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Numerical semigroups: suitable sets of pseudo Numerical semigroups: suitable sets of pseudo-Frobenius numbers Aureliano M. Robles-Perez´ Universidad de Granada A talk based on joint

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Text of Numerical semigroups: suitable sets of pseudo Numerical semigroups: suitable sets of...

  • Numerical semigroups: suitable sets of pseudo-Frobenius numbers

    Aureliano M. Robles-Pérez Universidad de Granada

    A talk based on joint works with Manuel Delgado, Pedro A. Garcı́a-Sánchez and José Carlos Rosales

    Meeting on Computer Algebra and Applications (EACA 2016)

    22-24th June 2016, Logroño

    A. M. Robles-Pérez (UGR) Suitable sets of pseudo-Frobenius numbers EACA 2016 1 / 25

  • Notation

    ∗ Z= {. . . ,−2,−1,0,1,2, . . .} ∗ N= {0,1,2, . . .}

    Definition I S numerical semigroup = S cofinite submonoid of (N,+).

    Definition I x < S is the Frobenius number of S if x + s ∈ S for all s ∈ N \ {0}. I x < S is a pseudo-Frobenius number of S if x + s ∈ S for all s ∈ S \ {0}.

    ∗ PF(S) = {Pseudo-Frobenius numbers of S}. ∗ F(S) = max(PF(S)) = max(Z \S) = Frobenius number of S. ∗ m(S) = min(S \ {0}).

    A. M. Robles-Pérez (UGR) Suitable sets of pseudo-Frobenius numbers EACA 2016 2 / 25

  • Remark

    ∗ t(S) = ]PF(S) = type of S.

    ∗∗ Let K[x] be a polynomial ring in one variable over a field K. We define the semigroup ring of S as the monomial subring

    K[S] := K[ts | s ∈ S].

    Then t(S) coincides with the Cohen-Macaulay type of K[S].

    A. M. Robles-Pérez (UGR) Suitable sets of pseudo-Frobenius numbers EACA 2016 3 / 25

  • Background

    ∗ f ∈ N \ {0}. ∗ S(f) = {S numerical semigroup | F(S) = f }. ∗ S(f) , ∅ for all f ∈ N \ {0}. ∗ There are algorithms to compute S(f)

    (Rosales, Garcı́a-Sánchez, Garcı́a-Garcı́a, Jiménez-Madrid, J. Pure Appl. Algebra 189, 2004)

    ∗∗ PF = {g1,g2, . . . ,gn−1,gn} ⊆ N \ {0}. ∗∗ S(PF) = {S numerical semigroup | PF(S) = PF}. ∗∗ S(PF) may be empty if n ≥ 1. ∗∗ S(PF) is finite when it is non-empty.

    A. M. Robles-Pérez (UGR) Suitable sets of pseudo-Frobenius numbers EACA 2016 4 / 25

  • Problems to solve

    Questions 1. Find conditions on the set PF that ensure that S(PF) , ∅. 2. Find an algorithm to compute S(PF).

    Solved cases I PF = {f }: symmetric numerical semigroups.

    (Blanco, Rosales, Forum Math. 25, 2013)

    I PF = {f , f2 }: pseudo-symmetric numerical semigroups. (Blanco, Rosales, Forum Math. 25, 2013)

    I Question 1 for PF = {g1,g2}. (R-P, Rosales, article to appear in Proc. Roy. Soc. Edinburgh Sect. A.)

    I Question 2 for general PF. (Delgado, Garcı́a-Sánchez, R-P, LMS J. Comput. Math. 19(1), 2016)

    A. M. Robles-Pérez (UGR) Suitable sets of pseudo-Frobenius numbers EACA 2016 5 / 25

  • Question 1 for n=2 (1) Proposition

    I Let g1,g2 be two positive integers such that g1 < g2 and g2 is odd. Let m = 2g1−g2. There exists a numerical semigroup S such that PF(S) = {g1,g2} if and only if m > 0, m - g1, m - g2, m - (g2−g1).

    S = ( 〈m〉∪

    { g2 + 1

    2 ,→

    }) \ ({

    g1−km | k ∈ N } ∪

    { g2−km | k ∈ N

    }) Example

    I PF = {17,21} is an admissible set of pseudo-Frobenius numbers. I m = 2 ·17−21 = 13. I S =

    ( 〈13〉∪

    { 21+1

    2 ,→ } ) \ ( {17,4}∪ {21,8}

    ) = {0,11,→}\ {17,21}=

    {0,9,15,16,17,18,21,22,23,24,25,26,27,28,30→}= 〈11,12,13,14,15,16,18,19,20〉 ∈ S({17,21}).

    I S = 〈5,9,13〉 ∈ S({17,21}).

    A. M. Robles-Pérez (UGR) Suitable sets of pseudo-Frobenius numbers EACA 2016 6 / 25

  • Question 1 for n=2 (2) Proposition

    I Let g1,g2 be two positive integers such that g1 < g2 and g2 is even. Let m = g1− g22 , 0. There exists a numerical semigroup S such that PF(S) = {g1,g2} if and only if m > 0, m - g1, m - g2, m - (g2−g1).

    S = ( 〈m〉∪

    {g2 2

    + 1,→ }) \ ({

    g1−km | k ∈ N } ∪

    { g2−km | k ∈ N

    }) Example

    I PF = {17,20} is an admissible set of pseudo-Frobenius numbers. I m = 17− 202 = 7. I S =

    ( 〈7〉∪

    { 20 2 + 1,→

    } ) \ ( {17}∪ {20,13}

    ) = {0,7,11,→}\ {13,17,20}=

    {0,7,11,12,14,15,16,18,19,21,→}= 〈7,11,12,15,16〉 ∈ S({17,20}).

