The Reflection Map and Infinitesimal Deformations of Sphere Mappings

The reflection map introduced by D’Angelo is applied to deduce simpler descriptions of nondegeneracy conditions for sphere maps and to the study of infinitesimal deformations of sphere maps. It is shown that the dimension of the space of infinitesimal deformations of a nondegenerate sphere map is bounded from above by the explicitly computed dimension of the space of infinitesimal deformations of the homogeneous sphere map. Moreover a characterization of the homogeneous sphere map in terms of infinitesimal deformations is provided.

For n = 2 write z = z 1 , w = z 2 . A lot is known about mappings of spheres, see the survey by D'Angelo [9] and the references therein. A prominent example of a sphere map is the homogeneous sphere map H d n of degree d from S 2n−1 into S 2K −1 for some K = K (n, d) ∈ N, which consists of all lexicographically ordered monomials in z = (z 1 , . . . , z n ) ∈ C n of degree d and is given by  The purpose of this article is to study the reflection map, which was introduced by D'Angelo [7] in the case of sphere mappings and further investigated by the same author in [8] in the case of maps of hyperquadrics. The reflection map of a mapping H allows to effectively compute and deduce several properties of the X-variety associated to H . The X-variety was introduced and studied by Forstnerič in [19] to extend CR maps satisfying certain smoothness assumptions. In the case of real-analytic CR maps of spheres it is shown that these maps are rational.
The homogeneous sphere map H d n plays a crucial role in the classification of polynomial maps, see the works of D'Angelo [3,5] and [10] for rational sphere maps. The homogeneous sphere map appears in the definition of the reflection map C H for a rational sphere map H = P/Q : S 2n−1 → S 2m−1 with Q = 0 on S 2n−1 : Let V H : C m → C K be a matrix with holomorphic entries, satisfying V H (X ) ·H d n /Q = X ·H on S 2n−1 for X ∈ C m , where · denotes the euclidean inner product. The previous identity is achieved by the homogenization technique of D'Angelo [3]. V H is referred to as reflection matrix and C H (X ):=V H (X ) ·H d n /Q for X ∈ C m . See Sect. 2.4 below for more details.
In this article the reflection matrix will be applied in two ways. In the first case it is shown that nondegeneracy conditions of sphere maps can be rephrased in terms of rank conditions on the reflection matrix. The nondegeneracy conditions considered here were introduced in [20] and [22] respectively. In the case of a sphere map H : S 2n−1 → S 2m−1 they are defined as follows: If denotes the set of real-analytic CR vector fields tangent to S 2n−1 , then H is called finitely nondegenerate at p ∈ S 2n−1 , if there is an integer ∈ N, such that, span C {L 1 · · · L k H (z)| z= p : L j ∈ , k ≤ } = C m .
The map H is called holomorphically nondegenerate if there is no nontrivial holomorphic vector field tangent to S 2m−1 along the image of H . These notions of nondegeneracy were originally defined for submanifolds and introduced by [24] and [1] respectively, see also the survey of Lamel [21]. For more details on nondegeneracy conditions for CR maps see also Sect. 2.2 below. Then the following theorem is shown: Theorem 1 Let H : S 2n−1 → S 2m−1 be a rational map of degree d. This has immediate consequences to show sufficient and necessary conditions in terms of nondegeneracy conditions for the X-variety of H to be an affine bundle or that it agrees with the graph of the map, see Sect. 5 and Theorem 3 below for more details.
In the second case, applications of the reflection matrix to the study of infinitesimal deformations are provided. For M ⊂ C N and M ⊂ C N real submanifolds consider the set H(M, M ) of all maps, which are holomorphic in a neighborhood of M and satisfying H (M) ⊂ M . In [14][15][16][17] locally rigid maps were studied. They correspond to isolated points in the quotient space of H(M, M ) under automorphisms. A sufficient linear condition was provided for local rigidity of a given map, which is formulated in terms of infinitesimal deformations. An infinitesimal deformation of a map H : M → M is a holomorphic vector, defined in a neighborhood of M, whose real part is tangent to M along the image of H . The set of infinitesimal deformations of a map H is denoted by hol(H ). Examples of infinitesimal deformations of a map H can be obtained from smooth curves of The results involving infinitesimal deformations are summarized in the following theorem:  [3,23] or [6, Sect.5.1.4, Theorem 3] and demonstrates a new method to compute infinitesimal deformations for sphere maps. While the article [15] contains examples which required computer-assistance, it is shown in several examples in this article that the reflection matrix allows for explicit and effective computations of infinitesimal deformations of sphere maps.

