Powers of sums and their homological invariants

Let R and S be standard graded algebras over a field k, and $I?R$ and $J?S$ homogeneous ideals. Denote by P the sum of the extensions of I and J to $R{?}_{k}S$. We investigate several important homological invariants of powers of P based on the information about I and J, with focus on finding the exact formulas for these invariants. Our investigation exploits certain Tor vanishing property of natural inclusion maps between consecutive powers of I and J. As a consequence, we provide fairly complete information about the depth and regularity of powers of P given that R and S are polynomial rings and either $\mathrm{char}k=0$ or I and J are generated by monomials.     

On the regularity of binomial edge ideals

We study the regularity of binomial edge ideals. For a closed graph G we show that the regularity of the binomial edge ideal coincides with the regularity of and can be expressed in terms of the combinatorial data of G. In addition, we give positive answers to Matsuda‐Murai conjecture 8 for some classes of graphs.     

The validity of the strong perfect-graph conjecture for (K4−e)-free graphs

Berge's strong perfect-graph conjecture states that a graph is perfect iff it has neither C_2n+1 nor $\text{C}{}_{\mathrm{2n+1}}$, n ≥ 2 as an induced subgraph. In this note we establish the validity of this conjecture for (K₄?e)-free graphs.     

A residual transcendental extension of a valuation on K to K(x1, … , xn)

Let v be a valuation of a field K, G_v its value group and k_v its residue field. Let w be an extension of v to K(x₁, … , x_n). w is called a residual transcendental extension of v if k_w/k_v is a transcendental extension. In this study a residual transcendental extension w of v to K(x₁, … , x_n) such that transdegk_w/k_v = n is defined and some considerations related with this valuation are given.     

On the Posets ( \user1Wk2 , < ){\left( {{\user1{\mathcal{W}}}^{k}_{2} , < } \right)} and their Connections with Some Homogeneous Inequalities of Degree 2

A class of ranked posets {(D _h ^k , ≪)} has been recently defined in order to analyse, from a combinatorial viewpoint, particular systems of real homogeneous inequalities between monomials. In the present paper we focus on the posets D ₂ ^k , which are related to systems of the form {x _a x _b * _abcd x _c x _d : 0 ≤ a, b, c, d ≤ k, * _abcd ∈ {<, >}, 0 < x ₀ < x ₁ < ...< x _k}. As a consequence of the general theory, the logical dependency among inequalities is adequately captured by the so-defined posets . These structures, whose elements are all the D ₂ ^k 's incomparable pairs, are thoroughly surveyed in the following pages. In particular, their order ideals – crucially significant in connection with logical consequence – are characterised in a rather simple way. In the second part of the paper, a class of antichains is shown to enjoy some arithmetical properties which make it an efficient tool for detecting incompatible systems, as well as for posing some compatibility questions in a purely combinatorial fashion.