Additive Splittings of Homogeneous Polynomials

In this thesis we study when a homogeneous polynomial \(f\) decomposes or "splits" additively. Up to base change this means that it is possible to write \(f = g + h\) where \(g\) and \(h\) are polynomials in independent sets of variables. This simple idea leads us to define a set \(M_f\) of matrices associated to \(f\). Surprisingly, \(M_f\) turns out to be a commutative matrix algebra when \(deg f \ge 3\). We show how all (regular) splittings \(f = g_1 + ... + g_n\) can be computed from \(M_f\). Next we show how to find the minimal free resolution of the graded Artinian Gorenstein quotient \(R/\ann f\), assuming the minimal free resolutions of its additive components \(R/\ann g_i\) are known. From this we get simple formulas for the Hilbert function \(H\) and the graded Betti numbers of \(R/\ann f\). We may use this to compute the dimension of a "splitting subfamily" of the parameter space \(\PGor (H)\). Its closure is quite often an irreducible component of \(\PGor (H)\). We will also study degenerations of polynomials that split and see how they relate to \(M_f\). This situation is more difficult, but we are able to prove several partial results that together cover many interesting cases. In particular, we prove that \(f\) has a regular or degenerate splitting if and only if the ideal \(\ann f\) has at least one generator in its socle degree. Finally, we look at some generalizations of \(M_f\).