Reducible varieties and valuated matroids

一个射影代数集X在射影n空间中是不可约的，如果不存在X分解为两个严格较小的射影代数集的并。在实际应用中，通常不容易确定一个射影代数集是否不可约。本课题的目的是证明一个猜想，该猜想的目的是说一个射影代数集的信息是可约的，它总是在与定义我们的射影代数集的理想的d次部分相对应的d次的赋值矩阵中找到。对于四种不同的射影代数集，我们已经得到了这个猜想的一个显式结果，下一步是检验我们是否能得出一个更一般的证明。由于这是一个数学项目，它属于工程和物理科学研究委员会的职权范围，并且没有官方的外部合作伙伴参与这个项目。

A projective algebraic set X in the projective n-space is irreducible if there does not exist a decomposition of X as a union of two strictly smaller projective algebraic sets. In practice, it is not usually easy to determine whether a projective algebraic set is irreducible or not. The aim of this project is to prove the conjecture which aims to say that the information of a projective algebraic set being reducible is always found in the valuated matroid of some degree d corresponding to the degree-d part of the ideal defining our projective algebraic set. We have an explicit result of this conjecture for four different families of projective algebraic sets, and the next step is to check to see if we can work out a proof in more generality. As this is a project in mathematics, it falls under the remit of the Engineering and Physical Sciences Research Council, and there are no official external partners for this project.