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 .