    I S = 〈7,8,11〉 ∈ S({17,20}).

    A. M. Robles-Pérez (UGR) Suitable sets of pseudo-Frobenius numbers EACA 2016 7 / 25

  • Question 1 for n=2 and n=3

    Example I PF = {9,15} is not a set of pseudo-Frobenius numbers.

    (g∗ = 2×9−15 = 3) I PF = {9,10,15} is a set of pseudo-Frobenius numbers.

    S = 〈4,13,14,19〉. (g∗1 = 2×9−15 = 3, g∗2 = 2×9−10 = 8, g∗3 = 2×10−15 = 5)

    Example I PF = {12,18} is not a set of pseudo-Frobenius numbers.

    (g∗ = 12− 182 = 3) I PF = {12,16,18} is a set of pseudo-Frobenius numbers.

    S = 〈7,10,13,15,19〉. (g∗1 = 12−

    18 2 = 3, g

    ∗ 2 = 12−

    16 2 = 4, g

    ∗ 3 = 16−

    18 2 = 7)

    A. M. Robles-Pérez (UGR) Suitable sets of pseudo-Frobenius numbers EACA 2016 8 / 25

  • General results: initial necessary condition

    Lemma I If S is a numerical semigroup, then x ∈ Z \S if and only if f −x ∈ S for

    some f ∈ PF(S).

    Lemma I Let PF(S) = PF = {g1 < · · · < gn}, with n ≥ 2. If i ∈ {2, . . . ,n} and

    g ∈ 〈PF〉 \ {0} with g < gi , then there exists k ∈ {1, . . . ,n} such that gk − (gi −g) ∈ S.

    Corollary I g1 ≥ gn −gn−1.

    Example I PF = {17,41} ⇒ 17 ≤ 41−17⇒S(PF) = ∅.

    A. M. Robles-Pérez (UGR) Suitable sets of pseudo-Frobenius numbers EACA 2016 9 / 25

  • General results

    Lemma I If S({g1,g2,g3 = g1 + g2}) , ∅, then g3 is odd.

    Proposition I Let α,β,g be positive integers.

    • S({g,αg,βg}) , ∅ if and only if g is odd, α = 2 and β = 3. • If (α+ 1)g +β is odd and α > β, then S({g,αg +β,(α+ 1)g +β}) , ∅.

    Proposition I If g1 + g2 ≥ 3g3+12 and g3 is odd, then S({g1,g2,g3}) , ∅.

    A. M. Robles-Pérez (UGR) Suitable sets of pseudo-Frobenius numbers EACA 2016 10 / 25

  • A simple idea to compute S(PF): by filtering S(max(PF))

    gap> pf := [17,21];;

    gap> nsf21 := NumericalSemigroupsWithFrobeniusNumber(21);;time;

    Length(nsf21);

    437

    1828

    gap> nspf1721 := Filtered(nsf21, s -> PseudoFrobeniusOfNumerical

    Semigroup(s) = pf);;time;Length(nspf1721);

    110

    6

    gap> pf := [17,31];;

    gap> nsf31 := NumericalSemigroupsWithFrobeniusNumber(31);;time;

    Length(nsf31);

    88937

    70854

    gap> nspf1731 := Filtered(nsf31, s -> PseudoFrobeniusOfNumerical

    Semigroup(s) = pf);;time;Length(nspf1731);

    4906

    1

    A. M. Robles-Pérez (UGR) Suitable sets of pseudo-Frobenius numbers EACA 2016 11 / 25

  • Alternative

    ∗ Determine the elements and the gaps of all numerical semigroups (“forced” integers) with the given set of pseudo-Frobenius numbers, obtaining a list of “free” integers.

    ∗ We then construct a tree in which branches correspond to assuming that free integers are either gaps or elements.

    Example I PF = {17,21}

    0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

    0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

    0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

    0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

    0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

    0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

    0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

    0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

    0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

    0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

    0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

    0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

    0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

    0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

    0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

    0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

    0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

    0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

    0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

    A. M. Robles-Pérez (UGR) Suitable sets of pseudo-Frobenius numbers EACA 2016 12 / 25

  • Forced and free integers

    ∗ PF = {g1,g2, . . . ,gn−1,gn} a given set (⊆ N \ {0}). ∗ ` ∈ GF (PF) if ` is a gap of any S ∈ S(PF). ∗ ` ∈ EF (PF) if ` ≤ gn + 1 and it is an element of any S ∈ S(PF).

    Proposition I S(PF) , ∅ if and only if GF (PF)∩EF (PF) = ∅.

    ∗∗ Free(PF) = {1, . . . ,gn} \ (GF (PF)∪EF (PF)).

    Example I PF = {17,21}. I GF (PF) = {1,2,3,4,7,8,17,21}. I EF (PF) = {0,13,14,18,19,20,22}. I Free(PF) = {5,6,9,10,11,12,15,16}.

    A. M. Robles-Pérez (UGR) Suitable sets of pseudo-Frobenius numbers EACA 2016 13 / 25

  • Starting forced gaps Lemma

    I If S is a numerical semigroup, then m(S) ≥ t(S) + 1.

    Lemma I Let PF(S) = {g1 < g2 < · · · < gn−1 < gn}, with n > 1. Let i ∈ {2, . . . , t(S)}

    and g ∈ 〈PF(S)〉 with g < gi . Then gi −g ∈ gaps(S) = N \S.

    ∗ PF = {g1 < g2 < · · · < gn−1 < gn} (n > 1). ∗ Let sfg(PF) be the set of positive divisors of

    PF∪{x ∈ N | 1 ≤ x ≤ n}∪ {gi −g | i ∈ {2, . . . ,n},g ∈ 〈P

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