Monomial Ideals and Planar Graphs

Ezra Miller and Bernd Sturmfels Department of Mathemati s, University of California, Berkeley CA 94720, USA enmiller math.berkeley.edu, bernd math.berkeley.edu Abstra t. Gr obner basis theory redu es questions about systems of polynomial equations to the ombinatorial study of monomial ideals, or stair ases. This arti le gives an elementary introdu tion to urrent resear h in this area. After reviewing the bivariate ase, a new orresponden e is established between planar graphs and minimal resolutions of monomial ideals in three variables. A brief guide is given to the literature on omplexity issues and monomial ideals in four or more variables. 1 Introdu tion A monomial ideal M is an ideal generated by monomials xi1 1 xi2 2 xin n in a polynomial ring K[x1 ; x2; : : : ; xn℄. Monomial ideals are ubiquitous in the study of Gr obner bases. For instan e, if I = hx4 + y4 1; x7 + y7 2i then its initial ideal with respe t to the total degree term order equalsM = hx4; x3y4; xy7; y10i. The ideal I has 28 distin t omplex roots, orresponding to the 28 monomials xiyj not in M , that is, to the 28 latti e points under the stair ase depi ting M :