Genera of numerical semigroups and polynomial identities for degrees of syzygies
aa r X i v : . [ m a t h . A C ] D ec Genera of numerical semigroups andpolynomial identities for degrees of syzygies
Leonid G. FelDepartment of Civil Engineering, Technion, Haifa 32000, Israel e-mail: [email protected]
Abstract
We derive polynomial identities of arbitrary degree n for syzygies degrees of numerical semi-groups S m = h d , . . . , d m i and show that for n ≥ m they contain higher genera G r = P s ∈ Z > \ S m s r of S m . We find a number g m = B m − m + 1 of algebraically independent genera G r and equations,related any of g m + 1 genera, where B m = P m − k =1 β k and β k denote the total and partial Betti num-bers of non-symmetric semigroups. The number g m is strongly dependent on symmetry of S m anddecreases for symmetric semigroups and complete intersections. Keywords: numerical semigroups, degrees of syzygies, the Frobenius number and genus
Primary – 20M14, Secondary – 11P81.
Two sets of polynomial and quasi-polynomial identities for degrees of syzygies in numerical semi-groups S m = h d , . . . , d m i were derived recently [7] when studying the rational representation (Rep)of the Hilbert series of S m and the quasi-polynomial Rep of the restricted partition function. A part ofpolynomial identities of degrees ≤ n ≤ m − were coincided with Herzog-K ¨uhl’s equations [13]for the Betti numbers of graded Cohen-Macaulay modules of codimension m − , but a new additionalpolynomial identity of degree n = m − turned out to be an important tool in a study of symmetric (notcomplete intersection) semigroups with small embedding dimension ( edim ) m = 4 , , [6, 8, 9].A further application [10] of polynomial identities for higher degrees of syzygies, n ≥ m , involvespower sums G n − m = P s ∈ ∆ m s n − m , which called genera of numerical semigroups, where ∆ m = Z > \ S m and G = m denote a set of semigroup gaps and its cardinality ( genus ), respectively. A set ∆ m isuniquely defined by semigroup generators d j . Albeit there are µ − explicitly known gaps ≤ s ≤ µ − ,where µ = min { d , . . . , d m } denotes a semigroup multiplicity, a most of gaps s > µ (including thelargest gap F m which called the Frobenius number) cannot be determined explicitly.By a fundamental theorem of symmetric polynomials [14], there exists a finite number g m of alge-braically independent genera G r . On the other hand, an involvement of G r into polynomial identities1or syzygies degrees may decrease this number. In the present paper, we study this question for arbitrarysemigroup S m and find how does the number g m depend on special characteristics of a semigroup (e.g., edim ) and its properties (non-symmetric, symmetric, complete intersection). For this purpose, we de-rive polynomial identities of higher degrees n ≥ m and find algebraic equalities related a finite number g m + 1 of genera.The paper is organized in six sections. In section 2 we obtain polynomial identities (9) of higher de-grees n ≥ m following an approach, suggested in [13]. The rest of this section is completely technical:we determine necessary expressions for all entries appeared in formula (9). In section 3 we derive linearequations (19) for alternating power sums C k ( S m ) and put forward a conjecture on the linear Rep of co-efficients K p in (22) by genera G r of a semigroup and special polynomials T r defined in (25). In section4 we prove the existence of a polynomial equation R G ( G , . . . , G g m ) = 0 for arbitrary non-symmetricsemigroups, where g m = B m − m + 1 , and B m = P m − k =1 β k and β k denote the total and partial Bettinumbers of S m , and find such equation for S . In section 5 we discuss supplementary relations for G k in symmetric semigroups and complete intersection (CI), and give formulas for g m in both cases, e.g.,in the latter case it looks much simple, g m = m − . In section 6 we give concluding remarks aboutcoefficients K p . Recall the basic facts on numerical semigroups and polynomial identities following [7]. Let a numericalsemigroup S m be minimally generated by a set of natural numbers { d , . . . , d m } , where µ ≥ m and gcd( d , . . . , d m ) = 1 . Its generating function H ( S m ; z ) , H ( S m ; z ) = X s ∈ S m z s , z < , ∈ S m , (1)is referred to as the Hilbert series of S m and has a rational Rep, H ( S m ; z ) = Q ( S m ; z ) Q mi =1 (1 − z d i ) , C k,j ∈ Z > , ≤ k ≤ m − , ≤ j ≤ β k , (2) Q ( S m ; z ) = 1 − β X j =1 z C ,j + β X j =1 z C ,j − · · · ± β m − X j =1 z C m − ,j , m − X k =0 ( − k β k = 0 , β = 1 , (3)where C k,j and β k stand for degrees of syzygies and partial Betti’s numbers, respectively. The largestdegree Q m of the ( m − -th syzygy is related to the Frobenius number F m of S m , Q m = F m + σ , Q m = max PF ( S m ) , PF ( S m ) = (cid:8) C m − , , . . . , C m − ,β m − (cid:9) , σ = m X j =1 d j , (4)where PF ( S m ) is called a set of pseudo-Frobenius numbers. Denote by C k ( S m ) the alternating powersum of syzygies degrees, C k ( S m ) = β X j =1 C k ,j − β X j =1 C k ,j + . . . − ( − m − β m − X j =1 C km − ,j , (5)2nd write the polynomial identities (Theorem 1 in [7]) for a semigroup S m , C ( S m ) = 1 , C r ( S m ) = 0 , ≤ r ≤ m − , C m − ( S m ) = ( − m ( m − π m , (6)where π m = Q mi =1 d i . Start with relation for the Hilbert series H ( S m ; z ) and a generating function Φ ( S m ; z ) for the