arXiv subjects

Ştefan O. Tohǎneanu

Publications and source records attributed to Ştefan O. Tohǎneanu.

3 recordsLinked to original sources

arXiv2026

Associated primes, witnesses, and omega invariants of monomial ideals

We introduce and study the omega invariant of a proper ideal in a Noetherian commutative ring, defined as the number of associated primes of the ideal. Our main objective is to investigate this invariant for monomial ideals and their powers. We characterize associated primes through monomial witnesses and provide an algorithmic procedure for constructing such witnesses from the exponent vectors of the minimal generators. These results lead to explicit formulas and bounds for the omega invariant without requiring the computation of a primary decomposition. We further establish alternative descriptions using irreducible decompositions and Alexander duality. A matrix-based approach is developed to detect associated primes of powers of monomial ideals directly from the exponent matrix of the original ideal. We also investigate the behavior of witnesses under passage from $I^n$ to $I^{n+1}$ and derive corresponding results for edge ideals of graphs.

math.AC↗

arXiv2026

Generalized Hamming weights and symbolic powers of Stanley-Reisner ideals of matroids

It is well-known that the first generalized Hamming weight of a linear code, more commonly called \textit{the minimum distance} of the linear code, corresponds to the initial degree of the Stanley-Reisner ideal of the matroid of the dual code. Our starting point in this paper is a generalization of this fact -- namely, the $r$-th generalized Hamming weight of a matroid is the smallest degree of a squarefree monomial in the $r$-th symbolic power of the Stanley-Reisner ideal of the matroid (in the appropriate range for $r$). We show that the squarefree monomials in successive symbolic powers of the Stanley-Reisner ideal of a matroid suffice to describe all symbolic powers of the Stanley-Reisner ideal. Hence, we provide explicit expressions for initial degree statistics of symbolic powers of the Stanley-Reisner ideal of a matroid in terms of its generalized Hamming weights. A key aspect of our approach is a careful study of duality. If the generalized Hamming weights of a matroid and its dual are both subadditive, we prove a simple expression for the initial degree of every symbolic power of the Stanley-Reisner ideal of the matroid, which closely mirrors that of a uniform matroid. This has unexpectedly far-reaching consequences - we prove the generalized Hamming weights of a matroid and its dual are both subadditive for many interesting classes of matroids and codes, including sparse paving matroids, perfect matroid designs, matroids arising from Steiner systems, first-order affine and projective Reed-Muller codes, constant weight codes, Griesmer codes, and perfect codes. As an application, we study the resurgence and asymptotic resurgence of the matroid configurations introduced by Geramita-Harbourne-Migliore-Nagel. In particular, we explicitly compute the asymptotic resurgence of a matroid configuration of points arising from a perfect matroid design.

math.AC↗

arXiv2020

Ideals generated by $a$-fold products of linear forms have linear graded free resolution

Given $Σ\subset R:=\mathbb K[x_1,\ldots,x_k]$, where $\mathbb K$ is a field of characteristic 0, any finite collection of linear forms, some possibly proportional, and any $1\leq a\leq |Σ|$, we prove that $I_a(Σ)$, the ideal generated by all $a$-fold products of $Σ$, has linear graded free resolution. This allows us to determine a generating set for the defining ideal of the Orlik-Terao algebra of the second order of a line arrangement in $\mathbb P_{\mathbb{K}}^2$, and to conclude that for the case $k=3$, and $Σ$ defining such a line arrangement, the ideal $I_{|Σ|-2}(Σ)$ is of fiber type. We also prove several conjectures of symbolic powers for defining ideals of star configurations of any codimension $c$.

math.AC↗