Semigroups, Frobenius number and M obius function J.L. Ram rez - PowerPoint PPT Presentation

Kannan’s result Theorem (Kannan, 1992) There is a polynomial time algorithm to compute g ( a 1 , . . . , a n ) when n ≥ 2 is fixed. Let P be a closed bounded convex set in R n and let L be a lattice of dimension n also in R n . The least positive real t so that tP + L equals R n is called the covering radius of P with respect to L (denoted by µ ( P , L )). J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Kannan’s result Theorem (Kannan, 1992) There is a polynomial time algorithm to compute g ( a 1 , . . . , a n ) when n ≥ 2 is fixed. Let P be a closed bounded convex set in R n and let L be a lattice of dimension n also in R n . The least positive real t so that tP + L equals R n is called the covering radius of P with respect to L (denoted by µ ( P , L )). J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Kannan’s result Theorem (Kannan, 1992) There is a polynomial time algorithm to compute g ( a 1 , . . . , a n ) when n ≥ 2 is fixed. Let P be a closed bounded convex set in R n and let L be a lattice of dimension n also in R n . The least positive real t so that tP + L equals R n is called the covering radius of P with respect to L (denoted by µ ( P , L )). J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Kannan’s result Theorem (Kannan, 1992) There is a polynomial time algorithm to compute g ( a 1 , . . . , a n ) when n ≥ 2 is fixed. Let P be a closed bounded convex set in R n and let L be a lattice of dimension n also in R n . The least positive real t so that tP + L equals R n is called the covering radius of P with respect to L (denoted by µ ( P , L )). J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Theorem (Kannan, 1992) Let n − 1 L = { ( x 1 , . . . , x n − 1 ) | x i integers and � a i x i ≡ 0 m´ od a n } i =1 and n − 1 S = { ( x 1 , . . . , x n − 1 ) | x i ≥ 0 reals and � a i x i ≤ 1 } . i =1 Then, µ ( S , L ) = g ( a 1 , . . . , a n ) + a 1 + · · · + a n J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Theorem (Kannan, 1992) Let n − 1 L = { ( x 1 , . . . , x n − 1 ) | x i integers and � a i x i ≡ 0 m´ od a n } i =1 and n − 1 S = { ( x 1 , . . . , x n − 1 ) | x i ≥ 0 reals and � a i x i ≤ 1 } . i =1 Then, µ ( S , L ) = g ( a 1 , . . . , a n ) + a 1 + · · · + a n J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Example 1: Let a 1 = 3 , a 2 = 4 and a 3 = 5. J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Example 1: Let a 1 = 3 , a 2 = 4 and a 3 = 5. x 2 7 6 5 4 3 2 S 1 x1 0 1 2 3 4 5 6 7 J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Example 1 cont ... then, g (3 , 4 , 5) = 2 and thus µ ( L , S ) = 14 Notice that (14) S covers the plane while (13) S does not. J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Example 1 cont ... then, g (3 , 4 , 5) = 2 and thus µ ( L , S ) = 14 x2 x2 7 7 6 6 5 5 4 4 3 3 2 2 1 1 x1 x1 0 0 1 2 3 4 5 6 7 1 2 3 4 5 6 7 Uncovered gaps Notice that (14) S covers the plane while (13) S does not. J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Example 1 cont ... then, g (3 , 4 , 5) = 2 and thus µ ( L , S ) = 14 x2 x2 7 7 6 6 5 5 4 4 3 3 2 2 1 1 x1 x1 0 0 1 2 3 4 5 6 7 1 2 3 4 5 6 7 Uncovered gaps Notice that (14) S covers the plane while (13) S does not. J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Example 2: Let a 1 , a 2 be positive integers with gcd ( a 1 , a 2 ) = 1. Minimum integer t such that tS covers the interval [0 , b ] is ab . Thus, g ( a , b ) = µ ( S , L ) − a − b = ab − a − b . J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Example 2: Let a 1 , a 2 be positive integers with gcd ( a 1 , a 2 ) = 1. S 0 b 2b 3b 1/a Minimum integer t such that tS covers the interval [0 , b ] is ab . Thus, g ( a , b ) = µ ( S , L ) − a − b = ab − a − b . J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Example 2: Let a 1 , a 2 be positive integers with gcd ( a 1 , a 2 ) = 1. S 0 b 2b 3b 1/a Minimum integer t such that tS covers the interval [0 , b ] is ab . Thus, g ( a , b ) = µ ( S , L ) − a − b = ab − a − b . J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Hilbert series and Ap´ ery set Let A [ S ] = K [ z 1 , . . . , z n ] be the polynomial ring over K (of characteristic 0) associated to the semigroup S = � a 1 , . . . , a n � . Then, de Hilbert series of A [ S ] is Q ( z ) z s = � H ( A [ S ] , z ) = (1 − z a 1 ) · · · (1 − z a n ) · i ∈ S g ( a 1 , . . . , a n ) = degree of H ( A [ S ] , z ) Theorem (Herzog 1970, Morales 1987) Formula for H ( A [ S ] , z ) when S = � a , b , c � J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Hilbert series and Ap´ ery set Let A [ S ] = K [ z 1 , . . . , z n ] be the polynomial ring over K (of characteristic 0) associated to the semigroup S = � a 1 , . . . , a n � . Then, de Hilbert series of A [ S ] is Q ( z ) z s = � H ( A [ S ] , z ) = (1 − z a 1 ) · · · (1 − z a n ) · i ∈ S g ( a 1 , . . . , a n ) = degree of H ( A [ S ] , z ) Theorem (Herzog 1970, Morales 1987) Formula for H ( A [ S ] , z ) when S = � a , b , c � J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Hilbert series and Ap´ ery set Let A [ S ] = K [ z 1 , . . . , z n ] be the polynomial ring over K (of characteristic 0) associated to the semigroup S = � a 1 , . . . , a n � . Then, de Hilbert series of A [ S ] is Q ( z ) z s = � H ( A [ S ] , z ) = (1 − z a 1 ) · · · (1 − z a n ) · i ∈ S g ( a 1 , . . . , a n ) = degree of H ( A [ S ] , z ) Theorem (Herzog 1970, Morales 1987) Formula for H ( A [ S ] , z ) when S = � a , b , c � J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