semigroupgaps s ∈ ∆ m , Φ ( S m ; z ) + H ( S m ; z ) = 11 − z , Φ ( S m ; z ) = X s ∈ ∆ m z s , (7)and present the numerator Q ( S m ; z ) in (2) as follows, Q ( S m ; z ) = (1 − z ) m − Π ( S m ; z ) , Π ( S m ; z ) = Ψ ( S m ; z ) [1 − (1 − z )Φ ( S m ; z )] , (8)where Ψ ( S m ; z ) is a product of cyclotomic polynomials Ψ j ( S m ; z ) , Ψ ( S m ; z ) = m Y j =1 Ψ j ( z ) , Ψ j ( z ) = d j − X k =0 z k , Ψ j (1) = d j , Ψ ( S m ; 1) = π m . Differentiating r times the first equality in (8), we obtain an infinite set of algebraic equations relatedsyzygies degrees C k,j of a semigroup S m with its generators d j and gaps s ∈ ∆ m , Q ( r ) z ( z ) = r X k =0 ( − k ( m − m − k − (cid:18) rk (cid:19) (1 − z ) m − k − Π ( r − k ) z ( z ) , r ≥ , (9)where Q ( r ) z ( z ) = d r Q ( S m ; z ) dz r , Π ( r ) z ( z ) = d r Π ( S m ; z ) dz r . Calculate separately derivatives Q ( r ) z ( z ) and Π ( r − k ) z ( z ) . According to expression (3), we obtain, Q ( r ) z ( z ) = − β X j =1 ( C ,j ) r z C ,j − r + β X j =1 ( C ,j ) r z C ,j − r − . . . + ( − m − β m − X j =1 ( C m − ,j ) r z C m − ,j − r , (10)where ( C i,j ) r = C i,j ( C i,j − × . . . × ( C i,j − r + 1) , if r ≤ C i,j and ( C i,j ) r = 0 , if r > C i,j , and ( x ) r = x ( x − × . . . × ( x − r + 1) denotes the falling factorial.Making use of alternating sums C k ( S m ) in (5), present the polynomial expansion (10) as follows, Q ( r ) z (1) = − r X k =0 S rk C k ( S m ) , S nk = ( − n − k h nk i , S nn = 1 , (11)3here S nk denote Stirling’s numbers of the 1st kind and symbols (cid:2) nk (cid:3) satisfy the recurrence relation, (cid:20) n + 1 k (cid:21) = n h nk i + (cid:20) nk − (cid:21) , ≤ k ≤ n, h nn i = 1 . (12)In Appendix A we present the first expressions for h nn − k i up to k = 9 .A straightforward calculation of the derivative Π ( r ) z ( z ) gives Π ( r ) z ( z ) = Ψ ( r ) z ( z ) + r X k =1 k (cid:18) rk (cid:19) Ψ ( r − k ) z ( z )Φ ( k − z ( z ) − (1 − z ) r X k =0 (cid:18) rk (cid:19) Ψ ( r − k ) z ( z )Φ ( k ) z ( z ) , (13)where Ψ ( r ) z ( z ) = r = k + ... + k m X k ,...,k m ≥ rk ! · · · k m ! m Y j =1 Ψ ( k j ) j,z ( z ) , Ψ ( k ) j,z ( z ) = d j − X l ≥ k ( l ) k z l − k , Φ ( k ) z ( z ) = X s ∈ ∆ ms ≥ k ( s ) k z s − k , and Ψ ( r ) z ( z ) = d r Ψ ( S m ; z ) dz r , Ψ ( r ) j,z ( z ) = d r Ψ j ( S m ; z ) dz r , Φ ( r ) z ( z ) = d r Φ ( S m ; z ) dz r . (14)Thus, we obtain an expression for the derivative Π ( r ) z ( z ) at z = 1 , Π ( r ) z =1 = Ψ ( r ) z =1 + r X k =1 k (cid:18) rk (cid:19) Ψ ( r − k ) z =1 Φ ( k − z =1 = Ψ ( r ) z =1 + r r − X k =0 (cid:18) r − k (cid:19) Ψ ( r − k − z =1 Φ ( k ) z =1 , (15) Π ( r ) z =1 = Π ( r ) z ( z = 1) , Ψ ( r ) z =1 = Ψ ( r ) z ( z = 1) , Ψ ( k ) j,z =1 = Ψ ( k ) j,z ( z = 1) , Φ ( k ) z =1 = Φ ( k ) z ( z = 1) . Φ ( r ) z =1 , Ψ ( r ) z =1 and Π ( r ) z =1 Derivatives Φ ( r ) z =1 may be calculated separately in accordance with (14), Φ ( r ) z =1 = r X k =0 S rk G k , G k = X s ∈ ∆ m s k , e.g. (16) Φ (0) z =1 = G , Φ (1) z =1 = G , Φ (2) z =1 = G − G , Φ (3) z =1 = G − G + 2 G , Φ (4) z =1 = G − G + 11 G − G , Φ (5) z =1 = G − G + 35 G − G + 24 G . The sums G k are known as genera of numerical semigroup S m and G denotes a semigroup genus .General formulas for derivatives Ψ ( r ) z =1 and Π ( r ) z =1 are given in (13) and (15) and cannot be simplifiedessentially for arbitrary r . Here we present the expressions for Ψ ( r ) z =1 and Π ( r ) z =1 for small r ≤ . Allnecessary calculations are given in Appendix B. Ψ (0) z =1 π m = 1 , Ψ (1) z =1 π m = σ − m , Ψ (2) z =1 π m = (cid:18) σ − m (cid:19) + σ − σ + 5 m , (17) Ψ (3) z =1 π m = (cid:18) σ − m − (cid:19) "(cid:18) σ − m (cid:19) + σ − σ + 3 m , Ψ (4) z =1 π m = 13 (cid:18) σ − σ + 5 m (cid:19) + (cid:18) σ − m (cid:19) σ − σ + 5 m (cid:18) σ − m (cid:19) − σ − m σ − σ + 3 m ) − σ − σ + 360 σ − m , σ k = P mj =1 d kj are power sums of generators d j . Substituting (16,17) into formulas (15), wearrive at expressions for Π ( r ) z =1 , Π (0) z =1 π m = 1 , Π (1) z =1 π m = σ − m G , (18) Π (2) z =1 π m = (cid:18) σ − m (cid:19) + σ − σ + 5 m
12 + ( σ − m ) G + 2 G , Π (3) z =1 π m = (cid:18) σ − m − (cid:19) "(cid:18) σ − m (cid:19) + σ − σ + 3 m + " (cid:18) σ − m (cid:19) + σ − σ + 5 m G + 3( σ − m ) G + 3( G − G ) , Π (4) z =1 π m = 13 (cid:18) σ − σ + 5 m (cid:19) + (cid:18) σ − m (cid:19) σ − σ + 5 m (cid:18) σ − m (cid:19) − σ − m σ − σ + 3 m ) − σ − σ + 360 σ − m
120 + (cid:18) σ − m − (cid:19) (cid:2) ( σ − m ) + σ − σ + 3 m (cid:3) G + (cid:2) σ − m ) + σ − σ + 5 m (cid:3) G + 6( σ − m )( G − G ) + 4( G − G + 2 G ) . C k ( S m ) In the right hand side of expression (9) for Q ( r ) z (1) , there survives a solely one term, namely, when k = m − . Combining the resulting expression in (9) with (11), we obtain, ( − m ( m − (cid:18) rm − (cid:19) Π ( r − m +1) z =1 = r X k = m − S rk C k ( S m ) , r ≥ m − . Represent the last equation in a more convenient way by shifting the variable r , i.e., r = m + p , m + p X k = m − S m + pk C k ( S m ) = ( − m ( m + p )!(1 + p )! Π ( p +1) z =1 , p ≥ − . (19)Thus, we arrive at the matrix equation with p + 2 variables C k ( S m ) , where k = m − , . . . , m + p , S m − m − . . . S mm − S mm . . . S m +1 m − S m +1 m S m +1 m +1 . . . . . . . . . . . . . . . S m + pm − S m + pm S m + pm +1 . . . S m + pm + p C m − ( S m ) C m ( S m ) C m +1 ( S m ) . . . C m + p ( S m ) = ( − m ( m − (0) z =1 m ! Π (1) z =1( m +1)!2! Π (2) z =1 . . . ( m + p )!