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By Spencer J. Bloch, R. Keith Dennis, Eric M. Friedlander, Micahel Stein (ed.)

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Extra resources for Applications of algebraic K-theory to algebraic geometry and number theory, Part 2

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First count the number of nodes that involve points at level d (all 0-points) or d − 1 (no 0-points). The number of nodes is d + 1 − k where k is the number of sign changes at the d − 1 level. Now look at two adjacent levels less than d. If the number of sign changes in one level is m and the number of sign changes in the lower level is n, then there must be at least |m−n| 2 nodes involving points only on these two levels. Now by induction plus adding the node at the bottom will complete the proof.

1: Let U ⊂ Cn be open and f : U → H be holomorphic with respect to one basis. Show that it is holomorphic with respect to any basis. 5. 2: Let U ⊂ Cn be open and f : U → H be holomorphic. Suppose that 0 ∈ U. Show that you can write f (z) = ∑α cα zα , where cα ∈ H . 32) where C is the matrix of coefficients and Z is an infinite vector of all the monomials. Now Z does not map all of U into 2 and C may not even be a bounded operator. We have to be a little careful. We can always rescale the variables to ensure that the series for r(z, z¯) has a domain of convergence that includes the point z = (1, 1, .

Let us prove this for degree two maps using the machinery we have developed. One direction is simple of course, so let us concentrate on the hard direction. Suppose that f and g are monomial degree-two maps that are spherically equivalent. 7. MONOMIAL DEGREE ESTIMATES 53 corresponding to f (z) 2 − 1 and g(z) 2 − 1 in homogeneous coordinates and find the matrix A f for f and Ag for g. We notice that these matrices must be diagonal. They are in canonical form, and the canonical form is canonical up to permutation of the blocks (hence permutation of variables).

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