#A44 INTEGERS 8 (8) SYMMETRIC (NOT COMPLETE INTERSECTION) NUMERICAL SEMIGROUPS GENERATED BY FIVE ELEMENTS Leonid G. Fel Department of Civil Engineering, Technion, Haifa, Israel lfel@technion.ac.il Received: 7//7, Revised: //8, Accepted: 5/5/8, Published: 5/8/8 Abstract We consider symmetric (not complete intersection) numerical semigroups S 5, generated by five elements, and derive inequalities for degrees of syzygies of S 5 and find the lower bound F 5 for their Frobenius numbers. We study a special case W 5 of such semigroups, which satisfy the Watanabe Lemma, and show that the lower bound F 5w for the Frobenius number of the semigroup W 5 is stronger than F 5.. Symmetric Numerical Semigroups Generated by Five Integers Let a numerical semigroup S m = hd,..., d m i be generated by a set of m integers {d,..., d m } such that gcd(d,..., d m ) =, where d and m denote a multiplicity and an embedding dimension (edim) of S m. There exist m polynomial identities [7] for degrees of syzygies associated with the semigroup ring k[s m ]. They are a source of various relations for semigroups of a di erent nature. In the case of complete intersection (CI) semigroups, such a relation for the degrees e j of the st syzygy was found in [7]. The next nontrivial case exhibits a symmetric (not CI) semigroup generated by m 4 integers. In [6] such semigroups with m = 4 were studied and the lower bound for the Frobenius number F (S 4 ) was found. In the present paper we deal with a more di cult case of symmetric semigroups S 5. Consider a symmetric numerical semigroup S 5, which is not CI, and generated by five elements d i. Its Hilbert series H (S 5 ; t) with the st Betti number reads, H (S 5 ; t) = Q 5 (t) Q 5 i= ( tdi ), Q 5 (t) = t xj + t yj + t g yj t g xj + t g, () where x j, y j, g Z >, x j, y j < g, and the Frobenius number is defined as follows: F (S 5 ) = g, where = P 5 d j. There are two more constraints, > 4 The research was partly supported by the Kamea Fellowship.
INTEGERS: 8 (8) and d > 5. The inequality > 4 holds since the semigroup S 5 is not CI, and the condition d > 5 is necessary since the numerical semigroup hm, d,..., d m i is never symmetric [5]. Polynomial identities for degrees of syzygies for numerical semigroups were derived in [7], Theorem. In the case of the symmetric (not CI) semigroup S 5, they read: x r j x 4 j y r j + (g y j ) r + y 4 j + (g y j ) 4 + (g x j ) r g r =, apple r apple, () (g x j ) 4 g 4 = 4 5, 5 = 5Y d j. () Only two of the four identities in (,) are not trivial; these are the identities with r =, 4, x j (g x j ) = x j(g x j ) + 5 = y j (g y j ), (4) y j (g y j ). (5) Denote by Z k the k-th power symmetric polynomial Z k (z,..., z n ) = P n zk j, z j, and recall the Newton-Maclaurin inequality [8], Z apple nz. (6) Applying (6) to the right-hand side of equality (5) and substituting into the resulting inequality an equality (4), we obtain @ x j (g x j ) A apple ( ) 4 x j(g x j ) + 5 5. (7) On the other hand, according to (6), another inequality holds: @ x j (g x j ) A apple x j(g x j ). (8) Proposition. Let a symmetric (not CI) semigroup S 5 be given with the Hilbert series H (S 5 ; z) according to (). Then the following inequality holds: s g g 5, g 5 = ( ) 4p 5, ( ) = 4 4. (9) 4 j
INTEGERS: 8 (8) Proof. Inequality (8) holds always, while inequality (7) is not valid for every set {x,..., x, g}. In order to make both inequalities consistent, we have to find a range for g where both inequalities (7) and (8) are satisfied. As we will see in (), this requirement also constrains the admissible values of x j. To provide both inequalities to be correct, it is enough to require x j (g x j ) A @ 5 x j (g x j ) A. () @ Making use of the notation in (6), rewrite inequality () for = P x j and = P x j, g C, C = p ( ) 5, and combine it with inequality (6), namely,. Thus, we obtain g + C apple. () Represent the last inequality () in the following way, g C + g, and obtain immediately the lower bound g 5 in accordance with (9). The lower bound for the Frobenius number is given by F 5 = g 5. Inequality () constrains the degrees x j of the st syzygy for the symmetric (not CI) semigroup S 5, s s g g5 @ A g apple apple g g5 @ + A g. () Below we present twelve symmetric (not CI) semigroups S 5 with di erent Betti s numbers = 7, 8, 9, and give a comparative Table for their largest degree g of syzygies and its lower bound g 5 : = 7, A 7 = h6,, 4, 5, 9i, A 7 = h6,, 4, 7, i, A 7 = h9,,,, 7i, = 8, A 8 = h6,, 4, 9, i, A 8 = h8,,, 4, 9i, A 8 = h8, 9,,, 9i, = 9, A 9 = h7,,, 8, i, A 9 = h9,,, 4, 9i, A 9 = h8,,, 5, 5i, =, A = h9,, 9,, 7i, A = h9, 7, 8,, i, A = h, 8,, 45, 54i.