( p +1)! Π ( p +1) z =1 , (20)where, according to definition of the Stirling numbers (12), we have in a diagonal S rr = 1 , r ≥ .5he general solution of matrix equation (20) may be written as follows, C m + p ( S m ) = ( − m ( m + p )!( p + 1)! Π ( p +1) z =1 − p +1 X j =1 ( − j (cid:20) m + pm + p − j (cid:21) C m + p − j ( S m ) , e.g., (21) C m − ( S m ) = ( − m ( m − (0) z =1 , C m ( S m ) = ( − m m ! Π (1) z =1 + (cid:20) mm − (cid:21) C m − ( S m ) , C m +1 ( S m ) = ( − m ( m + 1)!2 Π (2) z =1 + (cid:20) m + 1 m (cid:21) C m ( S m ) − (cid:20) m + 1 m − (cid:21) C m − ( S m ) . Calculating C m + p ( S m ) in (21) by consecutive substitution of C m + q − ( S m ) into C m + q ( S m ) , where q = 0 , . . . , p , we arrive at the final expression, C n ( S m ) = ( − m n !( n − m ) ! π m K n − m , C m − ( S m ) = ( − m ( m −
1) ! π m , (22)where a coefficient K p is a linear combination of genera G , . . . , G p . We present here expressions for K p when p ≤ , K = G + δ , δ p = σ p − p , (23) K = G + σ G + 3 δ + δ ,K = G + σ G + 3 σ + σ G + δ (cid:0) δ + δ (cid:1) ,K = G + 32 σ G + 3 σ + σ G + σ ( σ + σ )8 G + 15 δ + 30 δ δ + 5 δ − δ .K = G + 2 σ G + 3 σ + σ G + σ ( σ + σ )2 G +15 σ + 30 σ σ + 5 σ − σ G + δ δ + 10 δ δ + 5 δ − δ ,K = G + 52 σ G + 56 (cid:0) σ + σ (cid:1) G + 54 σ ( σ + σ ) G +15 σ + 30 σ σ + 5 σ − σ G + σ σ + 10 σ σ + 5 σ − σ G +63 δ + 315 δ δ + 315 δ δ − δ δ + 35 δ δ − δ + 16 δ K = G + 3 σ G + 54 (cid:0) σ + σ (cid:1) G + 52 σ ( σ + σ ) G +15 σ + 30 σ σ + 5 σ − σ G + σ σ + 10 σ σ + 5 σ − σ G +63 σ + 315 σ σ + 315 σ σ − σ σ + 35 σ σ − σ + 16 σ G + δ δ + 63 δ δ + 105 δ δ − δ δ + 35 δ − δ δ + 16 δ . Formulas for K , K , K , K were calculated by consecutive substitution of expressions (22) and (18) into (21). Theother three formulas for K , K , K were found in two steps: 1) by analytic derivations (with help of Mathematica software)of expressions for Ψ ( r ) z =1 and Π ( r ) z =1 , r = 5 , , , which are extremely lengthy to be disposed in the paper, 2) by consecutivesubstitution of the found expressions for Π ( r ) z =1 and C m + r − ( S m ) into (21). K p (even with help of Mathematica software) encounterswith enormous technical difficulties. On the other hand, a careful observation of formulas (23) allowsto put forward a conjecture about a general formula for K p for arbitrary p , which is related to a specialkind of symmetric polynomials P n = P n ( x , . . . , x m ) of degree n in m variables, discussed in [11], P n = m X j =1 x nj − m X j>r =1 ( x j + x r ) n + m X j>r>i =1 ( x j + x r + x i ) n − . . . − ( − m m X j =1 x j n . (24)In what follows, we make use of a remarkable property of polynomials P n : its factorization reads [11], P n ( x , . . . , x m ) = ( − m +1 n !( n − m )! χ m T n − m ( x , . . . , x m ) , χ m = m Y j =1 x j , T = 1 , (25)where T r ( x , . . . , x m ) is a symmetric polynomial of degree r in m variables. According to [11], thispolynomial satisfies inequality, T r ( x , . . . , x m ) > , x , . . . , x m > , (26)Denote X k = P mj =1 x kj and, by a fundamental theorem of symmetric polynomials [14], use suchpower sums as a basis for algebraic Rep of polynomials T r . In other words, instead of T r ( x , . . . , x m ) ,we make use in (25) of polynomials T r ( X ) = T r ( X , . . . , X r ) , which were derived in [11] and pre-sented in Appendix C. To pose a conjecture, define two polynomials T r ( σ ) = T r ( σ , . . . , σ r ) and T r ( δ ) = T r ( δ , . . . , δ r ) by replacing X k → σ k and X k → δ k in T r ( X , . . . , X r ) , where σ k and δ k aredefined in (17) and (23), respectively. Conjecture 1
Let a semigroup S m = h d , . . . , d m i be given and G r denote its genera according to (16).Then the alternating power sums C k ( S m ) in (5) are given by (22) with K p as follows K p = p X r =0 (cid:18) pr (cid:19) T p − r ( σ ) G r + 2 p +1 p + 1 T p +1 ( δ ) , (27)If (27) holds for any p , then, combining it with (26) and keeping in mind σ i , δ i > , we get K p > . K r in numerical semigroups Consider a numerical semigroup S m and write equalities (6) and (22) for alternating sums C k ( S m ) ,defined in (5), as a system of non-homogeneous polynomial equations for positive integer variables C k,j . For convenience, rename C k,j by one-index variable z i in such a way, that i runs through thetwo-index ( k, j ) table, enumerating elements of the 1st and following rows successively, z = C , , z = C , , . . . , z β = C ,β , z β +1 = C , , z β +2 = C , , . . . , z ζ m = C m − ,β m − , where a total number ζ m = { z i } of independent variables z i is dependent on inner properties of S m ,7.g., non-symmetric, symmetric (not CI), symmetric CI and others more sophisticated (Weierstrass’,Arf’s, hyperelliptic etc ). We consider here the three basic kind of semigroups mentioned above.A study of n + 1 non-homogeneous polynomial equations f j ( z , . . . , z n ) = 0 in n variables z i goesback to classical works of B´ezout, Sylvester, Caley and Macaulay [15], who has mostly considered anequivalent problem with n +1 homogeneous polynomial equations in n +1 variables. The use of a multi-variate resultant R es { f , f , . . . , f n } , which is an irreducible polynomial over a ring A [ f , f , . . . , f n ] ,generated by coefficients of f j , and vanishes whenever all polynomials f j have a common root, is astandard computational tool in the elimination theory. An interest in finding explicit formulas for resul-tants, extending Macaulay’s formulas as a quotient of two determinants, has been renewed in the 1990th(see [2] and references therein).Bearing in mind a special form of equations (6) and (22), we consider here the most general proper-ties of these equations: the existence of an algebraic relation among K p , which entered in (22), and itsrescaled version, given below. Rewrite polynomial equations (6) and (22) in new notations, Γ k ( z , . . . , z ζ m , L k ) = 0 , k ≥ , ζ m = B m , (28) L k = , if ≤ k ≤ m − , ( − m ( m − π m , if k = m − , ( − m k !