TAILIEUCHUNG - On the type and generators of monomial curves
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 .
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