The Ap´ ery set of S = � a 1 , . . . , a n � for m ∈ S is Ap ( S ; m ) = { s ∈ S | s − m �∈ S } 1 � z w S = Ap ( S ; m ) + m Z ≥ 0 , H ( S ; z ) = 1 − z m w ∈ Ap ( S ; m ) Theorem (R.A. and R¨ odseth, 2008) S = � a , a + d , . . . , a + kd , c � H ( S ; x ) = F s v ( a ; x )(1 − x c ( P v +1 − P v ) ) + F s v − s v +1 ( a ; x )( x c ( P v +1 − P v ) − x cP v +1 (1 − x a )(1 − x d )(1 − x a + kd )(1 − x c ) where s v , s v +1 , P v , P v +1 are some particular integers. Remark: Contains the case n = 3 when k = 1 and b = a + d . J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

The Ap´ ery set of S = � a 1 , . . . , a n � for m ∈ S is Ap ( S ; m ) = { s ∈ S | s − m �∈ S } 1 � z w S = Ap ( S ; m ) + m Z ≥ 0 , H ( S ; z ) = 1 − z m w ∈ Ap ( S ; m ) Theorem (R.A. and R¨ odseth, 2008) S = � a , a + d , . . . , a + kd , c � H ( S ; x ) = F s v ( a ; x )(1 − x c ( P v +1 − P v ) ) + F s v − s v +1 ( a ; x )( x c ( P v +1 − P v ) − x cP v +1 (1 − x a )(1 − x d )(1 − x a + kd )(1 − x c ) where s v , s v +1 , P v , P v +1 are some particular integers. Remark: Contains the case n = 3 when k = 1 and b = a + d . J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

The Ap´ ery set of S = � a 1 , . . . , a n � for m ∈ S is Ap ( S ; m ) = { s ∈ S | s − m �∈ S } 1 � z w S = Ap ( S ; m ) + m Z ≥ 0 , H ( S ; z ) = 1 − z m w ∈ Ap ( S ; m ) Theorem (R.A. and R¨ odseth, 2008) S = � a , a + d , . . . , a + kd , c � H ( S ; x ) = F s v ( a ; x )(1 − x c ( P v +1 − P v ) ) + F s v − s v +1 ( a ; x )( x c ( P v +1 − P v ) − x cP v +1 (1 − x a )(1 − x d )(1 − x a + kd )(1 − x c ) where s v , s v +1 , P v , P v +1 are some particular integers. Remark: Contains the case n = 3 when k = 1 and b = a + d . J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

