Linear rank inequalities on five or more variables

Ranks of subspaces of vector spaces satisfy all linear inequalities satisfied by entropies (including the standard Shannon inequalities) and an additional inequality due to Ingleton. It is known that the Shannon and Ingleton inequalities generate all such linear rank inequalities on up to four variables, but it has been an open question whether additional inequalities hold for the case of five or more variables. Here we give a list of 24 inequalities which, together with the Shannon and Ingleton inequalities, generate all linear rank inequalities on five variables. We also give a partial list of linear rank inequalities on six variables and general results which produce such inequalities on an arbitrary number of variables; we prove that there are essentially new inequalities at each number of variables beyond four (a result also proved recently by Kinser). This work was supported by the Institute for Defense Analyses and the National Science Foundation. R. Dougherty is with the Center for Communications Research, 4320 Westerra Court, San Diego, CA 92121-1969 ([email protected]). C. Freiling is with the Department of Mathematics, California State University, San Bernardino, 5500 University Parkway, San Bernardino, CA 92407-2397 ([email protected]). K. Zeger is with the Department of Electrical and Computer Engineering, University of California, San Diego, La Jolla, CA 92093-0407 ([email protected]). Dougherty-Freiling-Zeger August 13, 2010