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By connectedness K must contain a connected path from (a, 0) to (0, b). 9 we can change all the 0-points of D above this connected path into P and N-points without increasing the number of nodes. 8 to complete the proof. Further Reading Here we list useful books for extra reading. Papers cited appear in the next section. [AG] N. I. Akhiezer and I. M. , New York, 1993. MR1255973 [BER] M. Salah Baouendi, Peter Ebenfelt, and Linda Preiss Rothschild, Real submanifolds in complex space and their mappings, Princeton Mathematical Series, vol.

Thus if we take gf and set zn+1 = 1, we obtain a rational mapping that takes sphere to sphere. On the other hand, suppose that we have a rational mapping gf that takes S2n−1 ⊂ Cn to S2N−1 ⊂ CN . We homogenize f and g (call the homogenized versions f˜ and g). ˜ Then the map ( f˜, g) ˜ takes HQ(n, 1) to HQ(N, 1). 33) we note that r must have signature pair (N, 1). Thus we have an identification of rational maps of spheres with bihomogeneous polynomials vanishing on HQ(n, 1) with signature pair (N, 1).

We can then simply f or g and use the proposition. Note that we will only consider V where the number of positive and negative eigenvalues are not equal. Otherwise we would also want to consider maps that take V to −V . The V we will consider defines the ball and always has only one negative eigenvalue. For the hyperquadric HQ(a, b) we let V be the matrix with a ones and b negative ones on the diagonal. Thus for a sphere, V will always have N ones and one −1. Similarly for J. Let U(N, 1) be the set of matrices M such that M ∗V M = V .

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