In the paper under review, the authors introduce and develop the theory of Waring-like decompositions of polynomials. Let us present the main idea behind this project. Let \(R = \bigoplus_{i\geq 0} R_{i} = \in k[x_{1}, \dots,x_{n}]\) with \(k\) algebraically closed and \(n \geq 2\). The Waring decomposition of \(f \in R_{d}\) is \[ f = \sum_{i=1}^{s} L_{i}^{d}, \] where \(L_{i}\)’s are linear forms, and this is a classical subject of research. In the paper, the authors define a certain variation on the Waring decomposition. Fix an integer \(d\), we denote by \(\mathcal{P}_{r} = \mathbb{Z}[Z_{1},\dots,Z_{r}]\), and by \(\mathcal{M}_{r,d}\) the subset of all monomials such that \[ M= Z_{1}^{d_{1}} \cdots Z_{r}^{d_{r}} \] and
\((\bullet)\) \(d_{i} > 0\) for all \(i\);
\((\bullet \bullet)\) \(d_{1} + \cdots + d_{r} = d.\)
Of course, the above conditions imply that \(r \leq d\).
Let \(M\) be as above, and let \(f \in R_{d}\).
\((\bullet)\) An \(M\)-decomposition of \(f\) having length \(s\) is an expression of the form \[ f = \sum_{j=1}^{s} L_{1,j}^{d_{1}} \cdot L_{2,j}^{d_{2}} \cdots L_{r,j}^{d_{r}}, \] where \(L_{i,j}\)’s are linear forms.
\((\bullet \bullet)\) The \(M\)-rank of \(f\) is the least integer \(s\) such that \(f\) has an \(M\)-decomposition of length \(s\).
Observe that if \(M = Z_{1}^{d}\), then the \(M\)-rank of \(f\) is known as the Waring rank of \(f\), and it can be shown that every \(f \in R_{d}\) has an \(M\)-decomposition of finite length for any choice of \(M\)
There is a nice geometric way of considering the problem of finding \(M\)-rank of a polynomial \(f \in R_{d}\) with \(M \in \mathcal{M}_{r,d}\). Let \(\mathbb{P}(R_{1})\) be the projective space based on the \(k\)-vector space \(R_{1}\). We define the morphism \[ \phi_{M} : (\mathbb{P}(R_{1}))^{r} \rightarrow \mathbb{P}(R_{d}) \simeq \mathbb{P}^{{ d + n - 1 \choose n-1} - 1}, \] where \((\mathbb{P}(R_{1}))^{r}\) denotes the cartesian product of \(r\) copies of \(\mathbb{P}(R_{1})\), by \[ \phi_{M}([L_{1}],\dots,[L_{d}]) = [L_{1}^{d_{1}}L_{2}^{d_{2}} \cdots L_{r}^{d_{r}}], \] and denote the image of \(\phi_{M}\) by \(\mathbb{X}_{M}\).
If \(X \subset \mathbb{P}^{t}\) is any projective variety, then \[ \sigma_{s}(X) := \overline{ \{ P \in \mathbb{P}^{t} : P \in \langle P_{1}, \dots, P_{s} \rangle , P_{i} \in X \}} \] is defined as the \(s\)-secant variety of \(X\). A natural question behind this definition focuses on the dimension of \(\sigma_{s}(X)\) if \(s \geq 2\). It is easy to see that \[ \dim \sigma_{s}(X) \leq \min \{ s \dim X + s - 1,t\}, \] and if the above inequality is satisfied for some \(s\), then we say that \(\sigma_{s}(X)\) has the expected dimension, and if the above inequality is strict for some \(s\), then we say that \(\sigma_{s}(X)\) is defective. The difference between \[ \min \{ s \dim X + s - 1,t\} - \dim \sigma_{s}(X) \] is called the \(s\)-defect of \(X\).
Now we are ready to formulate the main results of the paper. The first one is a natural variation on Terracini’s Lemma.
Theorem 1. Let \(R = k[x_{1},\dots,x_{n}]\), \(M = Z_{1}^{d_{1}} \cdots L_{r}^{d_{r}} \in \mathcal{M}_{r,d}\), and let \(L_{1},\dots, L_{r}\) be general linear forms in \(R_{1}\) so that \(P = [L_{1}^{d_{1}} \cdots L_{r}^{d_{r}}]\) is a general point of \(\mathbb{X}_{M}= \phi_{M}((\mathbb{P}^{n-1})^{r})\). If \(F = L_{1}^{d_{1}} \cdots L_{r}^{d_{r}}\) and \(I_{p} = ( F/L_{1},\dots, F / L_{r}) = \bigoplus_{j \geq 0} (I_{P})_{j}\), then \[ T_{P}(\mathbb{X}_{M}) = \mathbb{P}((I_{P})_{d}). \]
For binary forms there is a nice classification result.
Theorem 2. Let \(R = k[x,y] = \bigoplus_{j\geq 0} R_{j}\) and let \(M = Z_{1}^{d_{1}} \cdots Z_{r}^{d_{r}} \in \mathcal{M}_{r,d}\) for any \(r\) and any \(d\) with \(r \leq d\). Then \(\sigma_{s}(\mathbb{X}_{M})\) has the expected dimension for every \(s\), i.e., \[ \dim \sigma_{s}(\mathbb{X}_{M}) = \min \{ s \dim \mathbb{X}_{M} + (s-1),d\} = \min \{sr + s - 1,d\} \] for every \(s\) and every \(M\).
The last result is devoted to secant line varieties to \(\mathbb{X}_{M}\) with \(n\geq 3\) variables.
Theorem 3. Let \(R = k[x_{1},\dots,x_{n}]\), \(M \in \mathcal{M}_{r,d}\) with \(n \geq 3\), \(r \geq 2\), \(d \geq 3\), and \[ M = Z_{1}^{d_{1}} \cdots Z_{r}^{d_{r}}. \] Then \(\sigma_{2}(\mathbb{X}_{M})\) is not defective, except for \(M = Z_{1}^{2}Z_{2}\) and \(n=3\). For this last case, \(\mathbb{X}_{M}\) has \(2\)-defect equal to \(1\).