Preliminaries
The purpose of this section is to introduce the necessary notions and notations needed throughout the article. These are only required for maps of spheres but without any effort and no loss of clarity the general case of maps of manifolds M and M is treated. To this end the following assumptions are made: Let M be a real-analytic generic submanifold of C N of codimension d. For a real-analytic CR submanifold M ⊂ C N of codimension d , let p ∈ M and ρ : of ρ j is given by The following notation is used: v · w:=v 1 w 1 + · · · + v n w n for vectors v = (v 1 , . . . , v n ) ∈ C n and w = (w 1 , . . . , w n ) ∈ C n .

Infinitesimal Deformations of CR Maps
One of the main objects of this article are infinitesimal deformations of a CR map.
for some open neighborhood U ⊂ C N of p.
In the case of sphere mappings, for a real-analytic CR map H :

Nondegeneracy Conditions for CR Maps
The purpose of this section is to provide the definitions of finite and holomorphic nondegeneracy for CR maps introduced by Lamel [20] and Lamel-Mir [22] respectively, and study some of their properties. Definition 2 A real-analytic CR map H : M → M is called holomorphically degenerate if there exists a real-analytic CR map Y : M → C N satisfying Y ≡ 0 and for every p ∈ M and every real-analytic mapping ρ = (ρ 1 , . . . , ρ d ) defined in a neighborhood of H ( p) vanishing on M , it holds that, for some open neighborhood U ⊂ C N of p. If a map is not holomorphically degenerate it is called holomorphically nondegenerate.
Simple examples of holomorphically degenerate maps are the following: is a holomorphically degenerate sphere map from S 2n−1 into S 2(n+k)−1 , since X = 0 ⊕ G for 0 ∈ C n and G a holomorphic function from C n into C k satisfies X ·H = 0.
Finite nondegeneracy is defined as follows: Definition 3 Consider a real-analytic CR map H : M → M . LetL 1 , . . . ,L n a basis of CR vector fields of M and for a multiindex α = (α 1 , . . . , α n ) ∈ N n denotē L α =L α 1 1 · · ·L α n n . Let p ∈ M. For each k ∈ N define the following subspaces of C N : for a real-analytic mapping ρ = (ρ 1 , . . . , ρ d ) defined in a neighborhood of H ( p) and vanishing on M . Define s( p): . Then for each k ∈ N also Y k :=z k 1 X is tangent to M along H (M) and these maps are complex-linearly independent. Since M is generic and the real part of a nontrivial holomorphic mapX , restricted to M, cannot vanish on M (the vanishing of Re(X )| M would imply thatX ≡ 0), the vector fields Re(Y k ) are real-linearly independent, hence dim R hol(H ) = ∞.
To prove (b) denote by X(H ) the set of holomorphic vector fields tangent to M along the image of H . Consider X = j a j (Z ) ∂ ∂ Z j ∈ X(H ) which, by the finite nondegeneracy of H and Proposition 1, satisfies a j ( p) = 0 and j a j (Z )ρ k Z j (H (Z ), H (Z )) = 0 for Z ∈ M, k = 1, . . . , d and a real-analytic mapping ρ = (ρ 1 , . . . , ρ d ) defined in a neighborhood of H ( p) and vanishing on M . Taking derivatives w.r.t. CR vector fields L of M one gets for multiindices β m ∈ N n for m = 1, . . . , N . Use coordinates as given in e.g., [2, Taking derivatives w.r.t. z and u one gets: where the expression "l.o.t." stands for terms vanishing at Z = p. Evaluating at Z = p one gets that Proceeding inductively shows that all derivatives of a j (Z ) have to vanish at p. This means that the holomorphic vector field X vanishes in a neighborhood of p on the generic submanifold M, hence X ≡ 0, which implies that H is holomorphically nondegenerate.
To show (c) let {X 1 , . . . , X s } be a basis of the space of holomorphic vector fields at p ∈ M tangent to M along the image of H and take X ∈ span C {X 1 , . . . , X s }. Consider the following equation for 1 ≤ k ≤ d : Take derivatives w.r.t. L and since X is holomorphic, it holds that, for any multiindex β ∈ N n . This implies that for any sequence β 1 , . . . , β N ∈ N n of multiindices and integers k m ∈ N with 1 ≤ k m ≤ d the vector field X belongs to the kernel of the matrix (L β m ρ k m Z (H (Z ), H (Z ))) 1≤m≤N . Hence outside a proper real-analytic variety Y of a neighborhood of p it holds that for any ∈ N one has dim C E (q) = N − s for q ∈ Y , such that the degeneracy of H is equal to s for all q ∈ Y .