The Ap´ ery set of S = � a 1 , . . . , a n � for m ∈ S is Ap ( S ; m ) = { s ∈ S | s − m �∈ S } 1 � z w S = Ap ( S ; m ) + m Z ≥ 0 , H ( S ; z ) = 1 − z m w ∈ Ap ( S ; m ) Theorem (R.A. and R¨ odseth, 2008) S = � a , a + d , . . . , a + kd , c � H ( S ; x ) = F s v ( a ; x )(1 − x c ( P v +1 − P v ) ) + F s v − s v +1 ( a ; x )( x c ( P v +1 − P v ) − x cP v +1 (1 − x a )(1 − x d )(1 − x a + kd )(1 − x c ) where s v , s v +1 , P v , P v +1 are some particular integers. Remark: Contains the case n = 3 when k = 1 and b = a + d . J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Input: a , d , c , k , s 0 Output: s v , s v +1 , P v , P v +1 1. r − 1 = a , r 0 = s 0 2. r i − 1 = κ i +1 r i + r i +1 , κ i +1 = ⌊ r i − 1 / r i ⌋ , 0 = r µ +1 < r µ < · · · < r − 1 3. p i +1 = κ i +1 p i + p i − 1 , p − 1 = 0 , p 0 = 1 4. T i +1 = − κ i +1 T i + T i − 1 , T − 1 = a + kd , T 0 = 1 a (( a + kd ) r 0 − kc ) 5. If there is a minimal u such that T 2 u +2 ≤ 0, Then � − T 2 u +2 � � � � � � � s v P v γ 1 r 2 u +1 − p 2 u +1 = , γ = + 1 γ − 1 1 s v +1 P v +1 r 2 u +2 p 2 u +2 T 2 u +1 6. Else s v = r µ , s v +1 = 0 , P v = p µ , P v +1 = p µ +1 . J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Fast Algorithms (computational algebraic methods) Einstein, Lichtblau, Strzebonski and Wagon Find g ( a 1 , . . . , a 4 ) involving 100-digit numbers in about one second Find g ( a 1 , . . . , a 10 ) involving 10-digit numbers in two days Roune Find g ( a 1 , . . . , a 4 ) involving 10, 000-digit numbers in few second Find g ( a 1 , . . . , a 13 ) involving 10-digit numbers in few days Package http://www.broune.com/frobby/ http://www.math.ruu.nl/people/beukers/frobenius/ http://cmup.fc.up.pt/cmup/mdelgado/numericalsgps/ http://reference.wolfram.com/mathematica/ref/FrobeniusNumber.html J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Fast Algorithms (computational algebraic methods) Einstein, Lichtblau, Strzebonski and Wagon Find g ( a 1 , . . . , a 4 ) involving 100-digit numbers in about one second Find g ( a 1 , . . . , a 10 ) involving 10-digit numbers in two days Roune Find g ( a 1 , . . . , a 4 ) involving 10, 000-digit numbers in few second Find g ( a 1 , . . . , a 13 ) involving 10-digit numbers in few days Package http://www.broune.com/frobby/ http://www.math.ruu.nl/people/beukers/frobenius/ http://cmup.fc.up.pt/cmup/mdelgado/numericalsgps/ http://reference.wolfram.com/mathematica/ref/FrobeniusNumber.html J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Fast Algorithms (computational algebraic methods) Einstein, Lichtblau, Strzebonski and Wagon Find g ( a 1 , . . . , a 4 ) involving 100-digit numbers in about one second Find g ( a 1 , . . . , a 10 ) involving 10-digit numbers in two days Roune Find g ( a 1 , . . . , a 4 ) involving 10, 000-digit numbers in few second Find g ( a 1 , . . . , a 13 ) involving 10-digit numbers in few days Package http://www.broune.com/frobby/ http://www.math.ruu.nl/people/beukers/frobenius/ http://cmup.fc.up.pt/cmup/mdelgado/numericalsgps/ http://reference.wolfram.com/mathematica/ref/FrobeniusNumber.html J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Applications A semigroup S is called symmetric if S ∪ ( g − S ) = Z . (Bresinsky, 1979) Monomial curves (Kunz, 1979, Herzog, 1970) Gorestein rings (Ap´ ery, 1945) Classification plane of algebraic branches (Buchweitz, 1981) Weierstrass semigroups (Pellikaan and Torres, 1999) Algebraic codes J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Applications A semigroup S is called symmetric if S ∪ ( g − S ) = Z . (Bresinsky, 1979) Monomial curves (Kunz, 1979, Herzog, 1970) Gorestein rings (Ap´ ery, 1945) Classification plane of algebraic branches (Buchweitz, 1981) Weierstrass semigroups (Pellikaan and Torres, 1999) Algebraic codes J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Shell-sort method 3,2,7,9,8,1,1,5,2,6 (increment sequence: 7,3,1) 7-sorted: 3,2,6,9,8,1,1,5,2,6 3-sorted: 1,2,1,3,5,2,7,8,6,9 1-sorted: 1,1,2,2,3,5,6,7,8,9 J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Shell-sort method 3,2,7,9,8,1,1,5,2,6 (increment sequence: 7,3,1) 7-sorted: 3,2,6,9,8,1,1,5,2,6 3-sorted: 1,2,1,3,5,2,7,8,6,9 1-sorted: 1,1,2,2,3,5,6,7,8,9 J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Shell-sort method 3,2,7,9,8,1,1,5,2,6 (increment sequence: 7,3,1) 7-sorted: 3,2,6,9,8,1,1,5,2,6 3-sorted: 1,2,1,3,5,2,7,8,6,9 1-sorted: 1,1,2,2,3,5,6,7,8,9 J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Shell-sort method 3,2,7,9,8,1,1,5,2,6 (increment sequence: 7,3,1) 7-sorted: 3,2,6,9,8,1,1,5,2,6 3-sorted: 1,2,1,3,5,2,7,8,6,9 1-sorted: 1,1,2,2,3,5,6,7,8,9 J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Shell-sort method 3,2,7,9,8,1,1,5,2,6 (increment sequence: 7,3,1) 7-sorted: 3,2,6,9,8,1,1,5,2,6 3-sorted: 1,2,1,3,5,2,7,8,6,9 1-sorted: 1,1,2,2,3,5,6,7,8,9 J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Shell-sort method 3,2,7,9,8,1,1,5,2,6 (increment sequence: 7,3,1) 7-sorted: 3,2,6,9,8,1,1,5,2,6 3-sorted: 1,2,1,3,5,2,7,8,6,9 1-sorted: 1,1,2,2,3,5,6,7,8,9 J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Lemme (Incerpi and Sedgewick, 1985)The number of steps required to h j -sort a set on N integers that is already h j +1 − h j +2 − · · · − h t -sorted is � Ng ( h j +1 , h j +2 , . . . , h t ) � O h j Theorem (Incerpi and Sedgewick, 1985)The running time of Shell-sort is O ( N 3 / 2 ) where N is the number of elements in the file (on average and in worst case). Conjecture (Gonnet, 1984)The asymptotic growth of the average case running time of Shell-sort is O ( N log N log log N ) where N is the number of elements in the file. J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Lemme (Incerpi and Sedgewick, 1985)The number of steps required to h j -sort a set on N integers that is already h j +1 − h j +2 − · · · − h t -sorted is � Ng ( h j +1 , h j +2 , . . . , h t ) � O h j Theorem (Incerpi and Sedgewick, 1985)The running time of Shell-sort is O ( N 3 / 2 ) where N is the number of elements in the file (on average and in worst case). Conjecture (Gonnet, 1984)The asymptotic growth of the average case running time of Shell-sort is O ( N log N log log N ) where N is the number of elements in the file. J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Lemme (Incerpi and Sedgewick, 1985)The number of steps required to h j -sort a set on N integers that is already h j +1 − h j +2 − · · · − h t -sorted is � Ng ( h j +1 , h j +2 , . . . , h t ) � O h j Theorem (Incerpi and Sedgewick, 1985)The running time of Shell-sort is O ( N 3 / 2 ) where N is the number of elements in the file (on average and in worst case). Conjecture (Gonnet, 1984)The asymptotic growth of the average case running time of Shell-sort is O ( N log N log log N ) where N is the number of elements in the file. J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Basics on posets Let ( P , ≤ ) be a locally finite poset , i.e, the set P is partially ordered by ≤ , and for every a , b ∈ P the set { c ∈ P | a ≤ c ≤ b } is finite. A chain of length l ≥ 0 between a , b ∈ P is { a = a 0 < a 1 < · · · < a l = b } ⊂ P . We denote by c l ( a , b ) the number of chains of length l between a and b . The M¨ obius function µ P is the function µ P : P × P − → Z � ( − 1) l c l ( a , b ) . µ P ( a , b ) = l ≥ 0 J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Basics on posets Let ( P , ≤ ) be a locally finite poset , i.e, the set P is partially ordered by ≤ , and for every a , b ∈ P the set { c ∈ P | a ≤ c ≤ b } is finite. A chain of length l ≥ 0 between a , b ∈ P is { a = a 0 < a 1 < · · · < a l = b } ⊂ P . We denote by c l ( a , b ) the number of chains of length l between a and b . The M¨ obius function µ P is the function µ P : P × P − → Z � ( − 1) l c l ( a , b ) . µ P ( a , b ) = l ≥ 0 J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Basics on posets Let ( P , ≤ ) be a locally finite poset , i.e, the set P is partially ordered by ≤ , and for every a , b ∈ P the set { c ∈ P | a ≤ c ≤ b } is finite. A chain of length l ≥ 0 between a , b ∈ P is { a = a 0 < a 1 < · · · < a l = b } ⊂ P . We denote by c l ( a , b ) the number of chains of length l between a and b . The M¨ obius function µ P is the function µ P : P × P − → Z � ( − 1) l c l ( a , b ) . µ P ( a , b ) = l ≥ 0 J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Consider the poset ( N , | ) of nonnegative integers ordered by divisibility , i.e., a | b ⇐ ⇒ a divides b . Let us compute µ N (2 , 36). We observe that { c ∈ N ; 2 | c | 36 } = { 2 , 4 , 6 , 12 , 18 , 36 } . Chains of 36 length 1 → { 2 , 36 }  { 2 , 4 , 36 }   { 2 , 6 , 36 }  12 18 length 2 { 2 , 12 , 36 }   { 2 , 18 , 36 }   4 6 { 2 , 4 , 12 , 36 }  { 2 , 6 , 12 , 26 } length 3 { 2 , 6 , 18 , 36 }  2 Thus, µ N (2 , 36) = − c 1 (2 , 36) + c 2 (2 , 36) − c 3 (2 , 36) = − 1 + 4 − 3 = 0 . J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

