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We prove two criteria for direct sum decomposability of homogeneous polynomials. For a homogeneous polynomial with a non‐zero discriminant, we interpret direct sum decomposability of the polynomial in terms of factorization properties of the Macaulay inverse system of its Milnor algebra. This leads to an if‐and‐only‐if criterion for direct sum decomposability of such a polynomial, and to an algorithm for computing direct sum decompositions over any field, either of characteristic 0 or of sufficiently large positive characteristic, for which polynomial factorization algorithms exist. For homogeneous forms over algebraically closed fields, we interpret direct sums and their limits as forms that cannot be reconstructed from their Jacobian ideal. We also give simple necessary criteria for direct sum decomposability of arbitrary homogeneous polynomials over arbitrary fields and apply them to prove that many interesting classes of homogeneous polynomials are not direct sums.
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We study the geometry of the morphism that sends a smooth hypersurface of degree d + 1 in P n−1 to its associated hypersurface of degree n(d − 1) in the dual space P n−1 ∨ .given by a categorical quotient of the locus of GIT semistable hypersurfaces. We call V m,n the GIT compactification of U m,n . The subject of this paper is a certain rational map V m,n V n(m−2),n , where n ≥ 2, m ≥ 3 and where we exclude the (trivial) case (n, m) = (2, 3). While this map has a purely algebraic construction, which we shall recall soon, it has several surprising geometric properties that we establish in this paper. In particular, this rational map restricts to a locally closed immersionĀ : U m,n → V n(m−2),n , and often Mathematics Subject Classification: 14L24, 13A50, 13H10.
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