The Multiplicity Conjecture is a deep problem relating the multiplicity (or degree) of a Cohen-Macaulay standard graded algebra with certain extremal graded Betti numbers in its minimal free resolution. In the case of level algebras of codimension three, Zanello has proposed a stronger conjecture. We prove this conjecture for the case of codimension three graded Gorenstein algebras.

Let k be a field, L ⊂ Zn be a lattice such that L∩Nn = {0}, and IL ⊂ k[x1, . . . , xn] the corresponding lattice ideal. We present the generalized Scarf complex of IL and show that it is indispensable in the sense that it is contained in every minimal free resolution of R/IL.

An irreducibly odd graph is a graph such that each vertex has odd degree and for every pair of vertices, a third vertex in the graph is adjacent to exactly one of the pair. This family of graphs was introduced recently by Manturov [1] in relation to free knots. In this paper, we show that every graph is the induced subgraph of an irreducibly odd graph. Furthermore, we prove that irreducibly odd...

In the theory of monomial ideals of a polynomial ring S over a field k, it is convenient that for each such ideal I there is a standard free resolution, so called Taylor resolution, that can be canonically constructed from the minimal system of monomial generators of I (see [7], p.439 and section 2). On the other hand no construction of a minimal resolution for an arbitrary monomial ideal has b...

The search for possible formalizations of logical calculi that are well suited for computer implementations is in the core of the methodological studies in the area of Automated Reasoning (cf. [1], [4], [5]). The theory of resolution proof systems developed in [4] provides a formal framework for the construction and study of the deductive proof systems based on the resolution principle. Resolut...

We survey some recent results on the minimal graded free resolution of a square-free monomial ideal. The theme uniting these results is the point-of-view that the generators of a monomial ideal correspond to the maximal faces (the facets) of a simplicial complex ∆. This correspondence gives us a new method, distinct from the Stanley-Reisner correspondence, to associate to a square-free monomial...

we define minimal cn-dominating graph $mathbf {mcn}(g)$, commonality minimal cn-dominating graph $mathbf {cmcn}(g)$ and vertex minimal cn-dominating graph $mathbf {m_{v}cn}(g)$, characterizations are given for graph $g$ for which the newly defined graphs are connected. further serval new results are developed relating to these graphs.