tailieunhanh - Báo cáo toán học: "The toric ideal of a matroid of rank 3 is generated by quadrics"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: The toric ideal of a matroid of rank 3 is generated by quadrics. | The toric ideal of a matroid of rank 3 is generated by quadrics Kenji Kashiwabara Department of General Systems Studies University of Tokyo 3-8-1 Komaba Meguroku Tokyo 153-8902 Japan. kashiwa@. .j p Submitted Aug 27 2008 Accepted Feb 1 2010 Published Feb 15 2010 Mathematics Subject Classification 52B40 Abstract White conjectured that the toric ideal associated with the basis of a matroid is generated by quadrics corresponding to symmetric exchanges. We present a combinatorial proof of White s conjecture for matroids of rank 3 by using a lemma proposed by Blasiak. 1 Introduction The bases of a matroid have many good properties. Combinatorial optimization problems among them can be effectively solved. In this paper we consider the conjecture about the bases of a matroid proposed by White 6 . While this conjecture has occasionally been stated in terms of algebraic expressions it is closely related to combinatorial problems. Our proof adopts a combinatorial approach. A matroid has several equivalent definitions. We define a matroid by a set of subsets that satisfies the exchange axiom. A family B of sets is the collection of bases of a matroid if it satisfies the exchange axiom given below. E For any X and Y in B for every a G X there exists b G Y such that X u b a is in B. An element of B is called a base. The exchange axiom is equivalent to the following stronger axiom known as the symmetric exchange axiom. SE For any X and Y in B for every a G X there exists b G Y such that Xu b a and Y u a b are in B. The pair X u b a and Y u a b of bases is said to be obtained from the pair X Y of bases by a symmetric exchange. THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R28 1 Let M be a matroid on a ground set E 1 2 . n . For each base B of M we consider a variable yB. Let Sm be the polynomial ring K yB B is a base of M where K is a field. Let Im be the kernel of the K-algebra homomorphism ỠM SM K xi . xn such that yB is sent to nxeBxi. IM is a toric ideal .