arXiv subjects

Stefan O. Tohaneanu

Publications and source records attributed to Stefan O. Tohaneanu.

13 recordsLinked to original sources

arXiv2024

Generalized Hamming weights and minimal shifts of Orlik-Terao algebras

In this note we show that the minimum distance of a linear code equals one plus the smallest shift in the first step of the minimal graded free resolution of the Orlik-Terao algebra (i.e., the initial degree of the Orlik-Tearo ideal) constructed from any parity-check matrix of the linear code. We move forward with this connection and we prove that the second generalized Hamming weight equals one or two plus the smallest shift at second step in the minimal graded free resolution of the same algebra. Via a couple of examples we show that this ambivalence is the best result one can get if one uses Orlik-Terao algebras to characterize the second generalized Hamming weight.

cs.IT↗

arXiv2022

Free arrangements with low exponents

In this article we show that any free hyperplane arrangement with exponents 1's and 2's is a supersolvable arrangement. We conjecture that any free arrangement with exponents 1's, 2's and exactly one 3, is also supersolvable, and we show this conjecture for hyperplane arrangements of ranks 4 and 5, and for inductively free arrangements of any rank.

math.CO↗

arXiv2019

Generalized minimum distance functions and algebraic invariants of Geramita ideals

Motivated by notions from coding theory, we study the generalized minimum distance (GMD) function $δ_I(d,r)$ of a graded ideal $I$ in a polynomial ring over an arbitrary field using commutative algebraic methods. It is shown that $δ_I$ is non-decreasing as a function of $r$ and non-increasing as a function of $d$. For vanishing ideals over finite fields, we show that $δ_I$ is strictly decreasing as a function of $d$ until it stabilizes. We also study algebraic invariants of Geramita ideals. Those ideals are graded, unmixed, $1$-dimensional and their associated primes are generated by linear forms. We also examine GMD functions of complete intersections and show some special cases of two conjectures of Tohăneanu--Van Tuyl and Eisenbud-Green-Harris.

math.AC↗

arXiv2019

On some ideals with linear free resolutions

Given $Σ\subset\mathbb K[x_1,\ldots,x_k]$, any finite collection of linear forms, some possibly proportional, and any $1\leq a\leq |Σ|$, it has been conjectured that $I_a(Σ)$, the ideal generated by all $a$-fold products of $Σ$, has linear graded free resolution. In this article we show the validity of this conjecture for two cases: the first one is when $a=d+1$ and $Σ$ is dual to the columns of a generating matrix of a linear code of minimum distance $d$; and the second one is when $k=3$ and $Σ$ defines a line arrangement in $\mathbb P^2$ (i.e., there are no proportional linear forms). For the second case we investigate what are the graded betti numbers of $I_a(Σ)$.

math.AC↗

arXiv2018

Equivalent matrices up to permutations

Given two $k\times n$ matrices $A$ and $B$, we describe a couple of methods to solve the matrix equation $XA=BY$, where $X$ is an invertible $k\times k$ matrix, and $Y$ is an $n\times n$ permutation matrix, both of which we want to determine. We are interested in pursuing those techniques that have algebraic geometric flavor. An application to solving such a matrix equation comes from the cryptanalysis of McEliece cryptosystem. By using codewords of minimum weight of a linear code, in concordance with these methods of solving $XA=BY$, we present an efficient way to determine the entire encryption keys for the McEliece cryptosystems built on Reed-Solomon codes.

math.CO↗

arXiv2013

Finding Inverse Systems from Coordinates

Let $I$ be a homogeneous ideal in $R=\mathbb K[x_0,\ldots,x_n]$, such that $R/I$ is an Artinian Gorenstein ring. A famous theorem of Macaulay says that in this instance $I$ is the ideal of polynomial differential operators with constant coefficients that cancel the same homogeneous polynomial $F$. A major question related to this result is to be able to describe $F$ in terms of the ideal $I$. In this note we give a partial answer to this question, by analyzing the case when $I$ is the Artinian reduction of the ideal of a reduced (arithmetically) Gorenstein zero-dimensional scheme $Γ\subset\mathbb P^n$. We obtain $F$ from the coordinates of the points of $Γ$.

math.AC↗

arXiv2012

On freeness of divisors on $\mathbb P^2$

Let $I\subset \mathbb C[x,y,z]$ be an ideal of height 2 and minimally generated by three homogeneous polynomials of the same degree. If $I$ is a locally complete intersection we give a criterion for $\mathbb C[x,y,z]/I$ to be arithmetically Cohen-Macaulay. Since the setup above is most commonly used when $I=J_F$ is the Jacobian ideal of the defining polynomial of a "quasihomogeneous" reduced curve $Y=V(F)$ in $\mathbb P^2$, our main result becomes a criterion for freeness of such divisors. As an application we give an upper bound for the degree of the reduced Jacobian scheme when $Y$ is a free rank 3 central essential arrangement, as well as we investigate the connections between the first syzygies on $J_F$, and the generators of $\sqrt{J_F}$.

