On Apolarity and Generic Canonical Forms
نویسندگان
چکیده
The theory of apolarity was first developed by Clebsch, Lasker, Richw x mond, Sylvester, and Wakeford 10, 17, 20 . They were first interested in studying homogeneous polynomials of degree p and in q variables, and in expressing them as sums of pth powers of linear terms. The problem is to minimize the number of pth powers which are required in such a sum. For instance, a result due to Sylvester is that a generic homogeneous polynomial in two variables of degree 2n y 1 may be written as a sum with n Ž . terms of 2n y 1 st powers. This is only true for a generic polynomial; for instance x 2 y cannot be written as the sum of two cubes. They then focused on the more general problem of finding canonical ways of expressing a generic homogeneous polynomial. For example, it was
منابع مشابه
On subcanonical Gorenstein varieties and apolarity
Let X be a codimension 1 subvariety of dimension > 1 of a variety of minimal degree Y . If X is subcanonical with Gorenstein canonical singularities admitting a crepant resolution, then X is Arithmetically Gorenstein and we characterise such subvarieties X of Y via apolarity as those whose apolar hypersurfaces are Fermat.
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