Rational Maps of Balls and their Associated Groups

Given a proper, rational map of balls, D'Angelo and Xiao introduced five natural groups encoding properties of the map. We study these groups using a recently discovered normal form for rational maps of balls. Using this normal form, we also provide several new groups associated to the map.


Introduction
Suppose f : B n → B N is a rational proper map, f = p g where p : C n → C N and g : C n → C are polynomials, p g is in lowest terms, and g(0) = 1.Further, suppose f is of degree d.We say that f is in normal form if f (0) = p(0) = 0, where g k are homogeneous polynomials of degree k.That is, f is in normal form if it fixes the origin, the denominator has no linear terms, and the quadratic terms are diagonalized.In [16] the second author proved that every rational proper map of balls can be put into the normal form above via spherical equivalence and that this is a normal form up to unitary automorphisms where the source automorphism fixes g 2 .
A natural problem is to study holomorphic maps invariant under subgroups of the automorphism group.This problem has a long history, going back to at least Cartan in [3].We refer the reader to [2, 4-6, 11-14, 18] and especially D'Angelo's books [8][9][10].Rudin [18] and Forstnerič [5] studied invariant proper maps to general domains in C n .In this context, one can construct proper maps invariant under any finite subgroup of the unitary group U(n).On the other hand, if the target domain is required to be a unit ball, then the problem has more structure.Forstnerič [7] showed that if the map is sufficiently smooth up to the boundary, then the map must be rational, and furthermore, in [6], he showed that for most groups, it is not possible to find a proper, rational map of balls invariant under that group.In [11], D'Angelo and Lichtblau gave the decisive result, completely characterizing which groups admit invariant, proper, rational maps between balls.
Moreover, given a proper map of balls, one can use various groups to detect various properties of the map.In particular, D'Angelo and Xiao [12,13] introduced the groups A f , G f , Γ f , T f , and H f , defined below.Using the normal form, we add several new groups, Given a polynomial ρ(z, z), let ρ (a,b) denote the monomials of ρ that are of degree a in the holomorphic z variables, and degree b in the antiholomorphic z variables.We refer ρ (a,b) as the bidegree-(a, b) part of ρ, and we write * instead of a or b to denote all degrees together.
. All the groups are closed, and therefore Lie subgroups.D'Angelo-Xiao [13] showed this for the groups A f , Γ f , G f , T f , H f , and it is immediate for the new groups we defined.As long as they are compact, it follows from standard theory that they can be conjugated to a subgroup of the unitary group.In fact, Γ f is noncompact if and only if f is linear fractional (an automorphism if n = N), see [13].Moreover, Lichtblau [17] (see also D'Angelo-Lichtblau [11]) has shown that G f must be finite, fixed-point-free, and cyclic.In the present paper, we prove is that once the map is in normal form, the relevant groups are all subgroups of the unitary group.Note that if f is in normal form and it is linear fractional, then it is in fact linear.
In particular, for a mapping that is not an automorphism, once the mapping is in normal form, all the groups are subgroups of the unitary group.One motivation for introducing the groups Σ f , D f , and ∆ (a,b) f is that they are often easier to compute.If the σ invariants are nonzero and distinct, namely, 0 < σ 1 < . . .< σ n , then Σ f ∼ = (Z 2 ) n .Therefore, a corollary is that G f must be cyclic and fixed-point-free for such f .In general, standard linear algebra says that Σ f is a direct sum of groups where if we have k zero σ's, the first factor is U(k), and for each set of k nonzero equal σs we get a factor of O(k) (real orthogonal group).In particular, when all σs are nonzero and distinct we get a direct sum of O(1) = {1, −1}.
Corollary 1.4.Suppose f = p g : B n → B N is a rational proper map in normal form such that 0 < σ 1 < . . .< σ n .Then G f is either trivial, or G f = {I, −I}.In particular, if G f = {I, −I}, then p(z) = p(−z) and g(z) = g(−z), and the degree of f is at least 4.
Furthermore, for any sufficiently small 0 < σ 1 < . . .< σ n , there exists a degree 4 map n and such that G f = {I, −I}.In particular, the corollary says that when σ 1 , . . ., σ n are nonzero and distinct, then for G f to be nontrivial, the map f must have degree at least 4. Hence, when the degree of f is 3, we have that G f is trivial, but we then note that D f = Σ f ∼ = (Z 2 ) n .It is not difficult to find a map of degree 4 or higher with a denominator of the form for small enough σ's and ǫ's (see Proposition 5.1 for example), in which case D f = {I}.
A natural problem is to determine what properties of the map f can be detected using the above groups.In particular, we focus on Γ f .D'Angelo and Xiao [12] proved that a rational proper map of balls whose Γ f group is not compact is spherically equivalent to the linear embedding and therefore is linear fractional, in which case Γ f = Aut(B n ).Gevorgyan, Wang, and Zimmer [19] extended this result to maps that are only C 2 up to the boundary.D'Angelo and Xiao [12] further proved that Γ f contains the torus if and only if f is spherically equivalent to a monomial map (a map where each component is a single monomial).They also prove that if the group Γ f contains the center of U(n) (the circle group {e iθ I}), then the map f is spherically equivalent to a polynomial map.They also show that for any finite subgroup Γ, there exists a rational f such that Γ f = Γ.We give a slightly stronger version of this result in the next theorem.We say a subgroup Γ ≤ U(n) is defined by finitely many invariant polynomials if there exists polynomials ρ 1 , . . ., ρ ℓ such that Theorem 1.5.Suppose that f : B n → B N is a rational proper map in normal form and f is not linear.Then Γ f ≤ U(n) is a subgroup that is defined by a single invariant polynomial.
Conversely, given any group Γ ≤ U(n) that is defined by finitely many invariant polynomials, there exists a rational, proper map f : B n → B N such that Γ f = Γ.Moreover, this map f can be chosen to be a polynomial that takes the origin to origin, and hence in normal form.
We conclude the introduction by outlining the results of the paper.In Section 2, we briefly recall the usual background on Hermitian forms.In Section 3, we give a sequence of lemmas to prove Theorem 1.3.In Section 4, we prove Theorem 1.5 and characterize the possible Γ f groups.In Section 5, we consider the problem of constructing maps with a given denominator.Finally, in Section 6, we prove Corollary, 1.4 and give an example.

