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a b s t r a c tLet S = {s 0 = 0 < s 1 < · · · < s i . . .} ⊆ N be a numerical non-ordinary semigroup; then set, for each i, ν i := #{(s i − s j , s j ) ∈ S 2 }. We find a non-negative integer m such thatwhere d ORD (i) denotes the order bound on the minimum distance of an algebraic geometry code associated to S. In several cases (including the acute ones, that have previously come up in the literature) we show that this integer m is the smallest one with the above property. Furthermore it is shown that every semigroup generated by an arithmetic sequence or generated by three elements is acute. For these semigroups, the value of m is also found.

a b s t r a c tIn this article we consider a semigroup ring R = K [[Γ ]] of a numerical semigroup Γ and study the Cohen-Macaulayness of the associated graded ring G(Γ ) := gr m (R) := ⊕ n∈N m n /m n+1 and the behaviour of the Hilbert function H R of R. We define a certain (finite) subset B(Γ ) ⊆ Γ and prove that G(Γ ) is Cohen-Macaulay if and only if B(Γ ) = ∅. Therefore the subset B(Γ ) is called the Cohen-Macaulay defect of G(Γ ). Further, we prove that if the degree sequence of elements of the standard basis of Γ is non-decreasing, then B(Γ ) = ∅ and hence G(Γ ) is Cohen-Macaulay. We consider a class of numerical semigroups Γ = ∑ 3 i=0 Nm i generated by 4 elements m 0 , m 1 , m 2 , m 3 such that m 1 + m 2 = m 0 + m 3 -so called ''balanced semigroups''. We study the structure of the Cohen-Macaulay defect B(Γ ) of Γ and particularly we give an estimate on the cardinality |B(Γ , r)| for every r ∈ N. We use these estimates to prove that the Hilbert function of R is nondecreasing. Further, we prove that every balanced ''unitary'' semigroup Γ is ''2-good'' and is not ''1-good'', in particular, in this case, G(Γ ) is not Cohen-Macaulay. We consider a certain special subclass of balanced semigroups Γ . For this subclass we try to determine the Cohen-Macaulay defect B(Γ ) using the explicit description of the standard basis of Γ ; in particular, we prove that these balanced semigroups are 2-good and determine when exactly G(Γ ) is Cohen-Macaulay.

In this paper we solve a problem posed by M.E. Rossi: Is the Hilbert function of a Gorenstein local ring of dimension one not decreasing? More precisely, for any integer h>1, hâ\u88\u8914+22k,35+46k|kâ\u88\u88N, 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

Communicated by A.V. Geramita MSC:20M14 94B35 a b s t r a c t Let S = {s i } i∈N ⊆ N be a numerical semigroup. For each i ∈ N, let ν(s i ) denote the number of pairs (s i − s j , s j ) ∈ S 2 : it is well-known that there exists an integer m such that the sequence {ν(s i )} i∈N is non-decreasing for i > 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 less than or equal to six. When S is the Weierstrass semigroup of a family {C i } i∈N of one-point algebraic geometry codes, these results give better estimates for the order bound on the minimum distance of the codes {C i }.

Let S = {si}i∈IN ⊆ IN be a numerical semigroup. For si ∈ S, let ν(si) denote the number of pairs (si −sj, sj) ∈ S 2 . When S is the Weierstrass semigroup of a family {Ci}i∈IN of one-point algebraicgeometric codes, a good bound for the minimum distance of the code Ci is the Feng and Rao order bound dORD(Ci) := min{ν(sj) : j ≥ i + 1}. It is well-known that there exists an integer m such that the sequence {ν(si)}i∈IN is non-decreasing for si > sm, therefore dORD(Ci) = ν(si+1) for i ≥ m. By way of some suitable parameters related to the semigroup S, we find upper bounds for sm, we evaluate sm exactly in many cases, further we give a lower bound for several classes of semigroups. Index Therms. Numerical semigroup, Weierstrass semigroup, AG code, order bound on the minimum distance, Cohen-Macaulay type. ). e := s 1 > 1, the multiplicity of S. c := min {r ∈ S | r + IN ⊆ S}, the conductor of S d := the greatest element in S preceding c, the dominant of S c ′ := max{s i ∈ S | s i ≤ d and s i − 1 / ∈ S}, the subconductor of S d ′ := the greatest element in S preceding c ′ , when c ′ > 0 k := d − c ′ q := d − d ′ ℓ := c − 1 − d, the number of gaps of S greater than d g := #(IN \ S), the genus of S (= the number of gaps of S) τ := #(S(1) \ S), the Cohen−M acaulay type of S Since c − e − 1 / ∈ S we have c − e ≤ c ′ ; we define the following sets H := [c − e, c ′ ] ∩ IN \ S ⊆ IN \ S S ′ := {s ∈ S | s ≤ d ′ } ⊆ S.

Given a one-dimensional semigroup ring R=k[[S]], in this article we study the behaviour of the Hilbert function HR. By means of the notion of support of the elements in S, for some classes of semigroup rings 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+3â\u89¤eâ\u89¤v+4, the decrease of HRgives an explicit description of the Apéry set of S. In particular for e=v+3, we prove that HRis non-decreasing if eâ\u89¤12 and we classify the semigroups with e=13 and HRdecreasing. Finally we deduce that HRis non-decreasing for every Gorenstein semigroup ring with eâ\u89¤v+4. This fact is not true in general: through numerical duplication and some of the above results another recent paper shows the existence of infinitely many one-dimensional Gorenstein rings with decreasing Hilbert function

Let p; m; d be positive integers, m i : m id, 0 i p and let n be positive integer such that gcdm; d; n 1 and m 0 < n. Let A be the coordinate ring of the algebroid monomial curve in the ane algebroid p 2-space A p2 K over a ®eld K, de®ned parametrically by X 1 T m 0 ; X 2 T m 1 ; . . . ; X p T m p ; X p1 T n : In this article assuming that the associated graded ring gr m A of A is Cohen-Maucaulay (and some more mild additional assumptions, see (2.4)), we give an explicit formula for the type of gr m A in terms of the standard basis of the semigroup generated by the almost arithmetic progression m 0 ; m 1 ; . . . ; m p ; n. Our special assumptions are satis®ed if p 1, that is, for the class of algebroid monomial space curves.