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Additional info for Topics in Classical Algebraic Geometry - I
Then a general homogeneous polynomial f ∈ C[t0 , . . , tn ]d can be written as a sum of dth powers of s linear forms unless (n, d, s) = (2, 2, 2), (2, 4, 5), (3, 4, 9), (4, 3, 7), (4, 4, 14). 5 The Waring problems The well-known Waring problem in number theory asks about the smallest number s(d) such that each natural number can be written as a sum of s(d) d-th powers of natural numbers. It also asks in how many ways it can be done. Its polynomial analog asks about the smallest number s(d, n) such that a general homogeneous polynomial of degree d in n + 1 variables can be written as a sum of s d-th powers of linear forms.
POLARITY Note that the kernel of the map S s E → S d−s E ∗ , ψ → Dψ (f ) is of dimension ≥ dim S s E − dim S d−s E ∗ = s + 1 − (d − s + 1) = 2s − d. Thus Dψ (f ) = 0 for some nonzero ψ ∈ S s E, whenever 2s > d. This shows that a f has always generilized polar s-polyhedron for s > d/2. If d is even, a binary form has an apolar d/2-form if and only det Catd/2 (f ) = 0. This is a divisor in the space of all binary d-forms. 1. Take d = 3. Assume that f admits a polar 2-polyhedron. Then f = (a1 t0 + b1 t1 )3 + (a2 t0 + b2 t1 )3 .
Suppose Z = Z . Choose a subscheme Z0 of Z of length Nk − 1 which is not a subscheme of Z . Since dim IZ0 (k) ≥ dim S k E ∗ − h0 (OZ ) = n+k − Nk + 1 = 1, k we can find a nonzero ψ ∈ IZ0 (k). The sheaf IZ /IZ0 is concentrated at one point x and is annihilated by the maximal ideal mx . Thus mx IZ0 ⊂ IZ . Let ξ ∈ E be a linear form on E ∗ vanishing at x but not vanishing at any subscheme of Z . This implies that ξψ ∈ IZ (k + 1) = IZ (k + 1) and hence ψ ∈ IZ (k) ⊂ APk (f ) contradicting the nondegeneracy of f .