M¨ obius classical arithmetic function Given n ∈ N the M¨ obius arithmetic function µ ( n ) is defined as  1 if n = 1   ( − 1) k if n = p 1 · · · p k with p i distincts primes  µ ( n ) = 0 otherwise (i.e; n admits at least one square   factor bigger than one)  Example: µ (2) = µ (7) = − 1 , µ (4) = µ (8) = 0 , µ (6) = µ (10) = 1 The inverse of the Riemann function ζ , s ∈ C , Re ( s ) > 0 � − 1 � + ∞ + ∞ µ ( n ) 1 ζ − 1 ( s ) = p − prime (1 − p − 1 ) = � = � � n 2 . n s n =1 n =1 J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

M¨ obius classical arithmetic function Given n ∈ N the M¨ obius arithmetic function µ ( n ) is defined as  1 if n = 1   ( − 1) k if n = p 1 · · · p k with p i distincts primes  µ ( n ) = 0 otherwise (i.e; n admits at least one square   factor bigger than one)  Example: µ (2) = µ (7) = − 1 , µ (4) = µ (8) = 0 , µ (6) = µ (10) = 1 The inverse of the Riemann function ζ , s ∈ C , Re ( s ) > 0 � − 1 � + ∞ + ∞ µ ( n ) 1 ζ − 1 ( s ) = p − prime (1 − p − 1 ) = � = � � n 2 . n s n =1 n =1 J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

M¨ obius classical arithmetic function Given n ∈ N the M¨ obius arithmetic function µ ( n ) is defined as  1 if n = 1   ( − 1) k if n = p 1 · · · p k with p i distincts primes  µ ( n ) = 0 otherwise (i.e; n admits at least one square   factor bigger than one)  Example: µ (2) = µ (7) = − 1 , µ (4) = µ (8) = 0 , µ (6) = µ (10) = 1 The inverse of the Riemann function ζ , s ∈ C , Re ( s ) > 0 � − 1 � + ∞ + ∞ µ ( n ) 1 ζ − 1 ( s ) = p − prime (1 − p − 1 ) = � = � � n 2 . n s n =1 n =1 J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