Hermitian forms
In this section, we briefly recall the standard setup to treat real-valued polynomials as Hermitian forms (see Chapter 1 of [10] for more details on this approach).A real-valued polynomial in C n can be written as a polynomial where α and β are multiindices.The coefficients c αβ can be put into a matrix [c αβ ] α,β by putting an order on the monomials (and hence the multiindices) and having α refer to rows and β to columns.The matrix [c αβ ] α,β is called the matrix of coefficients of r.The polynomial r is real-valued if and only if the matrix [c αβ ] α,β is Hermitan.Diagonalizing the matrix of coefficients then yields where • denotes the standard Hermitian norm, and P : C n → C a and G : C n → C b are polynomials whose components are linearly independent.As the rank is a+ b, this expansion cannot be done with fewer than a + b polynomials.In particular, if the matrix of coefficients of r is positive semidefinite, then r(z, z) = P (z) 2 , and conversely, if r is a Hermitian sum of squares, its matrix is positive semidefinite.The polynomial P in r(z, z) = P (z) 2 only needs to use those monomials that correspond to nonzero rows or columns of the matrix.Hence, if the entries in the matrix corresponding to purely holomorphic or purely antiholomorphic terms are all zero, then this corresponds to the row and column corresponding to the monomial 1.In other words, in this case, P can be chosen to have no constant term, that is, P (0) = 0.

Groups are subgroups of the unitaries
Given a proper rational map p g : Since rescaling r by a positive constant does not change the corresponding map, we will generally normalize r so that r(0, 0) = 1.The following observation was made in [15], and also used later by D'Angelo-Xiao [12].That is, two maps differ by a target automorphism if and only if the underlying forms are the same (up to rescaling).( The lemma says that the group Γ f is precisely the group that leaves the underlying form unchanged up to rescaling.We prove Theorem 1.3 in stages using the following lemmas. We briefly recall some properties of automorphisms and the normal form from [16].Let α ∈ B n and t = 1 − α 2 .Then every automorphism of B n is of the form Uϕ α where U ∈ U(n) and See Chapter 1 of [10] for background on automorphisms of B n .Now, we define the Λ f : In [16], the second author proves the following properties of the Λ f function.
Lemma 3.2 (From [16]).If f = p g : B n → B N is a rational proper map of degree d > 1, in lowest terms, then the following hold. ( (3) The Λ-function has a unique critical point in B n .
(4) There exists τ ∈ Aut(B N ) such that τ • f • ϕ α takes the origin to the origin and has no linear terms in its denominator when written in lowest terms if and only if α ∈ B n is the critical point of the corresponding Λ-function.
Lemma 3.3.Suppose f : B n → B N is a rational proper map in normal form and f is not linear.Then Proof.Suppose f = p g is a rational proper map in normal form that is not linear and let G(z) and let r(z, z) = |G(z)| 2 − P (z) 2 .Assume that r(0, 0) = 1.By Lemma 3.1, we have that r(z, z) = r(z, z).Since f of degree strictly greater than 1, Lemma 3.2 shows Λ f has a unique critical point.Since r = r, we find that Λ f = Λ F .As f is in normal form, the critical point of Λ f is at the origin.Again from the previous lemma, Λ for some constant C.But this means that ϕ fixes this critical point, that is, ϕ(0) = 0.An automorphism of the unit ball fixing the origin must be a unitary map (Corollary 1.6 of [10]), and hence ϕ ∈ U(n).Since f is in normal form, fixes the origin, and τ • f • ϕ −1 = f , we find that τ also fixes the origin and that τ ∈ U(N).
That A f is closed already follows from D'Angelo and Xiao [12,13], and it is also an immediate consequence of In particular, the lemma above gives Γ f ≤ U(n), but we can read even more from the proof.We thus have the following characterization of Γ f , where no rescaling is necessary.Lemma 3.4.Suppose f = p g : B n → B N is a proper rational map in normal form that is not linear.Then U ∈ Γ f ≤ U(n) if and only if In other words, ∆ Plugging into the underlying form and clearing denominators obtains Proof.First, put f = p g into normal form where g(0 Now plug in z = 0 to find Hence U ∈ D f .Next we note that which implies that if U ∈ ∆ ( * ,0) f , then U ∈ D f .The statement for Σ f follows analogously by considering only the quadratic terms.Lemma 3.6.Suppose f : B n → B N is a rational proper map in normal form and f is not linear.Then (i) Proof.The containment in the unitary follows since are closed follows as they are given by an invariant polynomial.The groups Γ f , G f , T f and H f are either equivalent to subgroups of A f or the projections onto the first or the second factor.In either case, it follows that they are all subgroups of the correct unitary groups.
The inclusions , and Theorem 1.3 follows from the lemmas above.

