arXiv subjects
Publications and source records attributed to Mesut Şahin.
We comprehensively study weighted projective Reed-Muller (WPRM) codes on weighted projective planes $\mathbb{P}(1,a,b)$. We provide the universal Gröbner basis for the vanishing ideal of the set $Y$ of $\mathbb{F}_q$--rational points of $\mathbb{P}(1,a,b)$ to get the dimension of the code. We determine the regularity set of $Y$ using a novel combinatorial approach. We employ footprint techniques to compute the minimum distance.
Motivated by applications to the theory of error-correcting codes, we give methods for computing a generating set for the ideal generated by $β$-graded polynomials vanishing on certain subsets of a simplicial complete toric variety $X$ over a finite field $\mathbb{F}_q$, where $β$ is a $d\times r$ matrix whose columns generate a subsemigroup $\mathbb{N}β$ of $\mathbb{N}^d$. We also give a method for computing the vanishing ideal of the set of $\mathbb{F}_q$-rational points of $X$. When $β=[w_1 \cdots w_r]$ is a row matrix corresponding to a numerical semigroup $\mathbb{N}β=\langle w_1,\dots,w_r \rangle$, $X$ is a weighted projective space and generators of the relevant vanishing ideal is given using generators of defining (toric) ideals of numerical semigroup rings corresponding to semigroups generated by subsets of $\{w_1,\dots,w_r\}$.
We obtain certain algebraic invariants relevant to study codes on subgroups of weighted projective tori inside an $n$-dimensional weighted projective space. As application, we compute all the main parameters of generalized toric codes on these subgroups of tori lying inside a weighted projective plane of the form $\Pp(1,1,a)$.
Weighted projective spaces are natural generalizations of projective spaces with a rich structure. Projective Reed-Muller codes are error-correcting codes that played an important role in reliably transmitting information on digital communication channels. In this case study, we explore the power of commutative and homological algebraic techniques to study weighted projective Reed-Muller (WPRM) codes on weighted projective spaces of the form $\mathbb{P}(1,1,a)$. We compute minimal free resolutions and thereby obtain Hilbert series for the vanishing ideal of the $\mathbb{F}_q$-rational points, and compute main parameters for these codes.
We show that the number of rational points of a subgroup inside a toric variety over a finite field defined by a homogeneous lattice ideal can be computed via Smith normal form of the matrix whose columns constitute a basis of the lattice. This generalizes and yields a concise toric geometric proof of the same fact proven purely algebraically by Lopez and Villarreal for the case of a projective space and a standard homogeneous lattice ideal of dimension one. We also prove a Nullstellensatz type theorem over a finite field establishing a one to one correspondence between subgroups of the dense split torus and certain homogeneous lattice ideals. As application, we compute the main parameters of generalized toric codes on subgroups of the torus of Hirzebruch surfaces, generalizing the existing literature.
Let $X$ be a complete simplicial toric variety over a finite field with a split torus $T_X$. For any matrix $Q$, we are interested in the subgroup $Y_Q$ of $T_X$ parameterized by the columns of $Q$. We give an algorithm for obtaining a basis for the unique lattice $L$ whose lattice ideal $I_L$ is $I(Y_Q)$. We also give two direct algorithmic methods to compute the order of $Y_Q$, which is the length of the corresponding code ${\cC}_{å,Y_Q}$. We share procedures implementing them in \verb|Macaulay2|. Finally, we give a lower bound for the minimum distance of ${\cC}_{å,Y_Q}$, taking advantage of the parametric description of the subgroup $Y_Q$. As an application, we compute the main parameters of the toric codes on Hirzebruch surfaces $\cl H_{\ell}$ generalizing the corresponding result given by Hansen.