math.AC↗

arXiv2012

From Spline Approximation to Roth's Equation and Schur Functors

Alfeld and Schumaker provide a formula for the dimension of the space of piecewise polynomial functions, called splines, of degree $d$ and smoothness $r$ on a generic triangulation of a planar simplicial complex $Δ$, for $d \geq 3r+1$. Schenck and Stiller conjectured that this formula actually holds for all $d \geq 2r+1$. Up to this moment there was not known a single example where one could show that the bound $d\geq 2r +1$ is sharp. However, in 2005, a possible such example was constructed to show that this bound is the best possible (i.e., the Alfeld-Schumaker formula does not hold if $d=2r$), except that the proof that this formula actually works if $d\geq 2r+1$ has been a challenge until now when we finally show it to be true. The interesting subtle connections with representation theory, matrix theory and commutative and homological algebra seem to explain why this example presented such a challenge. Thus in this paper we present the first example when it is known that the bound $d\geq 2r+1$ is sharp for asserting the validity of the Alfeld-Schumaker formula.

math.NA↗

arXiv2012

Homology of Homogeneous Divisors

One deals with arbitrary reduced free divisors in a polynomial ring over a field of characteristic zero, by stressing the ideal theoretic and homological behavior of the corresponding singular locus. A particular emphasis is given to both weighted homogeneous and homogeneous polynomials, allowing to introduce new families of free divisors which do not come from hyperplane arrangements nor as explicit discriminants from singularity theory.

math.AC↗

arXiv2012

A commutative algebraic approach to the fitting problem

Given a finite set of points $Γ$ in $\mathbb P^{k-1}$ not all contained in a hyperplane, the "fitting problem" asks what is the maximum number $hyp(Γ)$ of these points that can fit in some hyperplane and what is (are) the equation(s) of such hyperplane(s). If $Γ$ has the property that any $k-1$ of its points span a hyperplane, then $hyp(Γ)=nil(I)+k-2$, where $nil(I)$ is the index of nilpotency of an ideal constructed from the homogeneous coordinates of the points of $Γ$. Note that in $\mathbb P^2$ any two points span a line, and we find that the maximum number of collinear points of any given set of points $Γ\subset\mathbb P^2$ equals the index of nilpotency of the corresponding ideal, plus one.

math.AC↗

arXiv2012

Bounding invariants of fat points using a coding theory construction

Let $Z \subseteq \proj{n}$ be a fat points scheme, and let $d(Z)$ be the minimum distance of the linear code constructed from $Z$. We show that $d(Z)$ imposes constraints (i.e., upper bounds) on some specific shifts in the graded minimal free resolution of $I_Z$, the defining ideal of $Z$. We investigate this relation in the case that the support of $Z$ is a complete intersection; when $Z$ is reduced and a complete intersection we give lower bounds for $d(Z)$ that improve upon known bounds.

math.AC↗

arXiv2012

The minimum distance of sets of points and the minimum socle degree

Let $\mathbb K$ be a field of characteristic 0. Let $Γ\subset\mathbb P^n_{\mathbb K}$ be a reduced finite set of points, not all contained in a hyperplane. Let $hyp(Γ)$ be the maximum number of points of $Γ$ contained in any hyperplane, and let $d(Γ)=|Γ|-hyp(Γ)$. If $I\subset R=\mathbb K[x_0,...,x_n]$ is the ideal of $Γ$, then in \cite{t1} it is shown that for $n=2,3$, $d(Γ)$ has a lower bound expressed in terms of some shift in the graded minimal free resolution of $R/I$. In these notes we show that this behavior is true in general, for any $n\geq 2$: $d(Γ)\geq A_n$, where $A_n=\min\{a_i-n\}$ and $\oplus_i R(-a_i)$ is the last module in the graded minimal free resolution of $R/I$. In the end we also prove that this bound is sharp for a whole class of examples due to Juan Migliore (\cite{m}).

math.AC↗

arXiv2007

Freeness of Conic-Line Arrangements in $\mathbb P^2$

Let ${\mathcal C}= \bigcup_{i=1}^n C_i \subseteq \mathbb{P}^2$ be a collection of smooth rational plane curves. We prove that the addition-deletion operation used in the study of hyperplane arrangements has an extension which works for a large class of arrangements of smooth rational curves, giving an inductive tool for understanding the freeness of the module $Ω^1({\mathcal C})$ of logarithmic differential forms with pole along ${\mathcal C}$. We also show that the analog of Terao's conjecture (freeness of $Ω^1({\mathcal C})$ is combinatorially determined if ${\mathcal C}$ is a union of lines) is false in this setting.

math.AC↗