( k − m ) ! π m K k − m , if k ≥ m, where Γ k ( z , . . . , z ζ m , L k ) is a homogeneous polynomial of degree k with respect to all variables z i andlinear in L k , and B m = P m − k =1 β k denotes the total Betti number of non-symmetric semigroups.Making use of a homogeniety of the polynomial Γ k ( ξ , . . . , ξ ζ m , ℓ k ) , rescale the variables and thewhole equation (28) as follows, Γ k ( ξ , . . . , ξ ζ m , ℓ k ) = 0 , k ≥ , ξ j = z j υ m , υ m = π / ( m − m , (29) ℓ k = , if ≤ k ≤ m − , ( − m ( m − if k = m − , ( − m k !( k − m ) ! κ k − m , if k ≥ m, κ p = K p υ − ( p +1) m (30)where according to (5), the polynomial Γ k ( ξ , . . . , ξ ζ m , ℓ k ) for the arbitrary non-symmetric semigroup S m reads, Γ k ( ξ , . . . , ξ ζ m , ℓ k ) = β X j =1 ξ kj − β + β X j = β +1 ξ kj + . . . + ( − m ζ m X j = ζ m − β m − ξ kj − ℓ k . (31) Theorem 1
Let S m be a non-symmetric semigroup with the Hilbert series given in (1). Then there existsan algebraic equation in g m + 1 variables, κ , κ , . . . , κ g m , R K ( κ , κ , . . . , κ g m − , κ g m ) = 0 , g m = B m − m + 1 , (32) and the polynomial R K is irreducible over a ring A [ κ , κ , . . . , κ g m − , κ g m ] . roof Choose the first B m +1 polynomial equations (29) in B m variables ξ , . . . , ξ B m and build a newsystem of B m equations in B m − variables ξ , . . . , ξ B m by eliminating ξ in resultants R es { Γ , Γ j } , Γ j ( ξ , . . . , ξ B m , ℓ , ℓ j ) = 0 , j = 2 , . . . , B m + 1 , (33) Γ j ( ξ , . . . , ξ B m , ℓ , ℓ j ) = R es { Γ ( ξ , . . . , ξ B m , ℓ ) , Γ j ( ξ , . . . , ξ B m , ℓ j ) } . The polynomial Γ j in (33) is irreducible [2] over a ring A [ ξ , . . . , ξ B m , ℓ , ℓ j ] (see a detailed proof of aresultant irreducibility for two polynomials in [12]).At the 2nd step, choose the first B m polynomial equations (33) in B m − variables ξ , . . . , ξ B m andbuild B m − equations in B m − variables ξ , . . . , ξ B m by eliminating ξ in resultants R es n Γ j , Γ o , Γ , j ( ξ , . . . , ξ B m , ℓ , ℓ , ℓ j ) = 0 , j = 3 , . . . , B m + 1 , Γ , j ( ξ , . . . , ξ B m , ℓ , ℓ , ℓ j ) = R es (cid:8) Γ ( ξ , . . . , ξ B m , ℓ , ℓ ) , Γ j ( ξ , . . . , ξ B m , ℓ , ℓ j ) (cid:9) . The polynomial Γ , j is irreducible [2] over a ring A [ ξ , . . . , ξ B m , ℓ , ℓ , ℓ j ] by reasons mentioned above.Continuing to eliminate the variables ξ k successively and constructing the families of resultants, Γ ,...,kj ( ξ k +1 , . . . , ξ B m , ℓ , . . . , ℓ k , ℓ j ) = 0 , j = k + 1 , . . . , B m + 1 , Γ ,...,kj ( ξ k +1 , . . . , ξ B m , ℓ , . . . , ℓ k , ℓ j ) = R es k n Γ ,...,k − k , Γ ,...,k − j o , we arrive at the B m th step at one resultant equation R es B m n Γ ,...,B m − B m ( ξ B m , ℓ , . . . , ℓ B m − , ℓ B m ) , Γ ,...,B m − B m +1 ( ξ B m , ℓ , . . . , ℓ B m − , ℓ B m +1 ) o = 0 . (34) The polynomial R es B m in the l.h.s. of (34) is irreducible [2] over a ring A [ ℓ , . . . , ℓ B m − , ℓ B m , ℓ B m +1 ] as well as all resultants of two polynomials at previous steps.Equation (34) is free of any variable ξ i and involves only B m + 1 coefficients ℓ n , ≤ n ≤ B m + 1 .Keeping in mind the two first relations in (30), namely, ℓ n = 0 if ≤ n < m − , and an in-dependence of ℓ m − on κ j , we conclude that equation (34) is algebraic in B m − m + 2 variables ℓ m , ℓ m +1 , . . . , ℓ B m , ℓ B m +1 . However, by the 3rd relation in (30), such equation can be represented in κ , κ , . . . , κ B m − m , κ B m − m +1 as given in (32). (cid:3) Corollary 1
Let S m be a non-symmetric semigroup with the Hilbert series given in (1). Then thereexists g m algebraically independent genera. The set of such genera reads, { G , G , . . . , G g m − } . Proof
Combining Theorem 1 and formulas (23) as well as (27), by assumption that Conjecture 1 istrue, we arrive at algebraic equation R G ( G , G , . . . , G g m − , G g m ) = 0 , Note that inequality B m > m − holds, since, according to [18], we have β ≥ m − while the other β k are positive. R G over a ring A [ G , G , . . . , G g m − , G g m ] . Resolving the last equationwith respect to G g m and keeping in mind an irreduciblity of R G , we arrive at the algebraic function G g m = F ( G , . . . , G g m − ) , where the set { G , G , . . . , G g m − } comprises genera for any numericalsemigroup S m , which are algebraically independent. (cid:3) Theorem 1 may be extended on algebraic equations included κ n , n > g m , with a similar proof. Theorem 2
