论文标题
非产品Hessian等级1的不存在，$ \ Mathbb {r}^{r}^{n+1} $ in Dimension $ n \ geqslant 5 $
Inexistence of Non-Product Hessian Rank 1 Affinely Homogeneous Hypersurfaces $H^n$ in $\mathbb{R}^{n+1}$ in Dimension $n \geqslant 5$
论文作者
论文摘要
仿射组$ {\ rm aff}（\ mathbb {r}^3）的等价等价$ 1 $ 1 $ 1 $ 1 $ s^2 \ s^2 \ subset \ subset \ mathbb {r}^3 $，有时称为寄生虫，有时称为寄生虫，是由doubrov，komrakrov，rabinov，rabinov，rabinov，crabinove，rabinovichey，rabinove，crabinov，rabinovichew，rabinovich，rabinove， Arnaldsson，Valiquette。尤其是，已经完全了解了各个分支中差异不变的均匀模型和代数。 那更高的尺寸呢？我们考虑hypersurfaces $ h^n \ subset \ mathbb {r}^{n+1} $绘制为$ \ {u = f（x_1，\ dots，x_n）\} $，其Hessian Matrix $（f_ {f_ {x_i x_j}）有同质模型吗？ 作者在计算机上进行了完整的探索，$ n = 2、3、4、5、6、7 $。第一个预期的结果是在维度上获得均质模型的完整分类$ n = 2，3，4 $（即将发表的文章，案例$ n = 2 $已经知道）。第二个意外的结果是，在尺寸$ n = 5、6、7 $中，没有相当同质的模型！ （除非与$ \ mathbb {r}^m $的产品相当于均等模型的产品$ 2、3、4 $。 本文基于$ \ {u = f（x_1，\ dots，x_n）\} $ $ {\ rm aff}（\ rm aff}（\ rm aff}（\ rm aff}（\ mathbbbb {r}^n+1} $，n+1} $，尺寸$ n \ geqslant 2 $。
Equivalences under the affine group ${\rm Aff} (\mathbb{R}^3)$ of constant Hessian rank $1$ surfaces $S^2 \subset \mathbb{R}^3$, sometimes called parabolic, were, among other objects, studied by Doubrov, Komrakov, Rabinovich, Eastwood, Ezhov, Olver, Chen, Merker, Arnaldsson, Valiquette. Especially, homogeneous models and algebras of differential invariants in various branches have been fully understood. Then what about higher dimensions? We consider hypersurfaces $H^n \subset \mathbb{R}^{n+1}$ graphed as $\{ u = F(x_1, \dots, x_n) \}$ whose Hessian matrix $(F_{x_i x_j})$, a relative affine invariant, is, similarly, of constant rank $1$. Are there homogeneous models? Complete explorations were done by the author on a computer in dimensions $n = 2, 3, 4, 5, 6, 7$. The first, expected outcome, was to obtain a complete classification of homogeneous models in dimensions $n = 2, 3, 4$ (forthcoming article, case $n = 2$ already known). The second, unexpected outcome, was that in dimensions $n = 5, 6, 7$, there are no affinely homogenous models! (Except those that are affinely equivalent to a product of $\mathbb{R}^m$ with a homogeneous model in dimensions $2, 3, 4$.) The present article establishes such a non-existence result in every dimension $n \geqslant 5$, based on the production of a normal form for $\{ u = F(x_1, \dots, x_n) \}$ under ${\rm Aff} (\mathbb{R}^{n+1})$, up to order $\leqslant n+5$, valid in any dimension $n \geqslant 2$.