Infinitesimal Automorphisms of the Unit Sphere
In the following the well-known infinitesimal automorphisms of S 2n−1 , n ≥ 2 are listed for later reference. For A = (A 1 , . . . , A n ) ∈ hol(S 2n−1 ) the j-th component is given as follows: where α m , β m ∈ C and s m ∈ R and dim R hol(S 2n−1 ) = n(n + 2). The following notation is required: For a map H : S 2k−1 → S 2m−1 any T ∈ aut(H ) can be written as T = T 1 +. . .+T 4 , such that the T j are given as follows:

The Reflection Matrix
The following definition is a summary of [7, Definition 2.1, 2.2] introducing the homogenization and reflection map (which appears in the study of the X -variety, see also [8] for the case of hyperquadric maps): Denote by H(n, d) the complex vector space of homogeneous polynomials of degree d in n holomorphic variables z = (z 1 , . . . , z n ). WriteH(n, d) for the complex vector space with basis consisting of homogeneous polynomials of degree d in n anti-holomorphic variables z = (z 1 , . . . ,z n ).

Definition 4
Let H = P Q : U ⊂ C n → C m be a rational map of degree d (not necessarily a sphere map), where P = (P 1 , . . . , P m ) and Q : Since C H is linear and with holomorphic entries, such that and denote V :=QV H . The matrix V is referred to as reflection matrix of H .
Several properties of the reflection matrix and examples involving V are given in [7].

Examples and Constructions for Sphere Maps
In this section some particular examples of sphere maps and constructions of sphere maps are presented and their relation to the above nondegeneracy conditions are discussed.