There exists an algebraic equation in g m + 1 variables κ , κ , . . . , κ g m − and κ n , R K ( κ , κ , . . . , κ g m − , κ n ) = 0 , n > g m , (35) and the polynomial R K is irreducible over a ring A [ κ , κ , . . . , κ g m − , κ n ] . S In this section we consider the most simple case of non-symmetric numerical semigroups S generatedby three integers. The numerator in the rational Rep (3) of its Hilbert series H ( S ; z ) reads, Q ( S ; z ) = 1 − ( z x + z x + z x ) + z y + z y , β = 3 , β = 2 , g = 3 . Six polynomial equations (6) and (22) for five symmetric polynomials, X k = P j =1 x kj , k = 1 , , , and Y r = P j =1 y rj , r = 1 , , are given below, Y = X , Y = X + 2 π , Y = X + 6 π K ,Y = X + 24 π K , Y = X + 60 π K , Y = X + 120 π K , (36)where K i are given in (23). Bearing in mind the Newton identities [14] related symmetric polynomials, Y = 12 (cid:0) Y − Y (cid:1) Y , Y = 12 (cid:0) Y + 2 Y Y − Y (cid:1) , (37) Y = 14 (cid:0) Y − Y (cid:1) Y , Y = 14 (cid:0) Y + 6 Y Y − Y (cid:1) Y ,X = 16 (cid:0) X − X X + 8 X X + 3 X (cid:1) , X = 16 (cid:0) X − X X + 5 X X + 5 X X (cid:1) ,X = 112 (cid:0) X − X X − X X + 3 X + 4 X X + 12 X X X + 4 X (cid:1) , we present six equations (36) as follows Y (cid:0) Y − Y (cid:1) = X + 6 π K , Y = X , Y = X + 2 π , (38) (cid:0) Y + 2 Y Y − Y (cid:1) = X − X X + 8 X X + 3 X + 144 π K , Y (cid:0) Y − Y (cid:1) = X − X X + 5 X X + 5 X X + 360 π K , Y (cid:0) Y + 6 Y Y − Y (cid:1) = X − X X − X X + 3 X + 4 X X + 12 X X X +4 X + 1440 π K . Y − K Y + 12 K + π = Y , Y − K Y + 4 π K + 24 K = Y ( Y + 2 K ) ,Y − π Y + 8 π K Y + 8 π K − π + 80 K = Y − Y ( π − K Y ) . (39)Combining separately the 1st relation in (39) with the 2nd and 3rd relations, we get, respectively, (cid:16) K − K + π (cid:17) Y = 6 K − K K + π K , (40) (cid:16) K − K + π (cid:17) Y = 10 K − K + π K − π , and further, due to (36,37,39), X = 4 (cid:18) K − K K + 2 K K − K + π (cid:19) − (cid:0) K − K (cid:1) − π , X = Y ,X = Y − K Y + 3 (cid:16) K + π (cid:17) Y − K π , Y = X + 2 π . Rescaling K r = κ r q π r +13 , write a necessary condition to have non-trivial solutions for equations (40), κ = 2 κ − , otherwise, there exist three equalities κ = 23 κ − , κ = κ (cid:18) κ − (cid:19) , κ = 95 κ − κ + 1240 , (41)which define a special class of semigroups S . In section 5.2, we show that the last three formulas arerelated to symmetric 3-generated semigroups.Combining two equalities (40), we obtain, in accordance with Theorem 1, equation (32) in rescaledvariables κ , κ , κ , κ , (cid:18) κ − κ + κ − (cid:19) (cid:18) κ − κ + 14 (cid:19) = (cid:16) κ − κ κ + κ (cid:17) , (42)that manifests three algebraically independent genera G , G , G . κ n in S To derive the equation (35) for κ let us consider the power sums X , Y , X = X − X X + 7 X X + 21 X X + 28 X X , Y = ( Y − Y Y + 7 Y Y + 7 Y ) Y , and substitute them into equality Y = X + 840 π K . Making use of the last equality and five firstrelations in (38) and performing necessary calculations, we arrive at three equations for Y and Y . Y − K Y + 12 K + π = Y , Y − K Y + 4 π K + 24 K = Y ( Y + 2 K ) , K π Y + Y ( Y − Y )( Y − π ) + K ( Y − π + 4 π Y − Y Y − Y ) = − K , (43)11hich is similar to (39) by exception of the last one. Equations (43) can be resolved in Y as follows, (cid:16) K − K + π (cid:17) Y = 6 K − K K + π K , (cid:0) Y − K Y + 12 K (cid:1) (cid:16) K − K + π (cid:17) Y = 15 K − K (cid:16) π − K (cid:17) , (44)Combining two equations in (44) and rescaling the coefficients K r = κ r q π r +13 , we obtain, in accor-dance with Theorem 1, equation (35) in κ , κ , κ , κ variables, κ − κ (cid:18) κ − (cid:19) ! (cid:18) κ − κ + 14 (cid:19) = (cid:16) κ − κ κ + κ (cid:17) K ( κ , κ , κ ) , (45)where K ( κ , κ , κ ) is a positive definite function K = 2 (1+12 κ ) κ + 24 κ κ − κ (1+4 κ ) κ + 3 κ (1 − κ ) κ + 36 κ + 3 κ κ ) in the positive octant κ , κ , κ > . In order to prove that, we suppose, by way of contradiction, that K ( κ , κ , κ ) = 0 . The last equation may be resolved as quadratic in κ , κ ± = ± (cid:0) − κ + 12 κ (cid:1) q κ − κ + κ (cid:0) κ − − κ (cid:1) , κ − ≤ κ +2 . (46)Consider the largest real root κ +2 ≥ κ − and require κ ≤ κ . Combining (46) with the last inequality,we arrive for κ , κ > at the upper bound, κ +2 ≤ (cid:0) − κ (cid:1) q κ − κ − κ (cid:0) κ (cid:1) ≤ κ (cid:0) − κ − − κ (cid:1) ≤ − κ . Thus, the both roots κ ± are never positive. In other words, in the positive octant κ , κ , κ > thefunction K ( κ , κ , κ ) is never vanished. Since K ( κ , ,
0) = 2 κ , we conclude that the function K ( κ , κ , κ ) is always positive.Coming back to relation (44) and equations (45), we conclude that there exists one special case,when all (44,45) are satisfied identically, κ − κ + 14 = 0 , κ − κ κ + κ , κ − κ (cid:18) κ − (cid:19) = 0 , (47)This case is related to symmetric 3-generated semigroups and equalities (47) are coincided with thethree corresponding formulas for K , K , K in (67). K r and G r in symmetric semigroups If a numerical semigroup S m is symmetric, then degrees C k,j of syzygies and Betti’s numbers β k in therational Rep (2) of the Hilbert series are related as follows, β k = β m − k − , β m − = 1 , C k,j + C m − k − ,j = Q m , (48)12hile the number of gaps and non-gaps of S m are equal to G . Therefore, according to (4,23), we have F m = 2 G − , K = Q m = F m + σ , G X j =1 s nj + G X j =1 ( F m − s j ) n = F m X j =0 j n . (49)The last identity in (49) may be represented as follows G r + r X q =1 ( − q (cid:18) rq (cid:19) F qm G r − q = F r +1 m r + 1 + F rm r − X q =1 (cid:18) rq − (cid:19) B r − q +1 F qm q , (50)where B k denotes the Bernoulli number. Equality (50) reduces the number of independent genera G k twice, making G r dependent on genera with odd indices, G j − , j = 1 , . . . , r , G r F m = 12 r − X p =0 (cid:20)(cid:18) rp (cid:19) B r − p p + 1 + ( − p (cid:18) rp + 1 (cid:19) G r − p − (cid:21) F pm − r − r + 1 F rm , (51)e.g., G F m = G − F m − , F m −
