| Contents | Determining the tensor rank and Waring rank is a central problem in algebraic complexity theory. Recently, machine learning approaches, such as DeepMind's AlphaTensor and AlphaEvolve, have shown promise in finding tighter upper bounds for these ranks. In this study, we introduce a numerical approach framework based on sparse optimization and neural networks for improving the tensor rank and Waring rank of several polynomial tensors. Specifically, in the case of determinants, we prove that the rank of a four-by-four determinant is at most 12, and a five-by-five determinant is at most 42, by finding their exact new formulas over fields with a characteristic not equal to two. Furthermore, for an elementary symmetric polynomial of an even degree with multiple variables, we prove its exact Waring rank can be fully determined through a specific combinatorial calculation using binomial coefficients related to the number of variables and the degree.
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