On some invariants in numerical semigroups and estimations of the order bound

We study suitable parameters and relations in a numerical semigroup S. When S is the Weierstrass semigroup at a rational point P of a projective curve C, we evaluate the Feng-Rao order bound of the associated family of Goppa codes. Further we conjecture that the order bound is always greater than a fixed value easily deduced from the parameters of the semigroup: we also prove this inequality in several cases

On semigroup rings with decreasing Hilbert function

In this paper we study the Hilbert function HR of one-dimensional semigroup rings R = k[[S]]. For some classes of semigroups, by means of the notion of support of the elements in S, we give conditions on the generators of S in order to have decreasing HR. When the embedding dimension v and the multiplicity e verify v + 3 ? e ? v + 4, the decrease of HR gives explicit description of the Apery set of S. In particular for e = v+3, we classify the semigroups with e = 13 and HR decreasing, further we show that HR is non-decreasing if e < 12. Finally we deduce that HR is non-decreasing for every Gorenstein semigroup ring with e ? v + 4

One-dimensional Gorenstein local rings with decreasing Hilbert function

In this paper we solve a problem posed by M.E. Rossi: {\it Is the Hilbert function of a Gorenstein local ring of dimension one not decreasing? } More precisely, for any integer $h>1$, $h \notin\{14+22k, \, 35+46k \ | \ k\in\mathbb{N} \}$, we construct infinitely many one-dimensional Gorenstein local rings, included integral domains, reduced and non-reduced rings, whose Hilbert function decreases at level$h$; moreover we prove that there are no bounds to the decrease of the Hilbert function. The key tools are numerical semigroup theory, especially some necessary conditions to obtain decreasing Hilbert functions found by the first and the third author, and a construction developed by V. Barucci, M. D'Anna and the second author, that gives a family of quotients of the Rees algebra. Many examples are included

On the minimum distance of AG codes, on Weierstrass semigroups and the smoothability of certain monomial curves in 4-Space

In this paper we treat several topics regarding numerical Weierstrass semigroups and the theory of Algebraic Geometric Codes associated to a pair $(X, P)$, where $X$ is a projective curve defined over the algebraic closure of the finite field $F_q$ and P is a $F_q$-rational point of $X$. First we show how to evaluate the Feng-Rao Order Bound, which is a good estimation for the minimum distance of such codes. This bound is related to the classical Weierstrass semigroup of the curve $X$ at $P$. Further we focus our attention on the question to recognize the Weierstrass semigroups over fields of characteristic 0. After surveying the main tools (deformations and smoothability of monomial curves) we prove that the semigroups of embedding dimension four generated by an arithmetic sequence are Weierstrass.Comment: 30 pages, presented at CAAG 2010 (Bangalore, India

Syzygies of GS monomial curves and Weierstrass property.

We find a resolution for the coordinate ring R of an algebraic monomial curve associated to a G numerical semigroup (i.e. generated by a generalized arithmetic sequence), by extending a previous paper (Gimenez, Sengupta, Srinivasan) on arithmetic sequences . A consequence is the determinantal description of the first syzygy module of R. By this fact, via suitable deformations of the defining matrices, we can prove the smoothability of the curves associated to a large class of semigroups generated by arithmetic sequences, that is the Weierstrass property for such semigroups

On type sequences and Arf rings

In this article in Section~2 we give an explicit description to compute the type sequence $mathrm{t}_1,ldots,mathrm{t}_{n}$ of a semigroup $Gamma$ generated by an arithmetic sequence (see 2.7); we show that the $i$-th term $mathrm{t}_i$ is equal to $1$ or to the type $au_Gamma$, depending on its position. In Section 3, for analytically irreducible ring $R$ with the branch sequence&#13; R=R_0 subsetneq R_1 subsetneq ldotssubsetneq R_{m-1} subsetneq R_{m} =overline{R}, starting from a result proved in [4] we give a characterization (see 3.6) of the ``Arf'' property using the type sequence of $R$ and of the rings $R_j$, $1leq jleq m-1$. Further, we prove (see 3.9, 3.10) some relations among the integers $ell^*(R)$ and $ell^*(R_j)$, $1leq jleq m-1$. These relations and a result of [6] allow us to obtain a new characterization (see 3.12) of semigroup rings of minimal multiplicity with $ell^*(R)leq au(R)$ in terms of the Arf property, type sequences and relations between $ell^*(R)$ and $ell^*(R_j)$, $1leq jleq m-1$

On the order bound of one-point algebraic geometry codes

Let S ={si}i∈IN ⊆ IN be a numerical semigroup. For each i ∈ IN, let ν(si) denote the number of pairs (si−sj, sj) ∈ S 2 : it is well-known that there exists an integer m such that the sequence {ν(si)}i∈IN is non-decreasing for i &gt; m. The problem of finding m is solved only in special cases. By way of a suitable parameter t, we improve the known bounds for m and in several cases we determine m explicitly. In particular we give the value of m when the Cohen-Macaulay type of the semigroup is three or when the multiplicity is lower or equal to six. When S is the Weierstrass semigroup of a family {Ci}i∈IN of one-point algebraic geometry codes, these results give better estimates for the order bound on the minimum distance of the codes {Ci}. Index Terms. Numerical semigroup, Weierstrass semigroup, algebraic geometry code, order bound on the minimum distance

On some invariant ideals, and on extension of differentiations to seminormalization

AbstractLet A be a noetherian integral domain, D=(1,D1,…,Di…) be a differentation of A, and B be a ring such that A⊂B⊂Ā. In the paper we mainly prove (whenever Ā is finite over A): (a) if α is the conductor of A in B, then A√α is D-invariant. (b) D extends to the seminormalization +A of A in Ā

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