arXiv subjects
Publications and source records attributed to Emanuele Ventura.
In this paper, we study the real and the complex Waring rank of reducible cubic forms. In particular, we compute the complex rank of all reducible cubic forms. In the real case, for all reducible cubics, we either compute or bound the real rank depending on the signature of the degree two factor.
We study the partially ordered set $P(a_1,\ldots, a_n)$ of all multidegrees $(b_1,\dots,b_n)$ of monomials $x_1^{b_1}\cdots x_n^{b_n}$ which properly divide $x_1^{a_1}\cdots x_n^{a_n}$. We prove that the order complex $Δ(P(a_1,\dots,a_n))$ of $P(a_1,\ldots a_n)$ is (non-pure) shellable, by showing that the order dual of $P(a_1,\ldots,a_n)$ is $\mathrm{CL}$-shellable. Along the way, we exhibit the poset $P(4,4)$ as a new example of a poset with $\mathrm{CL}$-shellable order dual that is not $\mathrm{CL}$-shellable itself. For $n = 2$ we provide the rank of all homology groups of the order complex $Δ\left( P(a_1,a_2) \right)$. Furthermore, we give a succinct formula for the Euler characteristic of $Δ\left( P(a_1,a_2) \right)$.
Let $W_3(n)$ be the set of Waring ranks of reducible cubic forms in $n+1$ variables. We prove that $W_3(n)\subseteq \lbrace 1,..., 2n+1\rbrace$.