M¨ obius classical arithmetic function There are impressive results using µ , for instance for an integer n Pr ( n do not contain a square factor ) = 6 π 2 For ( N , | ) we have that for all a , b ∈ N ( − 1) r  if b / a is a product of r distinct primes  µ N ( a , b ) = 0 otherwise  µ N (2 , 36) = 0 because 36 / 2 = 18 = 2 · 3 2 J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

M¨ obius classical arithmetic function There are impressive results using µ , for instance for an integer n Pr ( n do not contain a square factor ) = 6 π 2 For ( N , | ) we have that for all a , b ∈ N ( − 1) r  if b / a is a product of r distinct primes  µ N ( a , b ) = 0 otherwise  µ N (2 , 36) = 0 because 36 / 2 = 18 = 2 · 3 2 J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

M¨ obius inversion formula Theorem Let ( P , ≤ ) be a poset, let p be an element of P and consider f : P → R a function such that f ( x ) = 0 for all x � p. Suppose that � g ( x ) = f ( y ) for all x ∈ P . y ≤ x Then, � g ( y ) µ P ( y , x ) for all x ∈ P . f ( x ) = y ≤ x J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Compute the Euler function φ ( n ) (the number of integers smaller or equal to n and coprime with n ) µ ( d ) � φ ( n ) = n d d | n Let D be a finite set and consider the poset ( P , ⊂ ) of multisets over D ordered by inclusion . Then, for all A , B multisets over D we have that ( − 1) | B \ A |  if A ⊂ B and B \ A is a set  µ P ( A , B ) = 0 otherwise  J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Compute the Euler function φ ( n ) (the number of integers smaller or equal to n and coprime with n ) µ ( d ) � φ ( n ) = n d d | n Let D be a finite set and consider the poset ( P , ⊂ ) of multisets over D ordered by inclusion . Then, for all A , B multisets over D we have that ( − 1) | B \ A |  if A ⊂ B and B \ A is a set  µ P ( A , B ) = 0 otherwise  J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Semigroup poset Let S := � a 1 , . . . , a n � ⊂ N m denote the subsemigroup of N m generated by a 1 , . . . , a n ∈ N m , i.e., S := � a 1 , . . . , a n � = { x 1 a 1 + · · · + x n a n | x 1 , . . . , x n ∈ N } . The semigroup S induces an partial order ≤ S on N m given by x ≤ S y ⇐ ⇒ y − x ∈ S . obius function associated to ( N m , ≤ S ). We denote by µ S the M¨ ∈ N m , or It is easy to check that µ S ( x , y ) = 0 if y − x / µ S ( x , y ) = µ S (0 , y − x ) otherwise. Hence we shall only consider obius function µ S : N m − → Z defined by the reduced M¨ for all x ∈ N m . µ S ( x ) := µ S (0 , x ) J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Semigroup poset Let S := � a 1 , . . . , a n � ⊂ N m denote the subsemigroup of N m generated by a 1 , . . . , a n ∈ N m , i.e., S := � a 1 , . . . , a n � = { x 1 a 1 + · · · + x n a n | x 1 , . . . , x n ∈ N } . The semigroup S induces an partial order ≤ S on N m given by x ≤ S y ⇐ ⇒ y − x ∈ S . obius function associated to ( N m , ≤ S ). We denote by µ S the M¨ ∈ N m , or It is easy to check that µ S ( x , y ) = 0 if y − x / µ S ( x , y ) = µ S (0 , y − x ) otherwise. Hence we shall only consider obius function µ S : N m − → Z defined by the reduced M¨ for all x ∈ N m . µ S ( x ) := µ S (0 , x ) J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Semigroup poset Let S := � a 1 , . . . , a n � ⊂ N m denote the subsemigroup of N m generated by a 1 , . . . , a n ∈ N m , i.e., S := � a 1 , . . . , a n � = { x 1 a 1 + · · · + x n a n | x 1 , . . . , x n ∈ N } . The semigroup S induces an partial order ≤ S on N m given by x ≤ S y ⇐ ⇒ y − x ∈ S . obius function associated to ( N m , ≤ S ). We denote by µ S the M¨ ∈ N m , or It is easy to check that µ S ( x , y ) = 0 if y − x / µ S ( x , y ) = µ S (0 , y − x ) otherwise. Hence we shall only consider obius function µ S : N m − → Z defined by the reduced M¨ for all x ∈ N m . µ S ( x ) := µ S (0 , x ) J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Known results about µ S 1 Deddens (1979). For S = � a , b � ⊂ N where a , b ∈ Z + are relatively prime:  1 if x ≡ 0 or a + b ( mod ab )  − 1 if x ≡ a or b ( mod ab ) µ S ( x ) = 0 otherwise  2 Chappelon and R.A. (2013). They provide a recursive formula for µ S when S = � a , a + d , . . . , a + kd � ⊂ N for some a , k , d ∈ Z + , and a semi-explicit formula for S = � a , a + d , a + 2 d � ⊂ N where a , d ∈ Z + , gcd { a , a + d , a + 2 d } = 1 and a is even. In both papers the authors approach the problem by a thorough study of the intrinsic properties of each semigroup . J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Known results about µ S 1 Deddens (1979). For S = � a , b � ⊂ N where a , b ∈ Z + are relatively prime:  1 if x ≡ 0 or a + b ( mod ab )  − 1 if x ≡ a or b ( mod ab ) µ S ( x ) = 0 otherwise  2 Chappelon and R.A. (2013). They provide a recursive formula for µ S when S = � a , a + d , . . . , a + kd � ⊂ N for some a , k , d ∈ Z + , and a semi-explicit formula for S = � a , a + d , a + 2 d � ⊂ N where a , d ∈ Z + , gcd { a , a + d , a + 2 d } = 1 and a is even. In both papers the authors approach the problem by a thorough study of the intrinsic properties of each semigroup . J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Main objectives 1 Provide general tools to study µ S for every semigroup S ⊂ N m . 2 Provide explicit formulas for certain families of semigroups S ⊂ N m . J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Main objectives 1 Provide general tools to study µ S for every semigroup S ⊂ N m . 2 Provide explicit formulas for certain families of semigroups S ⊂ N m . J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