112 = G ( G − , (52) G F m = 2 G − F m G + F m −
112 6 F m + 15 ,G F m = 3 G − F m G + 3 F m G − F m −
112 51 F m + 9 F m + 214 ,G F m = 4 G − F m G + 28 F m G − F m G + F m −
112 310 F m + 55 F m + 13 F m + 315 . To find a number g m of algebraically independent genera of symmetric semigroups, we apply The-orem 1 with a new number ζ m of independent variables, which differs from (31). For this purpose,represent formula (3) for the numerator Q ( S m ; t ) when m = 0 , separately and account forindependent variables x j , y j , . . . , z j , Q m in every of two cases,• m = 2 q, q ≥ , ζ q = P q − j =0 β j − t Q m − β X j =1 (cid:0) t x j − t Q m − x j (cid:1) + β X j =1 (cid:0) t y j − t Q m − y j (cid:1) − . . . − ( − q β q − X j =1 (cid:0) t z j − t Q m − z j (cid:1) , (53)• m = 2 q + 1 q ≥ , ζ q +1 = P qj =0 β j ( − q β q = P q − j =0 ( − j +1 β j t Q m − β X j =1 (cid:0) t x j + t Q m − x j (cid:1) + β X j =1 (cid:0) t y j + t Q m − y j (cid:1) − . . . + ( − q β q X j =1 (cid:0) t z j + t Q m − z j (cid:1) . (54)Note, that ζ q +1 = 0 (mod 2) since ζ q +1 may be presented for m = 1 , as follows ζ q +1 = 2 q X j =1 β j − , ζ q +3 = 2 q X j =1 β j + 1 . (55)13 heorem 3 Let S m be a symmetric (not CI) semigroup, then there are g m independent genera g q = ζ q − q, G , G , G , . . . G g q − , (56) g q +1 = ζ q +1 − q, G , G , G , . . . G g q +1 − . (57) Proof
First, consider symmetric semigroups S q with even edim . The total number of syzygies degrees(including those which are related in couples), appeared in (53), is given by ζ q − . Replacing by thisnumber the total Betti number B m in (32), we get e g q = 2 ζ q − − (2 q −
1) = 2( ζ q − q ) , which doesnot related to equalities (51). Keeping in mind the supplementary relations (51) for genera G k , we haveto decrease the last number twice, i.e., we arrive at (56).Next, consider symmetric semigroups S q +1 with odd edim . By similar considerations, as in thecase m = 2 q , we arrive at e g q +1 = 2 ζ q +1 − − (2 q + 1 −
1) = 2( ζ q +1 − q ) − , which does notrelated to equalities (51). Keeping in mind the supplementary relations (51) for genera G k , we have todecrease the number e g q +1 as follows: g q +1 = (1 + e g q +1 ) / , i.e., we arrive at (57). (cid:3) Combining Theorem 3 with (53,55) we may specify the number g m in more details, g q = q − X j =1 β j − q + 1 , g q +1 = 2 q X j =1 β j − − q , g q +3 = 2 q X j =1 β j − q + 1 . (58) S The numerator (3) in the rational Rep of its Hilbert series H ( S ; z ) reads [1], Q ( S ; z ) = 1 − X j =1 z x j + X j =1 z Q − x j − z Q , β = 5 , g = 4 . We present polynomial equations (6,22) for the ten first symmetric polynomials, X k = P j =1 x kj .Among them, equations of the 1st and 2nd degrees are coincided. Together with (49) they give X = 2 Q = 4 K . (59)The rest of eight equations might be decomposed in couples of the odd and even degrees, X − Q X + 2 Q = 3! π , X − Q X + 2 Q = 4!0! K π Q , X − Q X + 10 Q X − Q X + 6 Q = 5!1! K π , X − Q X + 203 Q X − Q X + 83 Q = 6!2! K π Q , X − Q X + 21 Q X − Q X + 35 Q X − Q X + 10 Q = 7!3! K π , X − Q X + 14 Q X − Q X + 14 Q X − Q X + 3 Q = 8!4! K π Q , X − Q X +36 Q X − Q X +126 Q X − Q X +84 Q X − Q X +14 Q = 9!5! K π , X − Q X +24 Q X − Q X + 2525 Q X − Q X +24 Q X − Q X + 165 Q = 10!6! K π Q Their successive solution gives, K = 2 (cid:18) K − K (cid:19) K , K = 4 (cid:18) K − K K + 45 K (cid:19) K ,K = 6 (cid:18) K − K K + 16 K K − K (cid:19) K , (60)where K , K , K , K are four independent coefficients, in accordance with (58). Formulas (60) and(23,52) are strongly related. Namely, the former may be obtained by a straightforward substitution of(52) into (23).The list of formulas (60) may be continued if we consider equations (22) for higher degrees, K r = 2 r (cid:16) K r − − ρ r K K r − + . . . − ( − r ρ r − r K r − K +( − r ρ r r K r (cid:17) K , ρ j r ∈ Q , where K j +1 , j ≥ , are algebraic (not polynomial) functions of K , K , K , K . K r and G r in symmetric CI semigroups This kind of numerical semigroups is described by a simple Hilbert series (2) with a numerator Q ( S m ; z ) Q ( S m ; z ) = (1 − z e ) (1 − z e ) · · · (1 − z e m − ) , built on m − degrees e j . The alternating power sum C k ( S m ) reads, C n ( S m ) = m − X j =1 e nj − m − X j>r =1 ( e j + e r ) n + . . . − ( − m − m − X j =1 e j n . (61)Then, according to the Rep (24,25), the expression in (61) may be written as follows, C n ( S m ) = ( − m n !( n − m + 1)! ε m − T n − m +1 ( E ) , C m − ( S m ) = ( − m ( m −
1) ! ε m − , (62)where polynomials T r ( E ) = T r ( E , . . . , E r ) are built by replacing X r → E r = P m − j =1 e rj in polyno-mials T r ( X , . . . , X r ) , defined in (25), and ε m − = Q m − j =1 e j .Combining formulas (22) and (62), we obtain, π m = ε m − , K p = T p +1 ( E ) p + 1 , p ≥ , (63)where the 1st equality was established earlier (see formula (5.4) in [7]) while the 2nd relation leads toan infinite number of equalities. An universality of (63) disappears if we consider symmetric (not CI)semigroups, see e.g., formula (59) for K . The number g m of independent genera G r is given by thenumber of degrees of syzygies, bearing in mind the 1st relation in (63),15 m = m − (64)In symmetric semigroups S m there holds a strict inequality µ > m (see [5]), that bounds a genus frombelow, G ≥ m + 1 , and leads, in combination with (64), to another inequality g m < G .Below we present three examples with symmetric CI semigroups S m , where m = 2 , , . Example 1