The Homogeneous Sphere Maps
The purpose of this section is to show some properties of the homogeneous sphere maps defined as follows: A direct computation or [ n} be a collection of vector fields. In order to denote powers of such vector fields the following notation is used: Define the set In the following lemma some basic facts about CR vector fields and their commutators are given. The proofs consist of straight forward calculations and are omitted.
Lemma 1 Assume n ≥ 3. In the following for 1 ≤ i, j, k, ≤ n assume i = j and k, / ∈ {i, j}. Define the following vector fields for S 2n−1 : Then T jk = −T k j and S i j = S ji and the following commutator relations hold: For n = 2 define the following vector field for S 3 : Then it holds that [S,L] = 2L.
Proof Set H :=H d n , fix 1 ≤ m ≤ n and define the following set of multiindices where the index set J from the beginning of Sect. 3.1 is used. It will be shown that is of full rank if z m = 0. This implies that H is d-nondegenerate at each point of S 2n−1 . The proof consists of two steps: on S 2n−1 , for all multiindices α, β ∈ J m with |α| < |β| and all γ ∈ J , δ ∈ J m , where J is defined in the beginning of Sect. 3.1.
Observe that the number of elements in D k m is equal to ). Thus, the two steps together imply that the matrix A m consists of linearly independent rows if z m = 0.
In order to show (A) one proceeds by induction on the length of α in (3): For |α| = 0 one needs to argue as follows. Since H ·H = 1 on S 2n−1 , it follows that p H ,β :=H ·L βH = 0 on S 2n−1 for all β ∈ J m . This means that p H ,β is a homogeneous polynomial vanishing on S 2n−1 , hence p H ,β vanishes in C n , see [5,Sect. II] Assume that (3) holds for |α| = k and |α| + 1 < |β|. If one applies L m j to (3) one obtains: the induction is completed and (A) is proved. In order to show (4), use the identities from Lemma 1, which imply that the expression of the left-hand side of (4) can be rewritten as a sum of terms of the form L α H ·L β T γ S δ L m jH and L α H ·L β T γ S δ H , where |β | ≥ |β|−1, γ ∈ J and δ ∈ J m . Hence using the induction hypothesis proves (4).
To prove (B) fix 0 ≤ k ≤ d and assume that the set D k m consists of vectors which are not linearly independent. By setting K :={α ∈ J m : |α| = k}, the linear dependence says that there are c α ∈ C, not all of them are zero, such that Note that in the vector R α H ∈ C K (n,d) each monomial of H d−k appears in exactly one component and it is of the following form: Consider the minimal α 0 ∈ K w.r.t. the lexicographic order. If m = 1, then α 0 = (2, . . . , 2), and otherwise α 0 = (1, . . . , 1). This implies that j 1 (α) ≥ j 1 (α 0 ) for all α > α 0 . Moreover t(α) > t(α 0 ) for all α > α 0 , which can be seen as follows: Denote the monomial in H d n at the k-th position by which implies that t(α 0 ) < t(α). Thus, in (6), considering the coefficient of z t(α 0 ) shows that c α 0 = 0.

The Group Invariant Sphere Maps
Another important class of sphere maps are the following, first introduced in [3]: where c k ≥ 0 for 1 ≤ k ≤ + 2 is given in [3] or [6, Sect. 5.2.2, Theorem 9].
The infinitesimal stabilizer of G consists of the vector field S 3 2 . The maps G are invariant under a fixed-point-free finite unitary group and appear in [12] as so-called sharp polynomials in the study of degree bounds for monomial maps.

The Tensor Product for Infinitesimal Deformations
Similar to the case of sphere maps ( [3], [5,Definition 4]) one can introduce a tensor operation for infinitesimal deformations.
Let A ⊆ C n be a linear subspace such that For vectors v = (v 1 , . . . , v n ) ∈ C n and w = (w 1 , . . . , w m ) ∈ C m the usual tensor product of v and w is denoted by v ⊗ w = (v 1 w 1 , . . . , v 1 w m , . . . , v n w 1 , . . . , v n w m ) ∈ C nm .

Definition 7
Let H : S 2n−1 → S 2m−1 and G : S 2n−1 → S 2 −1 be CR maps, X ∈ hol(H ) and A ⊆ C m be a linear subspace, then is called the tensor product of X by G on A.
We recall that the tensor product of mappings of spheres was introduced in [3] and [5,Definition 4]: For f : S 2n−1 → S 2m−1 and g : S 2n−1 → S 2 −1 CR maps and A ⊆ C m a linear subspace the tensor product of f by g on A given by The next result shows that holomorphic degeneracy is preserved by tensoring.

CR map. If H is holomorphically degenerate then E (A,G) H is holomorphically degenerate.
Proof Since H is holomorphically degenerate there exists a nontrivial holomorphic map W : C n → C m such that W ·H = 0 on S 2n−1 . Write H = E (A,G) H and consider W = T (A,G) W , which is a nontrivial holomorphic vector. Then the same computation (without taking the real part) as in the proof of Lemma 3 shows that W ·H = 0 on S 2n−1 , i.e., H is holomorphically degenerate.