S -graded Hilbert series Let k be a field. A semigroup S = � a 1 , . . . , a n � ⊂ N m induces a grading in the ring of polynomials k [ x 1 , . . . , x n ] by assigning deg S ( x i ) := a i for all i ∈ { 1 , . . . , n } . For all b ∈ N m , we denote by k [ x 1 , . . . , x n ] b the k -vector space formed by all polynomials S -homogeneous of S -degree b . Consider I ⊂ k [ x ] an ideal generated by S -homogeneous polynomials. For all b ∈ N m we denote by I b the k -vector space formed by the S -homogeneous polynomials of I of S -degree b . J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

S -graded Hilbert series Let k be a field. A semigroup S = � a 1 , . . . , a n � ⊂ N m induces a grading in the ring of polynomials k [ x 1 , . . . , x n ] by assigning deg S ( x i ) := a i for all i ∈ { 1 , . . . , n } . For all b ∈ N m , we denote by k [ x 1 , . . . , x n ] b the k -vector space formed by all polynomials S -homogeneous of S -degree b . Consider I ⊂ k [ x ] an ideal generated by S -homogeneous polynomials. For all b ∈ N m we denote by I b the k -vector space formed by the S -homogeneous polynomials of I of S -degree b . J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

S -graded Hilbert series Let k be a field. A semigroup S = � a 1 , . . . , a n � ⊂ N m induces a grading in the ring of polynomials k [ x 1 , . . . , x n ] by assigning deg S ( x i ) := a i for all i ∈ { 1 , . . . , n } . For all b ∈ N m , we denote by k [ x 1 , . . . , x n ] b the k -vector space formed by all polynomials S -homogeneous of S -degree b . Consider I ⊂ k [ x ] an ideal generated by S -homogeneous polynomials. For all b ∈ N m we denote by I b the k -vector space formed by the S -homogeneous polynomials of I of S -degree b . J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

The S -graded Hilbert function of M := k [ x 1 , . . . , x n ] / I is HF M : N m − → N , where HF M ( b ) := dim k ( k [ x 1 , . . . , x n ] b ) − dim k ( I b ) for all b ∈ N m . We define the S -graded Hilbert series of M as the formal power series in Z [[ t 1 , . . . , t m ]]: � HF M ( b ) t b H M ( t ) := b ∈ N m J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

The S -graded Hilbert function of M := k [ x 1 , . . . , x n ] / I is HF M : N m − → N , where HF M ( b ) := dim k ( k [ x 1 , . . . , x n ] b ) − dim k ( I b ) for all b ∈ N m . We define the S -graded Hilbert series of M as the formal power series in Z [[ t 1 , . . . , t m ]]: � HF M ( b ) t b H M ( t ) := b ∈ N m J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

We denote by I S the toric ideal of S , i.e., the kernel of the homomorphism of k -algebras ϕ : k [ x 1 , . . . , x n ] − → k [ t 1 , . . . , t m ] induced by ϕ ( x i ) = t a i for all i ∈ { 1 , . . . , n } . It is well known that I S is generated by S -homogeneous polynomials. Proposition: H k [ x 1 ,..., x n ] / I S ( t ) = H S ( t ) J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

We denote by I S the toric ideal of S , i.e., the kernel of the homomorphism of k -algebras ϕ : k [ x 1 , . . . , x n ] − → k [ t 1 , . . . , t m ] induced by ϕ ( x i ) = t a i for all i ∈ { 1 , . . . , n } . It is well known that I S is generated by S -homogeneous polynomials. Proposition: H k [ x 1 ,..., x n ] / I S ( t ) = H S ( t ) J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Hilbert series of a semigroup For every b = ( b 1 , . . . , b m ) ∈ N m , we denote t b := t b 1 1 · · · t b m m . Let S ⊂ N m be a semigroup, the Hilbert series of S is t b ∈ Z [[ t 1 , . . . , t m ]] � H S ( t ) := b ∈S Examples: (1) For S = � 2 , 3 � ⊂ N , we have that S = { 0 , 2 , 3 , 4 , 5 . . . } t 2 t 3 t 4 t 5 H S ( t ) = 1 + + + + + · · · t 2 H S ( t ) = t 2 t 4 t 5 + + + · · · Then, (1 − t 2 ) H S ( t ) = 1 + t 3 , and H S ( t ) = 1 + t 3 1 − t 2 J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Hilbert series of a semigroup For every b = ( b 1 , . . . , b m ) ∈ N m , we denote t b := t b 1 1 · · · t b m m . Let S ⊂ N m be a semigroup, the Hilbert series of S is t b ∈ Z [[ t 1 , . . . , t m ]] � H S ( t ) := b ∈S Examples: (1) For S = � 2 , 3 � ⊂ N , we have that S = { 0 , 2 , 3 , 4 , 5 . . . } t 2 t 3 t 4 t 5 H S ( t ) = 1 + + + + + · · · t 2 H S ( t ) = t 2 t 4 t 5 + + + · · · Then, (1 − t 2 ) H S ( t ) = 1 + t 3 , and H S ( t ) = 1 + t 3 1 − t 2 J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Hilbert series of a semigroup For every b = ( b 1 , . . . , b m ) ∈ N m , we denote t b := t b 1 1 · · · t b m m . Let S ⊂ N m be a semigroup, the Hilbert series of S is t b ∈ Z [[ t 1 , . . . , t m ]] � H S ( t ) := b ∈S Examples: (1) For S = � 2 , 3 � ⊂ N , we have that S = { 0 , 2 , 3 , 4 , 5 . . . } t 2 t 3 t 4 t 5 H S ( t ) = 1 + + + + + · · · t 2 H S ( t ) = t 2 t 4 t 5 + + + · · · Then, (1 − t 2 ) H S ( t ) = 1 + t 3 , and H S ( t ) = 1 + t 3 1 − t 2 J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

