In this paper we work from the algebraic point of view. From now on we do not need any hypothesis on the field k. In this paper we give a condition that implies that the minimal number of generators of the defining ideal I is bounded explicitly by its type. | Turk J Math (2018) 42: 2112 – 2124 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article On the type and generators of monomial curves Nguyen Thi DUNG∗, Thai Nguyen University of Agriculture and Forestry, Thai Nguyen, Vietnam Received: • Accepted/Published Online: • Final Version: Abstract: Let n1 , n2 , . . . , nd be positive integers and H be the numerical semigroup generated by n1 , n2 , . . . , nd . Let A := k[H] := k[tn1 , tn2 , . . . , tnd ] ∼ = k[x1 , x2 , . . . , xd ]/I be the numerical semigroup ring of H over k. In this paper we give a condition (∗) that implies that the minimal number of generators of the defining ideal I is bounded explicitly by its type. As a consequence for semigroups with d = 4 satisfying the condition (∗) we have µ(in(I)) ≤ 2(t(H)) + 1 . Key words: Frobenius number, pseudo-Frobenius number, almost Gorenstein ring, semigroup rings, monomial curve 1. Introduction Let n1 d monomials is bounded above by Ci,d,µ(J) the number of i -dimensional faces of the cyclic d-polytope with µ(J) vertices. For i = d − 1 this bound is strict. As a consequence we will prove the following theorem. 2115 DUNG/Turk J Math Theorem Suppose that the monomial ideal J ⊂ R is minimally generated by µ(J) monomials and rad(J) = m. Then µ(J) ≤ Cd−1,d,(t(R/J)+d) − 1. In particular for d = 3 we have µ(J) ≤ 2(t(R/J)) + 1 . By Theorem we have that J [a] is minimally generated by t(R/J) monomials since rad(J) = m . Proof Let m be an integer strictly bigger than the highest power of any variables appearing in the set of generators m of J [a] ; hence (J [a] )′ := J [a] + (xm 1 , . . . , xd )R is minimally generated by t(R/J) + d monomials. By Theorem (ii), the number of irreducible components of (J [a] )′ is the number of irreducible components of J [a] and the number of irreducible components of J [a] coincides with the number of generators of (J .
