On schemes evinced by generalized additive decompositions and their regularity (Q6573747)

This paper studies 0-dimensional schemes that are apolar to a given \(d\)-homogeneous polynomial \(F\), namely the 0-dimensional schemes defined by ideals annihilating \(F\) by derivation. Understanding the possible Hilbert functions of minimal apolar schemes is a deep and largely open question, which could give useful information on the nature of additive decompositions of polynomials and secant varieties, and whose grasp is challenging even in moderately small cases. The aim is to study when these Hilbert functions stabilize, and more specifically finding essential conditions for a given \(d\)-homogeneous polynomial to have minimal 0-dimensional apolar schemes that are regular in degree \(d\). While linked to classical Algebraic Geometry problems on secant varieties, from a complexity theory perspective, the knowledge of the regularity of minimal apolar schemes to a given polynomial might improve the efficiency of symbolic algorithms for computing ranks and minimal decomposition of polynomials.\N\NIf we restrict to apolar schemes that are locally contained in \((d +1)\)-fat points, then they correspond to \textit{generalized additive decompositions} (GADs) of \(F\), namely expressions as \(F = \sum_{i=1}^{r}L_i^{d-k_i}G_i\), where the \(L_i\)'s are pairwise non-proportional linear forms not dividing the corresponding \(G_i\in R_{k_i}\). In this view, GADs parameterize generic points of a joint variety of osculating varieties to a certain Veronese variety.\N\NAfter sections of partial but interesting and constructive results, the main achievement in the paper (Proposition 5.3) shows that \(d\)-regularity is granted by schemes evinced by GADs such that the \(L_i\)'s are linearly independent and the \(k_i\)'s are small enough, regardless of the scheme being minimal. However, all the assumptions of the Proposition are sharp. Drawing from the fact that schemes with components of low multiplicity usually exhibit low regularity, then it is proved that minimal tangential decompositions of degree-\(d\) forms always evince \(d\)-regular schemes.