Example 3 For a sphere map H its trivial infinitesimal deformations may give rise to nontrivial infinitesimal deformations of tensors of H : Let H be the map
and Note that j t (H , G) is holomorphically degenerate: The holomorphic vector field X = −t H ⊕ √ 1 − t 2 G satisfies X ·H = 0.

Nondegeneracy Conditions for Sphere Maps
In this section it is shown that holomorphic and finite degeneracy can be expressed in terms of rank conditions of the reflection matrix. Holomorphic nondegeneracy of a sphere map is equivalent to a generic rank condition of V : Proof For a map H = P Q the following equation holds on S 2n−1 : for any vector X ∈ C m . Let Y : U → C m , where U is a neighborhood of S 2n−1 , be a nontrivial holomorphic map such that Y ·H = 0 on S 2n−1 . By (7), this is equivalent to V Y ·H d n = 0 on S 2n−1 . Since H d n is holomorphically nondegenerate on S 2n−1 , the last equation is equivalent to V Y = 0 on S 2n−1 . From this consideration the equivalence of (a) and (b) follows.
The equivalence of (b) and (c) holds, since (b) is equivalent to the fact that V is injective on an open, dense subset of S 2n−1 , which is equivalent to (c).
The following proposition shows that finite nondegeneracy of a map H is equivalent to a pointwise rank condition of V H : In particular, the map H is finitely nondegenerate at p ∈ S 2n−1 if and only if the matrix V is of rank m at p ∈ S 2n−1 .

Theorem 1 follows from Propositions 3 and 4.
Proof Since X ·H = V H X ·H d n /Q on S 2n−1 for any X ∈ C m and V is holomorphic, it follows that on S 2n−1 . For any sequence of multiindices α = (α 1 , . . . , α ), α j ∈ N n and a sphere map F : S 2n−1 → S 2k−1 define the × k-matrix A α q (F):= L α jF | q 1≤ j≤ for q ∈ S 2n−1 . Then it holds that for any p ∈ S 2n−1 , any sequence of multiindices β = (β 1 , . . . , β r ), β j ∈ N n and r ∈ N. . . . , γ K (n,d) ), γ j ∈ N n , chosen according to the finite nondegeneracy of H d n given in the proof of Lemma 2, which can be seen as follows: One has that where c γ j ε ∈ C involves some constants, derivatives ofQ and terms of the form Q −m γ j ,ε for some m γ j ,ε ∈ N. Since the first row of A Now it is possible to prove the equivalence of (a) and (b). Assume (a), then there is k 0 ∈ N, such that H is (k 0 , s)-degenerate at p. This means that dim C E k 0 ( p) = N − s and hence, for any β = (β 1 , . . . , β r ), β j ∈ N n and r ∈ N, the kernel of the matrix A β p (H ) is at least of dimension s. Consider γ = (γ 1 , . . . , γ K (n,d) ), γ j ∈ N n according to the finite nondegeneracy of H d n given in the proof of Lemma 2. Let X j for 1 ≤ j ≤ s be linearly independent vectors in the kernel of A γ p (H ). By taking β = γ in (8) Assume that dim ker V ( p) = s > s, i.e., there are linearly independent vectors Y j ∈ ker V for 1 ≤ j ≤ s . Since H is of degeneracy s there exists a sequence of multiindices δ = (δ 1 , . . . , δ q ), δ j ∈ N n and q ∈ N, such that the kernel of For the other direction, assume (b) and argue similarly: If dim ker V ( p) = s, consider any sequence of multiindices = ( 1 , . . . , t ) for j ∈ N n and t ∈ N. Let X j for 1 ≤ j ≤ s be linearly independent vectors belonging to ker V ( p). By (8) it follows that X j ∈ ker A p (H ). Thus, the degeneracy of H is at least s.
Assume the degeneracy of H is equal to s > s. Argue as in the proof of the sufficient direction to conclude that dim ker V ( p) ≥ s, which is a contradiction.
The last statement follows immediately from the above shown equivalence.

