Current Trends on Monomial and Binomial Ideals

• Format: eBook
• Language: English
• Published: MDPI - Multidisciplinary Digital Publishing Institute 2020
• Subjects: (S2) condition; algebras with straightening laws; associated graded rings; Betti number; bipartite graph; Bipartite graphs; Castelnuovo-Mumford regularity; chain polytope; circuit; circulant graphs; Cohen Macaulay; Cohen-Macaulay; colon ideals; complete intersection; cover ideal; depth; depth of powers of bipartite graphs; directed cycle; distribuive lattice; dstab; edge ideal; edge ideals; edge polytope; edge ring; even cycle; flawless; graph; graphs; Gröbner bases; h-vector; integral closure; linear part; matching number; monomial ideal; multipartite graph; O-sequence; order and chain polytopes; order polytope; partially ordered set; polymatroidal ideal; projective dimension; Rees algebra; regular elements on powers of bipartite graphs; regularity; stable set polytope; stable set polytopes; Stanley depth; Stanley-Reisner ideal; Stanley-Reisner ring; Stanley's inequality; symbolic power; syzygy; toric ideals; toric ring
• Online Access: Open Access: DOAB: description of the publication; Open Access: DOAB, download the publication

 Historically, the study of monomial ideals became fashionable after the pioneering work by Richard Stanley in 1975 on the upper bound conjecture for spheres. On the other hand, since the early 1990s, under the strong influence of Gröbner bases, binomial ideals became gradually fashionable in commutative algebra. The last ten years have seen a surge of research work in the study of monomial and binomial ideals. Remarkable developments in, for example, finite free resolutions, syzygies, Hilbert functions, toric rings, as well as cohomological invariants of ordinary powers, and symbolic powers of monomial and binomial ideals, have been brought forward. The theory of monomial and binomial ideals has many benefits from combinatorics and Göbner bases. Simultaneously, monomial and binomial ideals have created new and exciting aspects of combinatorics and Göbner bases. In the present Special Issue, particular attention was paid to monomial and binomial ideals arising from combinatorial objects including finite graphs, simplicial complexes, lattice polytopes, and finite partially ordered sets, because there is a rich and intimate relationship between algebraic properties and invariants of these classes of ideals and the combinatorial structures of their combinatorial counterparts. This volume gives a brief summary of recent achievements in this area of research. It will stimulate further research that encourages breakthroughs in the theory of monomial and binomial ideals. This volume provides graduate students with fundamental materials in this research area. Furthermore, it will help researchers find exciting activities and avenues for further exploration of monomial and binomial ideals. The editors express our thanks to the contributors to the Special Issue. Funds for APC (article processing charge) were partially supported by JSPS (Japan Society for the Promotion of Science) Grants-in-Aid for Scientific Research (S) entitled ""The Birth of Modern Trends on Commutative Algebra and Convex Polytopes with Statistical and Computational Strategies"" (JP 26220701). The publication of this volume is one of the main activities of the grant.