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: On convexity of polynomial paths and generalized majorizations. | On convexity of polynomial paths and generalized majorizations Marija Dodigi Centro de Estruturas Lineares e Combinatorias CELC Universidade de Lisboa Av. Prof. Gama Pinto 2 1649-003 Lisboa Portugal dodig@ Marko Stosic Instituto de Sistemas e Robótica and CAMGSD Instituto Superior Tecnico Av. Rovisco Pais 1 1049-001 Lisbon Portugal mstosic@ Submitted Nov 15 2009 Accepted Apr 5 2010 Published Apr 19 2010 Mathematics Subject Classification 05A17 15A21 Abstract In this paper we give some useful combinatorial properties of polynomial paths. We also introduce generalized majorization between three sequences of integers and explore its combinatorics. In addition we give a new simple purely polynomial proof of the convexity lemma of E. M. de Sa and R. C. Thompson. All these results have applications in matrix completion theory. 1 Introduction and notation In this paper we prove some useful properties of polynomial paths and generalized majorization between three sequences of integers. All proofs are purely combinatorial and the presented results are used in matrix completion problems see . 2 4 7 10 11 . We study chains of monic polynomials and polynomial paths between them. Polynomial paths are combinatorial objects that are used in matrix completion problems see 7 9 11 . There is a certain convexity property of polynomial paths appeared for the first time in 5 . In Lemma 2 we give a simple direct polynomial proof of that result. We also show that no additional divisibility relations are needed. This work was done within the activities of CELC and was partially supported by FCT project ISFL-1-1431 and by the Ministry of Science of Serbia projects no. 144014 M. D. and 144032 M. S. . 1 Corresponding author. THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R61 1 Also we explore generalized majorization between three sequences of integers. It presents a natural generalization of a classical majorization in Hardy-Littlewood-Polya sense 6 and it .
