We consider non-degenerate graph immersions into affine space ${\mathbb{A}}^{n+1}$ whose cubic form is parallel with respect to the Levi-Civita connection of the affine metric. There exists a correspondence between such graph immersions and pairs $(J,\gamma )$, where J is an n-dimensional real Jordan algebra and γ is a non-degenerate trace form on J. Every graph immersion with parallel cubic form can be extended to an affine complete symmetric space covering the maximal connected component of zero in the set of quasi-regular elements in the algebra J. It is an improper affine hypersphere if and only if the corresponding Jordan algebra is nilpotent. In this case it is an affine complete, Euclidean complete graph immersion, with a polynomial as globally defining function. We classify all such hyperspheres up to dimension 5. As a special case we describe a connection between Cayley hypersurfaces and polynomial quotient algebras. Our algebraic approach can be used to study also other classes of hypersurfaces with parallel cubic form.

我们考虑将非退化图浸入仿射空间 ${\mathbb{一种}}^{ñ+\mathrm{1个}}$其立方形式相对于仿射度量的Levi-Civita连接是平行的。这样的图浸入和对之间存在对应关系$（Ĵ，\gamma ）$，其中J是n维实数Jordan代数，而γ是J上的非简并迹形式。可以将具有平行三次形式的每个图浸入扩展到一个仿射完全对称空间，该空间覆盖代数J中一组准正则元素的零的最大连接分量。。当且仅当相应的约旦代数是幂等的时，它才是不正确的仿射超球。在这种情况下，它是仿射完全，欧几里德完全图浸没，并且多项式作为全局定义函数。我们将所有此类超球面分类到维度5。作为一种特殊情况，我们描述了Cayley超曲面与多项式商代数之间的联系。我们的代数方法可用于研究具有平行立方形式的其他类别的超曲面。