CI semigroup S , g = 0 , e = π , E r = π r ,By formula (17) from [11] and (63) we obtain T r ( e ) = π r r + 1 , K r = π r +12 ( r + 1)( r + 2) . (65) Assuming that Conjecture 1 is true, substitute the last into (27), ( p + 1) p X r =0 (cid:18) pr (cid:19) T p − r ( σ ) G r = π p +12 p + 2 − p +1 T p +1 ( δ ) , which may be resolved with respect to G r if we make use of the inverse matrix (cid:0)(cid:0) pr (cid:1) T p − r ( σ ) (cid:1) − . We giveexplicit expressions for the five first genera G r , G = π − σ + 12 , G = π − σ + 12 2 π − σ − , (66) G = π − σ + 12 π ( π − σ )6 ,G = π − σ + 12 6 π + π (4 − σ ) + π ( σ − σ −
1) + ( σ + 1)( σ + 1)60 ,G = π − σ + 12 π ( π − σ )6 2 π − π (2 σ − − σ . Formulas (66) coincide with expressions for genera, derived [16] in terms of generators d , d , e.g., G = ( d − d − , G = ( d − d − d d − d − d − . Example 2
CI semigroup S , g = 1 , E = 2 K .There exists one independent power sum E , while the sums E r may be expressed as follows, E = E − π , E = E − π E + 2 π , E = E − π E + 9 π E − π . Substituting E k into expressions for T r ( E ) in Appendix C and subsequently into (63) we obtain K = 13 (cid:16) K − π (cid:17) , (67) K = 13 (cid:16) K − π (cid:17) K ,K = 15 (cid:18) K − π K + π (cid:19) ,K = 415 (cid:18) K − π K + π (cid:19) K ,K = 121 (cid:18) K − π K + 4 π K − π (cid:19) ,K = 17 (cid:18) K − π K + 10 π K − π (cid:19) K . ombining (67) with formulas (23) and (52), we obtain for genera G r the polynomial expressions in G . We present here only the four first formulas; expressions for G , G are extremely lengthy. G = 23 G + (cid:18) δ − (cid:19) G + γ , γ = δ − δ − π , (68) G = 23 G + 23 ( δ − G + 13 (cid:18) γ − δ + 12 (cid:19) G − γ ,G = 45 G + (cid:18) δ − (cid:19) G + (cid:18) γ δ
15 + π − δ + 13 (cid:19) G + (cid:20) γ (cid:18) δ − (cid:19) − δ (cid:18) δ − (cid:19)(cid:21) G + δ ( δ − π )20 − δ + 2 γδ − γ
12 + δ
30 + π G = 1615 G + 85 (cid:18) δ − (cid:19) G + 2 (cid:18) − δ δ + π
15 + 2 γ (cid:19) G + (cid:18) δ γ (cid:18) δ − (cid:19) − δ + 8 δ δ + π (cid:19) G + (cid:18) δ + 4 δ δ − δ + 2 δ − γ δ + 10 δ −
515 + π − (cid:19) G + 5 δ − δ − δ
15 + γδ π (cid:16) δ − π (cid:17) . In [4], formulas for G r were derived in terms of 3 diagonal elements of matrix of minimal relations forgenerators d j , that makes them less convenient from computational point of view than formulas (67,68). Example 3
CI semigroup S , g = 2 , E = 2 K , E = 12(2 K − K ) .There exist two independent power sums E , E , while the rest E k may be expressed as follows, E = 3 E E − E + 6 π , E = E + 2 E E − E + 8 π E , E = 5 E E − E + 10 π ( E + E ) , E = 6 E E + E − E E + 24 π E E + 12 π . Substituting E k into expressions for T r ( E ) in Appendix C and subsequently into (63) we obtain K = 2 (cid:18) K − K (cid:19) K , (69) K = 15 (cid:18) K + 2 K K − K − π K (cid:19) ,K = 15 (cid:18) K − K K + 163 K − π K (cid:19) K ,K = 17 (cid:18) K + 48 K K − K K + 643 K − π K K − π K + π (cid:19) ,K = 17 (cid:18) K − K K + 80 K K + 163 K − π K K + 2 π K + π (cid:19) K . By comparison (60) and (69), formulas for K and K in (69) may be obtained if we substitute K and K in (69) into K and K in (60). Combining (69) with formulas (23), we obtain for genera G r the olynomial expressions in G , G , e.g., G = 2 G − (cid:0) G − G + G (cid:1) , (70) G = 12 G G (cid:0)
10 + 8 G + 2 σ − σ − G (2 + σ ) − σ (cid:1) − G
15 + 2 G
15 + G (cid:0) σ + 3 σ − σ − (cid:1) − G (cid:0) σ ( σ − σ + 6 σ −
10) + 2 π + 2 σ − (cid:1) +1240 (cid:0) σ ( σ − σ + 2 σ −
2) + 2 π (cid:1) . We study polynomial identities of arbitrary degree n for syzygies degrees of numerical semigroups S m and show in (22) that for n ≥ m they contain higher genera G r = P s ∈ Z > \ S m s r of S m , β X j =1 C n ,j − β X j =1 C n ,j + . . . − ( − m − β m − X j =1 C nm − ,j = ( − m n !( n − m ) ! π m K n − m ( G , . . . , G p ) , where a coefficient K p ( G , . . . , G p ) is a linear combination of genera. We calculate explicitly severalfirst expressions (23) for K p , ≤ p ≤ , and put forward Conjecture 1 related K p and G , . . . , G p forany integer p ≥ . In symbolic calculus [17], this relationship (27) reads K p = ( T ( σ ) + G ) p + 2 p +1 p + 1 T p +1 ( δ ) , where after binomial expansion the symbolic powers T p − r ( σ ) G r are converted into T p − r ( σ ) G r . Sym-metric polynomials T r ( σ ) = T r ( σ , . . . , σ r ) and T r ( δ ) = T r ( δ , . . . , δ r ) are arisen in the theory of symmetric CI semigroups [7, 11] m − X j =1 e nj − m − X j>r =1 ( e j + e r ) n + . . . − ( − m − m − X j =1 e j n = ( − m n !( n − m + 1)! ε m − T n − m +1 ( E ) , where polynomials T r ( E ) = T r ( E , . . . , E r ) are related (see [11], formula (19)) to the polynomial partof the partition function, that gives a number of partitions of s ≥ into m − positive integers. Thus,in the relationship (23,27), there coexist genera G r of non-symmetric semigroup S m and characteristicpolynomials T r ( σ ) associated with semigroup generators d , . . . , d m .Based on a finite number of syzygies degrees and homogeneity of the m − first polynomial iden-tities (6), we find a number g m of algebraically independent coefficients K p for different kinds of semi-groups. Due to the relationship (27), this leads to g m algebraically independent genera G p . However,the polynomial equations, related K p , p ≥ g m , with independent coefficients K j , ≤ j < g m , readmuch shorter than their countpartners, related G p , p ≥ g m , with independent genera G j . It can be seenfor non-symmetric and symmetric (not CI) semigroups, (see section 5.2), but, in particular, for symmet-ric CI semigroups comparing relations (65), (67) and (69) with (66), (68) and (70). These observationsmake us to suppose that K p has deeper algebraic meaning than a simple combination of G p .18 Appendix: Stirling numbers of the 1st kind
