In this article we explore an approach to the problem of Albert described by U. Umirbaev. We characterize the space of invariant bilinear forms of the polarization algebra of a polynomial endomorphism $F:{\mathfrak{K}}^n\to {\mathfrak{K}}^n$, where $\mathfrak{K}$ is a field with characteristic different from two. We obtain some conjectures expressed in the language of polynomial endomorphisms, which are equivalent to the existence of invariant bilinear forms in a finite-dimensional commutative algebra. We give a characterization of the space of invariant bilinear forms in terms of differential forms in the ring $\mathfrak{K}[x_1,\dots ,x_n]$. We also introduce a new kind of algebra, we call them totally symmetric algebras, and we establish the relationship between these algebras and the existence of invariant bilinear forms in any commutative algebra.

在本文中，我们探讨了U. Umirbaev描述的解决Albert问题的方法。我们表征多项式内同态的极化代数的不变双线性形式的空间$F：{\mathfrak{ķ}}^{ñ}\to {\mathfrak{ķ}}^{ñ}$，在哪里 $\mathfrak{ķ}$是一个具有两个不同特征的领域。我们得到了用多项式内同态语言表达的一些猜想，这些猜想等同于有限维可交换代数中不变双线性形式的存在。我们用环中的微分形式来描述不变双线性形式的空间$\mathfrak{ķ}[X_{\mathrm{1个}}，\dots ，X_{ñ}]$。我们还介绍了一种新型的代数，称它们为完全对称代数，并建立了这些代数与任何可交换代数中不变双线性形式的存在之间的关系。