Let $X$ be a complete simplicial toric variety over a finite field $\mathbb{F}_q$ with homogeneous coordinate ring $S=\mathbb{F}_q[x_1,\dots,x_r]$ and split torus $T_X\cong (\mathbb{F}^*_q)^n$. We prove that vanishing ideal of a subset $Y$ of the torus $T_X$ is a lattice ideal if and only if $Y$ is a subgroup. We show that these subgroups are exactly those subsets that are parameterized by Laurents monomials. We give an algorithm for determining this parametrization if the subgroup is the zero locus of a lattice ideal in the torus. We also show that vanishing ideals of subgroups of $T_X$ are radical homogeneous lattice ideals of dimension $r-n$. We identify the lattice corresponding to a degenerate torus in $X$ and completely characterize when its lattice ideal is a complete intersection. We compute dimension and length of some generalized toric codes defined on these degenerate tori.
We study an operation, that we call lifting, creating non-isomorphic monomial curves from a single monomial curve. Our main result says that all but finitely many liftings of a monomial curve have Cohen-Macaulay tangent cones even if the tangent cone of the original curve is not Cohen-Macaulay. This implies that the Betti sequence of the tangent cone is eventually constant under this operation. Moreover, all liftings have Cohen-Macaulay tangent cones when the original monomial curve has a Cohen-Macaulay tangent cone. In this case, all the Betti sequences are nothing but the Betti sequence of the original curve.
In this article, we compute Betti numbers for a Cohen-Macaulay tangent cone of a monomial curve in the affine $4$-space corresponding to a pseudo symmetric numerical semigroup. As a byproduct, we also show that for these semigroups, being of homogeneous type and homogeneous are equivalent properties.
In this article, we study monomial curves, toric ideals and monomial algebras associated to $4$-generated pseudo symmetric numerical semigroups. Namely, we determine indispensable binomials of these toric ideals, give a characterization for these monomial algebras to have strongly indispensable minimal graded free resolutions. We also characterize when the tangent cones of these monomial curves at the origin are Cohen-Macaulay.
We study strong indispensability of minimal free resolutions of semigroup rings. We focus on two operations, gluing and extending, used in literature to produce more examples with a special property from the existing ones. We give a naive condition to determine whether gluing of two semigroup rings has a strongly indispensable minimal free resolution. As applications, we determine extensions of $3$-generated non-symmetric, $4$-generated symmetric and pseudo symmetric numerical semigroups as well as obtain infinitely many complete intersection semigroups of any embedding dimension, having strongly indispensable minimal free resolutions.
Based on the fact that projective monomial curves in the plane are complete intersections, we give an effective inductive method for creating infinitely many monomial curves in the projective $n$-space that are set theoretic complete intersections. We illustrate our main result by giving different infinite families of examples. Our proof is constructive and provides one binomial and $(n-2)$ polynomial explicit equations for the hypersurfaces cutting out the curve in question.
In this paper, we study a class of toric ideals obtained by using some geometric data of ADE trees which are the minimal resolution graphs of rational surface singularities. We compute explicit Gröbner bases for these toric ideals that are also minimal generating sets consisting of large number of binomials of degree $\leq 4$. In particular, they give rise to squarefree initial ideals as well.
Let $X$ be a complete $n$-dimensional simplicial toric variety with homogeneous coordinate ring $S$. We study the multigraded Hilbert function $H_Y$ of reduced $0$-dimensional subschemes $Y$ in $X$. We provide explicit formulas and prove non-decreasing and stabilization properties of $H_Y$ when $Y$ is a $0$-dimensional complete intersection in $X$. We apply our results to computing the dimension of some evaluation codes on $0$-dimensional complete intersection in simplicial toric varieties.
In this article, by using the technique of gluing semigroups, we give infinitely many families of 1-dimensional local rings with non-decreasing Hilbert functions. More significantly, these are local rings whose associated graded rings are not necessarily Cohen-Macaulay. In this sense, we give an effective technique to construct large families of 1-dimensional Gorenstein local rings associated to monomial curves, which support Rossi's conjecture saying that every Gorenstein local ring has non-decreasing Hilbert function.