Let $\mathbb{N}$ be the set of non-negative integers. Let $n\geq 2, d$ be positive integers. I would like a lower bound on the largest integer $r$ for which the following property holds:
For any tuples $J_1, J_2,K_1, K_2 \in \mathbb{N}^n$ of non-negative integers each summing to $d$ with $J_1\neq J_2$ and $K_1 \neq K_2$, there exist tuples $J_3,\dots,J_r, K_3,\dots,K_r \in\mathbb{N}^n$ of non-negative integers each summing to $d$ for which the following three properties hold:
- $J_1,\dots,J_r$ are distinct,
- $K_1,\dots,K_r$ are distinct, and
- $J_i+K_i \notin \{J_j+K_k : 1\leq j,k < i\}$ for all $i=3,\dots,r$.
My motivation for this question comes from looking at a natural generalization of the Catalecticant matrix for binary forms to $n$-ary forms (for binary forms, it is a Hankel matrix-- see here). The above statement would imply that the $2\times 2$ determinants of this matrix are in the span of the partial derivatives of the $r\times r$ determinants of this matrix. In turn, I am interested in this question in the context of studying prolongation ideals and geometry of secants (e.g. here).