INTEGERS: 8 (8) 4 S 5 A 7 A 7 A 7 A 8 A 8 A 8 A 9 A 9 A 9 A A A g 87 9 85 99 89 84 96 4 6 g 5 79. 8.8 77.5 88.6 8.6 77.4 9.7 89.4 9.7 5. 4. 6.9 Table. The largest degree g of syzygies and its lower bound g 5 for symmetric (not CI) semigroups S 5 with di erent Betti s numbers = 7, 8, 9,. The semigroups A i, i =,,, were studied by H. Bresinsky []; the other semigroups were generated numerically with the help of the package NumericalSgps [4] by M. Delgado. Comparing F 5 with the two known lower bounds of the Frobenius numbers F CI5 and F NS5 for the symmetric CI [7] and nonsymmetric [] semigroups generated by five elements, respectively, we have F CI5 = 4 4p 5, F NS5 = 4p 4 5, F NS5 < F 5 < F CI5.. Symmetric Numerical (Not CI) Semigroups S 5 With W Property Watanabe [] gave a construction of the numerical semigroup S m generated by m elements starting with a semigroup S m generated by m elements and proved the following lemma. Lemma. ([]). Let a numerical semigroup S m = h,..., m i be given and a Z >, a >, d m > m, such that gcd(a, d m ) =, d m S m. Consider a numerical semigroup S m =ha,..., a m, d m i and denote it by S m =has m, d m i. Then S m is symmetric if and only if S m is symmetric, and S m is symmetric CI if and only if S m is symmetric CI. For our purpose the following Corollary of Lemma is important. Corollary. Let a numerical semigroup S m = h,..., m i be given and a Z >, a >, d m > m, such that gcd(a, d m ) =, d m S m. Consider a semigroup S m =has m, d m i. Then S m is symmetric (not CI) if and only if S m is symmetric (not CI). To utilize this construction we define the following property. Definition. A symmetric (not CI) semigroup S m has the property W if there exists another symmetric (not CI) semigroup S m giving rise to S m by the construction, described in Corollary. Note that in Definition the semigroup S m does not necessarily possess the property W. The symbol W stands for Kei-ichi Watanabe. The next Proposition distinguishes the minimal edim of symmetric CI and symmetric (not CI) numerical semigroups with the property W.
INTEGERS: 8 (8) 5 Proposition. A minimal edim of symmetric (not CI) semigroup S m with the property W is m = 5. Proof. All numerical semigroups of edim = are symmetric CI, and all symmetric numerical semigroups of edim = are CI [9]. Therefore, according to Definition, a minimal edim of symmetric CI semigroups with the property W is edim =. A minimal edim of numerical semigroups, which are symmetric (not CI), is edim = 4. Therefore, according to Definition and Corollary, a minimal edim of symmetric (not CI) semigroups with the property W is edim = 5. According to Proposition, let us choose a symmetric (not CI) semigroup of edim = 5 with the property W and denote it by W 5 in order to distinguish it from other symmetric (not CI) semigroups S 5 (irrespective of the W property). Then the following containment holds: {W 5 } {S 5 }. A minimal free resolution associated with semigroups W 5 was described recently ([], section 4), where the degrees of all syzygies were also derived ([], Corollary ), e.g., its st Betti number is = 6. Lemma. Let two symmetric (not CI) semigroups W 5 = has 4, d 5 i and S 4 = h,,, 4i be given and gcd(a, d 5 ) =, d 5 S 4. Let the lower bound F 5w of the Frobenius number F (W 5 ) of the semigroup W 5 be represented as, F 5w = g 5w a P 4 j + d 5. Then p 4Y g 5w = a 5 4 ( ) + d 5, 4 ( ) = j. () Proof. Consider a symmetric (not CI) numerical semigroup S 4 generated by four integers (without the W property), and apply the recent result [6] on the lower bound F 4 of its Frobenius number, F (S 4 ), to obtain F (S 4 ) F 4, F 4 = h 4 4 j, h 4 = p 5 4 ( ). (4) The following relationship between the Frobenius numbers F (W 5 ) and F (S 4 ) was derived in []: F (W 5 ) = af (S 4 ) + (a )d 5. (5) Substituting two representations F (S 4 )=h P 4 j and F (W 5 )=g a P 4 into equality (5), we obtain j d 5 g a 4 j d 5 = ah a 4 j + (a )d 5! g = a(h + d 5 ). (6) Comparing the last equality in (6) with the lower bound of h in (4), we arrive at ().
