|
Mathematician Gian-Carlo Rota conjectured in 1971 that, for every finite field, the family of matroids that can be represented over that field has finitely many excluded minors.〔.〕 A proof of the conjecture has been announced by Geelen, Gerards, and Whittle. ==Statement of the conjecture== If is a set of points in a vector space defined over a field , then the linearly independent subsets of form the independent sets of a matroid ; is said to be a representation of any matroid isomorphic to . Not every matroid has a representation over every field, for instance, the Fano plane is representable only over fields of characteristic two. Other matroids are representable over no fields at all. The matroids that are representable over a particular field form a proper subclass of all matroids. A minor of a matroid is another matroid formed by a sequence of two operations: deletion and contraction. In the case of points from a vector space, deleting a point is simply the removal of that point from ; contraction is a dual operation in which a point is removed and the remaining points are projected a hyperplane that does not contain the removed points. It follows from this if a matroid is representable over a field, then so are all its minors. A matroid that is not representable over , and is minor-minimal with that property, is called an "excluded minor"; a matroid is representable over if and only if it does not contain one of the forbidden minors. For representability over the real numbers, there are infinitely many forbidden minors.〔.〕 Rota's conjecture is that, for every finite field , there is only a finite number of forbidden minors. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Rota's conjecture」の詳細全文を読む スポンサード リンク
|