] may be identified with elements of C[V ] by elimination: the map f * defined by be the monomial basis of C[w, z] V ; we order it as in Example 3.1.(It is straightforward to show that the ordering is compatible with A.) The transfinite diameter d A,B (K) obtained here is different to the one above.

( 3 )
Let K ⊂ V , let A = {C[w] ≤k }, and let B = {w α }.Note that C[w] ≤k = A k ∩ C[w] with A k as above.Similar to case 1, d A,B (K) = d(K w ).