Making use of recurrence equation (12), we calculate the first formulas for h nn − k i up to k = 9 . h nn i = 1 , (cid:20) nn − (cid:21) = (cid:18) n (cid:19) , (cid:20) nn − (cid:21) = (cid:18) n (cid:19) (cid:18) n − (cid:19) , (cid:20) nn − (cid:21) = (cid:18) n (cid:19)(cid:18) n (cid:19) , (cid:20) nn − (cid:21) = (cid:18) n (cid:19) (cid:18) n − n n
48 + 124 (cid:19) , (cid:20) nn − (cid:21) = (cid:18) n (cid:19) (cid:18) n − n n
16 + 18 (cid:19) n, (A1) (cid:20) nn − (cid:21) = (cid:18) n (cid:19) (cid:18) n − n
64 + 35 n
64 + 91 n − n − (cid:19) , (cid:20) nn − (cid:21) = (cid:18) n (cid:19) (cid:18) n − n
16 + 35 n
48 + 7 n − n − (cid:19) n. (cid:20) nn − (cid:21) = (cid:18) n (cid:19) (cid:18) n − n
64 + 105 n − n − n − n
64 + 101 n
960 + 380 (cid:19)(cid:20) nn − (cid:21) = (cid:18) n (cid:19) (cid:18) n − n
64 + 105 n − n − n
768 + 25 n
192 + 101 n
192 + 316 (cid:19) n. B Appendix: Derivatives Ψ ( r ) z =1 Find expressions for ratio of derivatives Ψ ( r ) z =1 / Ψ (0) z =1 , r ≤ , Ψ (1) z =1 Ψ (0) z =1 = m X j =1 Ψ (1) j,z =1 Ψ j,z =1 , Ψ (2) z =1 Ψ (0) z =1 = m X j =1 Ψ (1) j,z =1 Ψ j,z =1 + m X j =1 Ψ (2) j,z =1 Ψ j,z =1 − m X j =1 Ψ (1) j,z =1 Ψ j,z =1 ! , Ψ (3) z =1 Ψ (0) z =1 = m X j =1 Ψ (1) j,z =1 Ψ j,z =1 + m X j =1 Ψ (3) j,z =1 Ψ j,z =1 + 2 m X j =1 Ψ (1) j,z =1 )Ψ j,z =1 ! + 3 m X j =1 Ψ (1) j,z =1 Ψ j,z =1 m X j =1 Ψ (2) j,z =1 Ψ j,z =1 − m X j =1 Ψ (1) j,z =1 Ψ j,z =1 m X j =1 Ψ (1) j,z =1 Ψ j,z =1 ! − m X j =1 Ψ (1) j,z =1 Ψ (2) j,z =1 (Ψ j,z =1 ) , (B1) Ψ (4) z =1 Ψ (0) z =1 = m X j =1 Ψ (1) j,z =1 Ψ j,z =1 + m X j =1 Ψ (4) j,z =1 Ψ j,z =1 − m X j =1 Ψ (1) j,z =1 Ψ j,z =1 ! + 3 m X j =1 Ψ (1) j,z =1 Ψ j,z =1 ! − m X j =1 Ψ (1) j,z =1 Ψ j,z =1 ! m X j =1 Ψ (1) j,z =1 Ψ j,z =1 + 8 m X j =1 Ψ (1) j,z =1 Ψ j,z =1 m X j =1 Ψ (1) j,z =1 Ψ j,z =1 ! + 6 m X j =1 Ψ (1) j,z =1 Ψ j,z =1 m X j =1 Ψ (2) j,z =1 Ψ j,z =1 + 12 m X j =1 Ψ (1) j,z =1 Ψ j,z =1 ! Ψ (2) j,z =1 Ψ j,z =1 + 3 m X j =1 Ψ (2) j,z =1 Ψ j,z =1 − m X j =1 Ψ (1) j,z =1 Ψ j,z =1 ! m X j =1 Ψ (2) j,z =1 Ψ j,z =1 − m X j =1 Ψ (2) j,z =1 Ψ j,z =1 ! − m X j =1 Ψ (1) j,z =1 Ψ (3) j,z =1 (Ψ j,z =1 ) m X j =1 Ψ (1) j,z =1 Ψ j,z =1 m X j =1 Ψ (1) j,z =1 Ψ (2) j,z =1 (Ψ j,z =1 ) + 4 m X j =1 Ψ (1) j,z =1 Ψ j,z =1 m X j =1 Ψ (3) j,z =1 Ψ j,z =1 . Using a summation rule in a finite calculus (see formula (2.50) in [3]), we find a ratio Ψ ( r ) j,z =1 / Ψ j,z =1 , d − X l =0 ( l ) r = ( d ) r +1 r + 1 −→ Ψ ( r ) j,z =1 Ψ j,z =1 = ( d j − r r + 1 . (B2)Substituting (B2) into (B1) we arrive at formulas (17). C Appendix: Symmetric polynomials T k ( X , . . . , X m ) We present formulas for the first symmetric polynomials T k ( X , . . . , X m ) up to k = 7 . T = 1 , (C1) T = 12 X ,T = 13 3 X + X ,T = 14 X + X X ,T = 15 15 X + 30 X X + 5 X − X ,T = 16 3 X + 10 X X + 5 X − X X ,T = 17 63 X + 315 X X + 315 X X − X X + 35 X − X X + 16 X ,T = 18 9 X + 63 X X + 105 X X − X X + 35 X − X X + 16 X X . References [1] H. Bresinsky,
Symmetric semigroups of integers generated by four elements , Manuscripta Math., , 205-219 (1975)[2] I.M. Gelfand, M.M. Kapranov, A.V. Zelevinski, Discriminants, Resultants and MultidimensionalDeterminants , BirkhWauser, Boston, 1994.[3] R.L. Graham, D.E. Knuth & O. Patashnik,
Concrete Mathematics: a foundation for computer sci-ence , Addison-Wesley, NY, 2nd ed., 1994[4] L.G. Fel and B.Y. Rubinstein,
Power sums related to semigroups S ( d , d , d ) , Semigroup Forum, , 93-98 (2007)[5] L.G. Fel, Duality relation for the Hilbert series of almost symmetric numerical semigroups , IsraelJ. Math. , 413-444 (2011). 206] L.G. Fel,
On Frobenius numbers for symmetric (not complete intersection) semigroups generatedby four elements
Semigroup Forum, , 423-426 (2016).[7] L.G. Fel, Restricted partition functions and identities for degrees of syzygies in numerical semi-groups , Ramanujan J., , 465-491 (2017)[8] L.G. Fel, Symmetric (not complete intersection) semigroups generated by five elements , Integers:The Electronic J. of Comb. Number Theory, 18 (2018),
Symmetric (not complete intersection) semigroups generated by six elements , In
Numer-ical Semigroups , Springer INdAM Series , 93-109 (2020)[10] L.G. Fel, Symmetric (not complete intersection) semigroups generated by 4,5,6 elements
Symmetric polynomials associated with numerical semigroups , preprint, 2020https://arxiv.org/pdf/2010.03363.pdf[12] B. Hejmej,
A note about irreducibility of a resultant , Bulletin dela Soci´et´e des Sciences et desLetters de L ´od´z, , 27-32 (2018)[13] J. Herzog and M. K ¨uhl, On the Betti numbers of finite pure and linear resolutions , Comm. Algebra, , 1627-1646, (1984)[14] I.G. Macdonald, Symmetric functions and Hall polynomials , Oxford: Clarendon Press, 1995[15] F. S. Macaulay,