(2) For S = N m , we have that b ∈ N m t b = � ( b 1 ,..., b m ) ∈ N m t b 1 1 · · · t b m H S ( t ) = � m (1 + t 1 + t 2 1 + · · · ) · · · (1 + t m + t 2 = m + · · · ) = 1 = (1 − t 1 ) ··· (1 − t m ) J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

M¨ obius function via Hilbert series Assume that one can write b ∈ ∆ f b t b � H S ( t ) = (1 − t c 1 ) · · · (1 − t c k ) for some finite set ∆ ⊂ N m and some c 1 , . . . , c k ∈ N m . Theorem (1) � f b µ S ( x − b ) = 0 b ∈ ∆ for all x / ∈ { � i ∈ A c i | A ⊂ { 1 , . . . , k }} . J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

M¨ obius function via Hilbert series Assume that one can write b ∈ ∆ f b t b � H S ( t ) = (1 − t c 1 ) · · · (1 − t c k ) for some finite set ∆ ⊂ N m and some c 1 , . . . , c k ∈ N m . Theorem (1) � f b µ S ( x − b ) = 0 b ∈ ∆ for all x / ∈ { � i ∈ A c i | A ⊂ { 1 , . . . , k }} . J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Example: S = � 2 , 3 � We know that, H S ( t ) = 1 + t 3 1 − t 2 . By Theorem (1) we have that µ S ( x ) + µ S ( x − 3) = 0 for all x / ∈ { 0 , 2 } . It is evident that µ S (0) = 1. A direct computation yields µ S (2) = − 1. Hence  1 if x ≡ 0 or 5 ( mod 6)  − 1 if x ≡ 2 or 3 ( mod 6) µ S ( x ) = 0 otherwise  J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Example: S = � 2 , 3 � We know that, H S ( t ) = 1 + t 3 1 − t 2 . By Theorem (1) we have that µ S ( x ) + µ S ( x − 3) = 0 for all x / ∈ { 0 , 2 } . It is evident that µ S (0) = 1. A direct computation yields µ S (2) = − 1. Hence  1 if x ≡ 0 or 5 ( mod 6)  − 1 if x ≡ 2 or 3 ( mod 6) µ S ( x ) = 0 otherwise  J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

M¨ obius function via Hilbert series We consider G S the generating function of the M¨ obius function, which is � µ S ( b ) t b . G S ( t ) := b ∈ N m Theorem (2) H S ( t ) G S ( t ) = 1 . J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Example: S = N m We denote { e 1 , . . . , e m } the canonical basis of N m , i.e., e 1 = (1 , 0 , . . . , 0) , . . . , e m = (0 , . . . , 0 , 1) ∈ N m . We know that 1 H N m ( t ) = (1 − t 1 ) · · · (1 − t m ) By Theorem (2) we have that ( − 1) | A | t � � i ∈ A e i . G N m ( t ) = (1 − t 1 ) · · · (1 − t m ) = A ⊂{ 1 ,..., m } Hence,  ( − 1) | A | if x = � i ∈ A e i for some A ⊂ { 1 , . . . , m }  µ N m ( x ) = 0 otherwise  J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Example: S = N m We denote { e 1 , . . . , e m } the canonical basis of N m , i.e., e 1 = (1 , 0 , . . . , 0) , . . . , e m = (0 , . . . , 0 , 1) ∈ N m . We know that 1 H N m ( t ) = (1 − t 1 ) · · · (1 − t m ) By Theorem (2) we have that ( − 1) | A | t � � i ∈ A e i . G N m ( t ) = (1 − t 1 ) · · · (1 − t m ) = A ⊂{ 1 ,..., m } Hence,  ( − 1) | A | if x = � i ∈ A e i for some A ⊂ { 1 , . . . , m }  µ N m ( x ) = 0 otherwise  J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Definition We say that a semigroup S = � a 1 , . . . , a n � ⊂ N m is a complete intersection semigroup if its corresponding toric ideal I S is a complete intersection. Moreover, I S is a complete intersection if there exists a system of s = n − dim ( Q { a 1 , . . . , a n } ) S -homogeneous polynomials f 1 , . . . , f s such that I S = ( f 1 , . . . , f s ) . Whenever I S is a complete intersection generated S -homogeneous polynomials of S -degrees b 1 , . . . , b s ∈ N m , then H S ( t ) = (1 − t b 1 ) · · · (1 − t b s ) (1 − t a 1 ) · · · (1 − t a n ) J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Definition We say that a semigroup S = � a 1 , . . . , a n � ⊂ N m is a complete intersection semigroup if its corresponding toric ideal I S is a complete intersection. Moreover, I S is a complete intersection if there exists a system of s = n − dim ( Q { a 1 , . . . , a n } ) S -homogeneous polynomials f 1 , . . . , f s such that I S = ( f 1 , . . . , f s ) . Whenever I S is a complete intersection generated S -homogeneous polynomials of S -degrees b 1 , . . . , b s ∈ N m , then H S ( t ) = (1 − t b 1 ) · · · (1 − t b s ) (1 − t a 1 ) · · · (1 − t a n ) J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Explicit formulas for µ S A semigroup S ⊂ N m is said to be a semigroup with a unique Betti element b ∈ N m if I S is generated by S -homogeneous polynomials of S -degree b . Theorem Set r := dim ( Q { a 1 , . . . , a n } ) . Then, t � k j + n − r − 1 � � ( − 1) | A j | µ S ( x ) = , k j j =1 if x = � i ∈ A 1 a i + k 1 b = · · · = � i ∈ A t a i + k t b for k 1 , . . . , k t ∈ N . J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