Characterization of Γ f
We prove Theorem 1.5 in the following two lemmas.First, we observe that Lemma 3.4 quickly yields the first part of Theorem 1.5.Lemma 4.1.Suppose that f : B n → B N is a rational proper map in normal form and f is not linear.Then Γ f ≤ U(n) is a subgroup that is defined by a single invariant polynomial.
Proof.Let f = p g and apply Lemma 3.4.In particular, if ρ(z, z) = |g(z)| 2 − p(z) 2 , then the lemma implies that We now prove the second part of Theorem 1.5.
Lemma 4.2.Given any group Γ ≤ U(n) that is defined by finitely many invariant polynomials, there exists a polynomial proper map f : We note that the N required depends on the inviariant polynomials given.
Proof.Suppose that Γ ≤ U(n) is the group given by the invariant polynomials ρ 1 , . . ., ρ ℓ : We can ensure that ρ j (0) = 0 for each j.We will construct a single polynomial that defines Γ. Suppose that ρ j is of bidegree (d j , d j ).Write Note that Uz 2 = z 2 for any unitary U, and that each polynomial z 2k j ρ j (z, z) only has monomials of degree 2k j−1 + 1 through 2 k j .In particular, no two z 2k j ρ j (z, z) have monomials of the same degree.Therefore, ρ(Uz, Uz) = ρ(z, z) if and only if ρ j (Uz, Uz) = ρ j (z, z) for each j.The first factor z 2 ensures that ρ has no holomorphic or antiholomorphic terms.Suppose that ρ is of bidegree (d, d).Consider We have that R(z, z) = 1 when z 2 = 1 and moreover its matrix of coefficients is positive definite as it is a sum of squares of holomorphic polynomials.These polynomials thus give a polynomial proper map of balls.For a small ǫ > 0 write Note that the matrix of coefficients of R ǫ has zeros at all the entries for holomorphic or antiholomorphic terms.Moreover, except for the constant, the diagonal terms of the matrix for R are positive and at least 1 d .Thus, for a small enough ǫ, the matrix for R ǫ is still positive definite and hence a sum of squares of holomorphic polynomials Since the matrix of coefficients is zero at the entries for holomorphic and antiholomorphic terms, the polynomials f j can be picked so that f j (0) = 0 for all j.As R ǫ = 1 when z = 1, we have a proper map of balls f = (f 1 , . . ., f N ).We have that U ∈ Γ f if and only if That is, U ∈ Γ f if and only if ρ(Uz, Uz) = ρ(z, z) and that defines Γ.The map f is the desired map.