Example 5
For each ≥ 0 the map G is finitely nondegenerate at p ∈ S 3 : In this case the reflection matrix V is the following (2 + 2) × ( + 2)-matrix: where tH k = (z k , k 1 z k−1 w, . . . , w k ) ∈ C k+1 , blank spaces are filled up with zeros, D 1 is the (2 +2)×(2 +2)-diagonal matrix whose nonzero entries are the reciprocals of the coefficients of H 2 +2 and D 2 is the ( + 2) × ( + 2)-diagonal matrix which consists of the coefficients of G on the diagonal. It follows that V is of full rank on S 3 .

Example 6
The map H (z, w) = (z 4 , z 3 w, √ 3zw, w 3 ), which sends S 3 into S 7 , is listed in [3]. The reflection matrix is given by which is of full rank if and only if w = 0 and if w = 0 the kernel is of dimension 1, hence by Proposition 4 the map is finitely nondegenerate for w = 0 and of degeneracy 1 when w = 0. A direct computation (as in Definition 3) shows that the map is 3-nondegenerate at points {z = 0, w = 0} ∩ S 3 and 4-nondegenerate when {z = 0, |w| = 1}. If w = 0 and |z| = 1, the map is (3, 1)-degenerate.
The following example gives a map, for which the set of points in S 3 where the map is 2-degenerate consists of one isolated point.

Example 7
The map H (z, w) = (az − bzw)z, (az − bzw)w,bz +āzw, w 2 , for a, b ∈ C satisfying |a| 2 + |b| 2 = 1, sends S 3 into S 7 . The matrix V is given by where D is the 4 × 4-diagonal matrix whose nonzero entries are the reciprocals of the coefficients of The following result gives conditions to guarantee that a sphere map is finitely degenerate: Corollary 1 If a rational sphere map H : S 2n−1 → S 2m−1 of degree d satisfies K (n, d) < m, then H is finitely degenerate at any p ∈ S 2n−1 . In particular the map is holomorphically degenerate. (K (n, d), m) = K (n, d) on S 2n−1 . If H would be finitely nondegenerate at p ∈ S 2n−1 , by Proposition 4, V would be injective at p, hence rk V = m at p, a contradiction. By Proposition 3 it follows that H is holomorphically degenerate.

Example 8
The map H (z, w) = (z, cos(t)w, sin(t)zw, sin(t)w 2 ), t ∈ [0, 2π), sends S 3 to S 7 and is holomorphically degenerate by Corollary 1. The reflection matrix V is given as follows: If sin(t) = 0, then X (z, w) = (0, 1, − cot(t)z, − cot(t)w) is a holomorphic vector field tangent to S 7 along the image of H . One can check that if cos(t), sin(t) = 0 the map is of degeneracy 1 for z = 0 and of degeneracy 2 if z = 0. If cos(t) = 0 and sin(t) = 0 the map is of degeneracy 2 and when cos(t) = 0 the map is 1-degenerate.
The set of points where the map is finitely degenerate can be described by using Proposition 4: Proof The set D of points where H is of degeneracy s > s is the complement Y of the set where H is of generic degeneracy s, which is given by the union of the zero sets of any minor of V of size strictly less than rk V . Since V consists of holomorphic polynomial entries, Y is a complex algebraic variety and, by Proposition 4, agrees with D.