Explicit formulas for µ S A semigroup S ⊂ N m is said to be a semigroup with a unique Betti element b ∈ N m if I S is generated by S -homogeneous polynomials of S -degree b . Theorem Set r := dim ( Q { a 1 , . . . , a n } ) . Then, t � k j + n − r − 1 � � ( − 1) | A j | µ S ( x ) = , k j j =1 if x = � i ∈ A 1 a i + k 1 b = · · · = � i ∈ A t a i + k t b for k 1 , . . . , k t ∈ N . J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

When S = � a 1 , . . . , a n � ⊂ N is a semigroup with a unique Betti element and gcd { a 1 , . . . , a n } = 1, It is known that there exist pairwise relatively prime different integers b 1 , . . . , b n ≥ 2 such that a i := � j � = i b j for all i ∈ { 1 , . . . , n } . In this setting we can refine the previous Theorem. Corollary Set b := � n i =1 b i , then  if x = � i ∈ A a i + k b ( − 1) | A | � k + n − 2 �   k for some A ⊂ { 1 , . . . , n } , k ∈ N  µ S ( x ) =   0  otherwise J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

When S = � a 1 , . . . , a n � ⊂ N is a semigroup with a unique Betti element and gcd { a 1 , . . . , a n } = 1, It is known that there exist pairwise relatively prime different integers b 1 , . . . , b n ≥ 2 such that a i := � j � = i b j for all i ∈ { 1 , . . . , n } . In this setting we can refine the previous Theorem. Corollary Set b := � n i =1 b i , then  if x = � i ∈ A a i + k b ( − 1) | A | � k + n − 2 �   k for some A ⊂ { 1 , . . . , n } , k ∈ N  µ S ( x ) =   0  otherwise J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

S = � a 1 , a 2 , a 3 � ⊂ N complete intersection For every x ∈ Z we denote by α ( x ) the only integer such that 0 ≤ α ( x ) ≤ d − 1 and α ( x ) a 1 ≡ x ( mod d ). For every x ∈ Z and every B = ( b 1 , . . . , b k ) ⊂ ( Z + ) k , the Sylvester denumerant d B ( x ) is the number of non-negative integer solutions ( x 1 , . . . , x k ) ∈ N k to the equation x = � k i =1 x i b i . For S = � a 1 , a 2 , a 3 � complete intersection and gcd { a 1 , a 2 , a 3 } = 1, we have the following result. Theorem µ S ( x ) = 0 if α ( x ) ≥ 2 , or µ S ( x ) = ( − 1) α ( d B ( x ′ ) − d B ( x ′ − a 2 ) − d B ( x ′ − a 3 )+ d B ( x ′ − a 2 − a 3 )) otherwise, where x ′ := x − α ( x ) a 1 and B := ( da 1 , a 2 a 3 / d ) . J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function

S = � a 1 , a 2 , a 3 � ⊂ N complete intersection For every x ∈ Z we denote by α ( x ) the only integer such that 0 ≤ α ( x ) ≤ d − 1 and α ( x ) a 1 ≡ x ( mod d ). For every x ∈ Z and every B = ( b 1 , . . . , b k ) ⊂ ( Z + ) k , the Sylvester denumerant d B ( x ) is the number of non-negative integer solutions ( x 1 , . . . , x k ) ∈ N k to the equation x = � k i =1 x i b i . For S = � a 1 , a 2 , a 3 � complete intersection and gcd { a 1 , a 2 , a 3 } = 1, we have the following result. Theorem µ S ( x ) = 0 if α ( x ) ≥ 2 , or µ S ( x ) = ( − 1) α ( d B ( x ′ ) − d B ( x ′ − a 2 ) − d B ( x ′ − a 3 )+ d B ( x ′ − a 2 − a 3 )) otherwise, where x ′ := x − α ( x ) a 1 and B := ( da 1 , a 2 a 3 / d ) . J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Semigroups, Frobenius’ number and Moebius function