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By Jiří Lebl
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Extra info for Hermitian Forms Meet Several Complex Variables: Minicourse on CR Geometry Using Hermitian Forms
Sample text
It is easy to see that if the map takes sphere to sphere, then no negative exponents are allowed. 2: A monomial CR map taking sphere to sphere has no negative exponents. So if (J, A) is simultaneously diagonalizable, we are done. Therefore, let us see what happens in the case when we cannot simultaneously diagonalize J and A. From now on, assume that at least one block in the canonical form is larger than 1 × 1. As J defines the sphere, all blocks except one in the canonical form for J must be 1 × 1 blocks.
Hence p-degree is d. 5. Let Q be the homogeneous polynomial Q(x, y,t) = P(x, y,t) . 61) We define the function D : Z2 → {0, P, N} as follows. Let c j,k be the coefficient of x j ykt d− j−k in Q. 62) N if c j,k < 0. We call D the Newton diagram of Q, and we will say that D is the Newton diagram corresponding to P. We will call the m ∈ Z2 points of D, and we will call m a 0-point if D(m) = 0, a P-point if D(m) = P and an N-point if D(m) = N. Therefore, the Newton diagram of Q is an array of Ps, Ns, and 0s one for each coefficient of Q.
That is, every component is a single monomial. For higher n, some information is known. The following theorem is a result that is a culmination of successive improvements of theorems of Webster ’79 [W], Faran ’86 [F2], Cima-Suffridge ’83 [CS], Huang ’99 [H], Huang-Ji ’01 [HJ], Hamada ’05 [H], and Huang-Ji-Xu ’05 and ’06 [HJX]. Many of the later developments reduce the regularity to the boundary necessary for rationality. For simplicity we will state the theorem for rational maps. 8. Let F : Bn → BN be rational proper and holomorphic.