In order to better understand the structure of classical rings of invariants for binary forms, Dixmier proposed, as a conjectural homogeneous system of parameters, an explicit collection of invariants previously studied by Hilbert. We generalize Dixmier’s collection and show that a particular subfamily is algebraically independent. Our proof relies on showing certain alternating sums of products of binomial coefficients are nonzero. Along the way we provide a very elementary proof à la Racah, namely, only using the Chu–Vandermonde Theorem, for Dixon’s Summation Theorem. We also provide explicit computations of invariants, for the binary octavic, which can serve as ideal introductory examples to Gordan’s 1868 method in classical invariant theory.

中文翻译：

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与Dixmier猜想关于二元形式不变量的代数独立性结果

为了更好地理解二进制形式的不变式经典环的结构，Dixmier提出了一个希尔伯特先前研究的不变式的显式集合，作为参数的猜想齐次系统。我们对Dixmier的集合进行了概括，并证明了一个特定的子族是代数独立的。我们的证明依赖于证明二项式系数的乘积的某些交替和为非零。在此过程中，我们为狄克逊的求和定理提供了一个非常基本的àla Racah证明，即仅使用Chu-Vandermonde定理。我们还提供了二进制八进制的不变量的显式计算，可以作为经典不变理论中Gordan 1868年方法的理想入门实例。