Constructions
The following basic result is useful for constructing maps with a given denominator.The result we prove below is a straightforward construction that gives a numerator of a specific degree, for denominators close to 1.There is a far deeper result, see D'Angelo [10] or Catlin-D'Angelo [4], that any polynomial that does not vanish on the closed ball is a denominator of a proper map of balls.In that case however, the degree cannot be bounded.Proposition 5.1.Given any polynomial G : C n → C of degree d − 1 such that G(0) = 0, there exists an ǫ > 0, N ∈ N, and a polynomial P : C n → C N of degree d such that P (0) = 0 and P 1+ǫG is a proper map of B n to B N .The N can be taken to be one less than the number of different monomials of degree d or less in n variables.
A proof for d = 3 was given in [16] and must be somewhat modified for higher degree.
Remark 5.2.It may be convenient in some constructions to consider 1 + G(ǫz) instead of 1 + ǫG(z), and the proof follows in exactly the same way.That is, given an affine variety, there is a proper map with this variety as the pole set provided we can we can dilate it sufficiently far away from the origin.
Proof.One starts with the polynomial If we ignore the row and column corresponding to purely holomorphic and purely antiholomorphic terms (which is a single row and column), its matrix of coefficients is positive definite.Set N to be the rank of this matrix, which is one less than the number of all holomorphic monomials in n variables of degree d or less.Moreover R = 1 when z = 1.Now consider The rank of this matrix is N + 1, as we will have a 1 on the coefficient of the matrix for the constant term, the off diagonal elements are all small if ǫ > 0 is small, and all other diagonal elements are negative and of size roughly 1 d or larger (assuming ǫ is small).Thus, for a small ǫ, the matrix of coefficient will have 1 positive and N negative eigenvalues.The matrix of coefficients for r(z, z) has zeros at all the terms corresponding to the pure holomorphic and pure antiholomorphic terms, and hence is of rank N and therefore negative semidefinite.In particular, there exists a polynomial map P : The degree of P is at most d as that is the maximal degree of all monomials, and as there were no holomorphic or antiholomorphic terms in the matrix from which we constructed P , we can chose P to not include the constant monomial, and hence P (0) = 0.In fact, since 1 + ǫG(z) 2 has no terms of bidegree (d, d) and r does, we must conclude that P is in fact of degree d and not less.Since we also get that r(z, z) = 0 when z = 1, we find that P 1+ǫG is a proper map.
It is left to show that the map is in lowest terms.If not, that is, if there was a common multiple h of the components of P and 1+ǫG, then we would have r(z, z) = |h(z)| 2 A(z, z) for some real polynomial A. Consider the top degree part of r, that is, we look at the bidegree (d, d) part of r which is simply The function h, were it nonconstant, would be zero at some points arbitrarily far away from the origin, while r (d,d) and hence r must become arbitrarily negative as z becomes large.So h is a constant and P 1+ǫG is in lowest terms.Using the previous proposition, we can construct a map of degree 4 with denominator which then necessarily has the trivial group D f = {I}, and hence Γ f = {I}.
If the denominator is of the form 1 + σ If we want G f to also be {I, −I}, then we would need the numerator to be invariant, but that requires degree 4, see Lemma 6.1.For small σ we can always construct such a degree 4 map.Proposition 5.3.Given n ∈ N, there exists an N and an ǫ > 0 such that whenever 0 < σ 1 < • • • < σ n < ǫ then there exists a rational proper map of balls of degree 4 in normal form f = p g : B n → B N (in lowest terms) such that and such that p(z) = p(−z).In other words, G f = {I, −I}.
We note that the construction in Example 6.2 can be adapted to arbitrary n and ensure the existence of such an example whenever σ √ n would suffice.The advantage of the approach given here is that it easily generalizes to more complicated denominators.
Proof.We adapt the proof of Proposition 5.1 from above, however we need to work with forms where only holomorphic and antiholomorphic monomials of even degree arise.Let N be the number of monomials in n variables of degree 2 and 4. Thus start with Again R = 1 if z = 1.Write G(z) = n j=1 σ j z 2 j and construct Note the z 4 in the formula.The matrix of coefficients has nonzero rows and columns only for monomials of even degree, and taking this submatrix we find a full rank matrix of rank N + 1.As the on diagonal elements are roughly of size 1 2 or larger as long as σs are small, and all off diagonal elements are of size proportional to the σs, we find that the number of negative eigenvalues is N and there is one positive eigenvalue corresponding the constant 1.The matrix of |1 + G(z)| 2 also only has nonzero rows and columns for the monomials of even degree, and hence so does r(z, z) whose matrix again has no elements for the row and column corresponding to the constant.Hence its rank is N and it must be a negative semidefinite matrix so there is a polynomial p(z) only using degree 2 and 4 monomials such that Again r = 0 on z = 1 and we are finished, p 1+G is the desired map.It is in lowest terms by the same argument as in Proposition 5.1.

Group invariance of maps with generic σ
We prove the first part of Corollary 1.4 in the next lemma.The construction part of the corollary we have already done in Proposition 5.3.Lemma 6.1.Suppose f = p g : B n → B N is a rational proper map in normal form such that 0 < σ 1 < . . .< σ n .Then G f is either trivial, or G f = {I, −I}, in which case p(z) = p(−z) and g(z) = g(−z).In particular, if G f = {I, −I} then the degree of f is at least 4.
and H f are closed was proved by D'Angelo and Xiao in their work, and it also follows rather quickly once they are subgroups of the unitary group.That D f , Σ f , ∆ (a,b) f