The X-variety of a Sphere Map
In this section sufficient and necessary conditions in terms of nondegeneracy conditions are provided to guarantee that the X-variety of a sphere map satisfies certain properties, such as agreeing with the graph of the map or being an affine bundle.
First, the general definition of the X-variety of a map is repeated for the reader's convenience, see [19] and [7]: Since H maps M into M it follows that (Z , H (Z )) ∈ X H , i.e., the graph of H is contained in X H . In [7,Theorem 4.1] it is shown in the case when M ⊂ C n and M ⊂ C m are unit spheres that for any z = 0 it holds that (z, z ) ∈ X H if and only if z − H (z) ∈ ker V (z). X H has an exceptional fiber at p ∈ S 2n−1 if the dimension of the fiber { p ∈ C N : ( p, p ) ∈ X H } exceeds its generic value. In [7,Corollary 4.2] it is argued that the set of points over which X H has an exceptional fiber agrees with the set of points p ∈ S 2n−1 where the rank of V ( p) drops. Moreover

Infinitesimal Deformations of Sphere Maps
In this section infinitesimal deformations of rational sphere maps are studied. It turns out that similarly as in the case of sphere maps, where each sphere map is related to the homogeneous sphere map by tensoring, infinitesimal deformations of a sphere map are related to infinitesimal deformations of the homogeneous sphere map by the reflection matrix.

Lemma 5
Let H = P Q : U → S 2m−1 be a holomorphically nondegenerate rational sphere map of degree d, where U is a neighborhood of S 2k−1 . Then each X ∈ hol(H ) is of the form X = X Q , where X is a holomorphic polynomial of degree at most 2d satisfying Re(X ·P) = 0 on S 2k−1 .
Proof Let H be given as in the assumption of the lemma, where P = (P 1 , . . . , P m ) and Q : U → C with Q = 0 on U , a neighborhood of S 2k−1 . Then |Q| 2 Re(X ·H ) = Re(Q X ·P). Set X :=Q X. Considering homogeneous expansions of X = ≥0 X and P = d j=0 P j one obtains the following equation: After setting Z → Ze it for t ∈ R collect the Fourier coefficient of degree d + 0 for 0 ≥ 1 to get: By the holomorphic nondegeneracy of H this implies that X ≡ 0 for ≥ 2d + 1, i.e., deg X ≤ 2d.  Proof Let H = P Q : S 2n−1 → S 2m−1 be a rational map with Q = 0 on S 2n−1 . Consider as in Definition 4 the matrix V : C m → C K (n,d) whose entries are holomorphic polynomials in z ∈ C n . Then it holds on S 2n−1 that |Q| 2 X ·H = V X ·H d n for X ∈ C m as in (7). Thus on S 2n−1 one obtains, By Lemma 5 there are polynomials X 1 , . . . , X k ∈ P 2d (n, m) such that {X j = X j Q : 1 ≤ j ≤ k} is a basis of hol(H ). From Proposition 3 it follows that {V X j : 1 ≤ j ≤ k} is a set of linearly independent polynomials in hol(H d n ) which implies k ≤ dim hol(H d n ). To show the nontrivial implication of the second claim, assume that H = P is polynomial of degree d and dim hol(P) = dim hol(H d n ). By Proposition 3, since the reflection matrix V is injective on a dense, open subset S of S 2n−1 as a map from hol(P) to hol(H d n ), it follows by the rank theorem that dim V (hol(P)) = dim hol(P). Using the assumption dim hol(P) = dim hol(H d n ) this implies that V is invertible as a map from hol(P) to hol(H d n ), for z ∈ S. Thus, for any Y ∈ hol(H d n ) there exists X ∈ hol(P) with V X = Y , such that on S, the following equation holds: Consider z → e it z and a homogeneous expansion of F = d−1 j=0 F j and collect the coefficient of e idt to get that H d n ·F 0 = 0, hence by the holomorphic nondegeneracy of H d n one obtains F 0 = 0. Proceed inductively to show that F k = 0 for k ≤ d − 1.
Assume that F = 0 for all 0 ≤ ≤ k − 1. Collect the coefficient of e i(d−k)t in (11) to obtain that, which, by using the induction hypothesis and the holomorphic nondegeneracy of H d n , implies that F k = 0. In total one obtains that P is unitarily equivalent to H d n in S, hence they are equivalent everywhere on S 2n−1 .
One has the following inequality for the dimension of the space of infinitesimal deformations, when the tensor product is involved. Proof If F is holomorphically degenerate, by Proposition 2, the inequality is satisfied. Assume that F is holomorphically nondegenerate, then the same holds for H by Lemma 4. Instead of using V as in the proof of Theorem 4, one considers V a linear map defined by V (X ): It holds that if there exists Y ∈ hol(H ) with V (Y ) = 0 on S 2n−1 , then 0 = V (Y )·F = Y ·H on S 2n−1 , which implies, since H is holomorphically nondegenerate, that Y ≡ 0. From this it follows that the set { V (X j ) : 1 ≤ j ≤ k}, for X 1 , . . . , X k a basis of hol(H ), is linearly independent in hol(F), which gives the claimed inequality.
For the equality, assume that dim hol(F) = dim hol(H ) < ∞. Then dim hol(H ) = dim V (hol(H )) ≤ dim hol(F) = dim hol(H ), which implies, as in the proof of Theorem 4, that on S 2n−1 the map V , as a map from hol(H ) to hol(F), is invertible. Using a similar argument as in the proof of Theorem 4 (replacing V byṼ , H d n by F and P by H and using X ·H =Ṽ (X ) ·F) it follows that H and F are unitarily equivalent, which can only happen, when the complex subspace A is trivial. This concludes the proof.
The remainder of this section is a collection of lemmas concerning some properties of V H and its transpose and provide sufficient and necessary conditions for infinitesimal rigidity in terms of V H and its adjoint.
for all X ∈ C n , which concludes the proof.
The following example shows that a similar relation as the second identity in Proposition 5 does not hold for infinitesimal deformations in general:

