Let $C(\boldsymbol{x})$ be a cubic form in $n$ variables $\boldsymbol{x}=(x_1,...,x_n)$ over $\mathbb{Z}$. We can write $C(\boldsymbol{x})=\sum_{i,j,k=1}^{n}c_{ijk}x_ix_jx_k$, where the $c_{ijk}$ are symmetric with respect to permutations on their indices. Let $M(\boldsymbol{x})$ be the $n\times n$-matrix with $(i,j)$-entry given by $\sum_{k=1}^{n}c_{ijk}x_k$. Is there any explicit example of a cubic form $C$ for which there exists an $r=0,1,...,n$ and some $\varepsilon>0$ such that for infinitely many $H>0$, we have $\#\left\{\boldsymbol{x}\in\mathbb{Z}^n:|\boldsymbol{x}|\leq H,\;\text{rank}\;M(\boldsymbol{x})=r\right\}\gg H^{r+\varepsilon}$?
cubic forms with a rank condition
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polynomials
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0(i) A better way of defining $M(\boldsymbol{x})$ is: $M(\boldsymbol{x})=\nabla^2C(\boldsymbol{x})/3!$, where $\nabla^2C(\boldsymbol{x})$ is the Jacobian matrix of $C(\boldsymbol{x})$. (ii) Don't worry about the $c_{ijk}$... (iii) $|\boldsymbol{x}|=\max\left\{|x_j|:j=1,...,n\right\}$. – 2012-05-04