INTEGERS: 8 (8) 6 Proposition. Let two symmetric (not CI) semigroups W 5 = has 4, d 5 i and S 4 = h,,, 4i, where gcd(a, d 5 ) =, d 5 S 4, be given with the Hilbert series H (W 5 ; z) according to (). Then g 5w > g 5. (7) Proof. Keeping in mind that = 6 (see [], Corollary ), we determine g 5 for the semigroup W 5 according to (9), g 5 = a (6) 4p 4 ( )d 5, (6) = 4 4p 5/8 '.556, and calculate the following ratio: = g 5w g 5 = (6) p 5 4 ( ) + d 5 p = 4 4 ( )d 5 (6) p 5 +, = 4 s The function ( ) is positive for > and has an absolute minimum ( m ) '., m '.94, 4 ( ) '.8d 5. In other words, we arrive at ( m ) >, which proves the Proposition. 4 ( ) d. 5 We present ten symmetric (not CI) semigroups A 6 j with the W property and the Betti number = 6. All semigroups A 6 j are built according to Lemma and based on symmetric (not CI) semigroups S 4, generated by four integers [7]. We give a comparative Table for their g, lower bounds g 5w and g 5, and the parameter. A 6 =h4, 5, 6, 8, 6i, A 6 =h,, 4, 7, 9i, A 6 =h,,, 4, 6i, A 6 4 =h4, 6, 7, 8, 6i, A 6 5 = h6,, 6,, 4i, A 6 6 = h, 4, 9, 45, 5i, A 6 7 = h5, 4, 4, 45, 65i, A 6 8 = h, 5, 8, 4, 6i, A 6 9 = h, 8, 4, 5, 6i, A 6 = h45, 46, 469, 47, 474i, W 5 A 6 A 6 A 6 A 6 4 A 6 5 A 6 6 A 6 7 A 6 8 A 6 9 A 6 g 4 8 9 46 8 485 9 94 47 g 5w 9.4 4. 9.5 4.4 6.4.6 478.6 5478. 5498. 879. g 5 5.9. 8.9 9.9 94. 98. 44.8 466.8 4644. 7695..8.95.6.75..5.555..7.58 Table. The largest degree g of syzygies, its lower bounds g 5 and g 5w and the parameter for symmetric (not CI) semigroups W 5 with the Betti number = 6. Acknowledgement. The author is thankful to M. Delgado for providing him with various examples of symmetric semigroups generated by five elements which were found numerically with the help of the package NumericalSgps [4]. The author is grateful to the managing editor for valuable suggestions to improve the manuscript.
INTEGERS: 8 (8) 7 References [] V. Barucci and R. Fröberg, The minimal graded resolution of some Gorenstein rings, http://lanl.arxiv.org/pdf/.4747v [] A. Brauer, J.E. Shockley, On a problem of Frobenius, J. Reine Angew. Math., (96), 5-. [] H. Bresinsky, Monomial Gorenstein ideals, Manuscripta Math., 9 (979), 59-8. [4] M. Delgado, P.A. García-Sánchez and J. Morais, GAP Package NumericalSgps, Version..5, https://gap-packages.github.io/numericalsgps/ [5] L.G. Fel, Duality relation for the Hilbert series of almost symmetric numerical semigroups, Israel J. Math., 85 (), 4-444. [6] L.G. Fel, On Frobenius numbers for symmetric (not complete intersection) semigroups generated by four elements, Semigroup Forum, 9 (6), 4-46. [7] L.G. Fel, Restricted partition functions and identities for degrees of syzygies in numerical semigroups, Ramanujan J., 4 (7), 465-49. [8] G.H. Hardy, J.E. Littlewood and G. Polya, Inequalities, Univ. Press, Cambridge, 959. [9] J. Herzog, Generators and relations of Abelian semigroups and semigroup rings, Manuscripta Math., (97), 75-9. [] H. G. KillingbergtrØ, Betjening av figur i Frobenius problem, Normat (Nordisk Matematisk Tidskrift), (), 75-8, (In Norwegian) [] K. Watanabe, Some examples of -dim Gorenstein domains, Nagoya Math. J., 49 (97), -9.