An ideal of polynomials is symmetric if it is closed under permutations of variables. We relate general symmetric ideals to the so called Specht ideals generated by all Specht polynomials of a given shape. We show a connection between the leading monomials of polynomials in the ideal and the Specht polynomials contained in the ideal. This provides applications in several contexts. Most notably, this connection gives information about the solutions of the corresponding set of equations. From another perspective, it restricts the isotypic decomposition of the ideal viewed as a representation of the symmetric group.

中文翻译：

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对称理想，Specht多项式和对称方程组的解

如果多项式的理想在变量的排列下是封闭的，则它是对称的。我们将一般对称理想与给定形状的所有Specht多项式生成的所谓Specht理想相关联。我们展示了理想中多项式的主要多项式与理想中包含的Specht多项式之间的联系。这在多种情况下提供了应用程序。最值得注意的是，该连接提供了有关相应方程组解的信息。从另一个角度看，它限制了被视为对称基团表示形式的理想分子的同型分解。