Lemma 6
Let H : S 2n−1 → S 2m−1 be a polynomial map of degree d. In (b) the necessary direction need not be true as Example 9 shows.
Proof By Proposition 5 the following holds on S 2n−1 : which shows (a). In (b) assume X := tV H Y is holomorphic. By Proposition 5 one has, on S 2n−1 and taking the real part shows that X ∈ hol(H ). Consider the above equation and note that one has X ·H = V H X ·H d n , such that, since V H X is holomorphic, the holomorphic nondegeneracy of H d n implies Y = V H X ∈ V H (hol(H )).
Some of the nontrivial infinitesimal deformations of H d n originate from curves passing through the map as the following example shows: In the following a family of finitely nondegenerate rational sphere maps is constructed, which contains the homogeneous sphere map H d n for d odd. It is well-known that families of sphere maps exist, see the examples in [4] and [18,Examples 4.1,4.2], which motivated the construction. See also [11] for a study of homotopies of sphere maps.

Theorem 6
For k ≥ 1 the map H 2k+1 2 : S 3 → S 2k+2 is not locally rigid. More precisely, there exists a family F k s : S 3 → S 2k+2 of (2k + 1)-nondegenerate rational maps, where s ∈ R is sufficiently close to 0, with F k 0 = H 2k+1 2 and each F k s is not equivalent to H 2k+1 where r 1 :=s(s −2) and r 2 :=r 1 +2. WriteF = (F 1 , . . . ,F d ,Q), then F k s is equivalent to a polynomial map if and only if there exist a 1 , a 2 , A 1 , . . . , A d ∈ C and C ∈ C \{0} such that d j=1F j A j +Q = C(a 1 z + a 2 w + t) d .
The claim is that under the assumption s = 0 the Eq. (18) has no solution. Comparing the coefficient of z 2 t d−2 one obtains that a 1 = 0. Then the coefficient of zt d−1 gives r 1 = 0, which cannot be satisfied for 0 = s < 2. This finishes the proof.
Note that for k = 1 the vector d ds | s=0 F 1 s is a nontrivial infinitesimal deformation of H 3 2 from Example 10, when the parameter k ∈ C used there is taken to be real.