Numerical semigroups: suitable sets of pseudo-frobenius numbers

Transcription

1 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. García-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 / 25

2 Notation Z = {..., 2, 1,0,1,2,...} N = {0,1,2,...} Definition S numerical semigroup = S cofinite submonoid of (N,+). Definition x S is the Frobenius number of S if x + s S for all s N \ {0}. 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 / 25

3 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[t s 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 / 25

4 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, García-Sánchez, García-García, Jiménez-Madrid, J. Pure Appl. Algebra 189, 2004) PF = {g 1,g 2,...,g n 1,g n } 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 / 25

5 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 PF = {f}: symmetric numerical semigroups. (Blanco, Rosales, Forum Math. 25, 2013) PF = {f, f 2 }: pseudo-symmetric numerical semigroups. (Blanco, Rosales, Forum Math. 25, 2013) Question 1 for PF = {g 1,g 2 }. (R-P, Rosales, article to appear in Proc. Roy. Soc. Edinburgh Sect. A.) Question 2 for general PF. (Delgado, García-Sánchez, R-P, LMS J. Comput. Math. 19(1), 2016) A. M. Robles-Pérez (UGR) Suitable sets of pseudo-frobenius numbers EACA / 25

6 Question 1 for n=2 (1) Proposition Let g 1,g 2 be two positive integers such that g 1 < g 2 and g 2 is odd. Let m = 2g 1 g 2. There exists a numerical semigroup S such that PF(S) = {g 1,g 2 } if and only if m > 0, m g 1, m g 2, m (g 2 g 1 ). Example ( { }) ( g2 + 1 {g1 S = m, \ km k N } { g 2 km k N }) 2 PF = {17,21} is an admissible set of pseudo-frobenius numbers. m = = 13. S = ( 13 { , }) \ ( {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}). S = 5,9,13 S({17,21}). A. M. Robles-Pérez (UGR) Suitable sets of pseudo-frobenius numbers EACA / 25

7 Question 1 for n=2 (2) Proposition Let g 1,g 2 be two positive integers such that g 1 < g 2 and g 2 is even. Let m = g 1 g There exists a numerical semigroup S such that PF(S) = {g 1,g 2 } if and only if m > 0, m g 1, m g 2, m (g 2 g 1 ). Example ( { g2 }) ( {g1 S = m 2 + 1, \ km k N } { g 2 km k N }) PF = {17,20} is an admissible set of pseudo-frobenius numbers. m = = 7. S = ( 7 { , }) \ ( {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}). S = 7,8,11 S({17,20}). A. M. Robles-Pérez (UGR) Suitable sets of pseudo-frobenius numbers EACA / 25

8 Question 1 for n=2 and n=3 Example PF = {9,15} is not a set of pseudo-frobenius numbers. (g = = 3) PF = {9,10,15} is a set of pseudo-frobenius numbers. S = 4,13,14,19. (g 1 = = 3, g 2 = = 8, g 3 = = 5) Example PF = {12,18} is not a set of pseudo-frobenius numbers. (g = = 3) PF = {12,16,18} is a set of pseudo-frobenius numbers. S = 7,10,13,15,19. (g 1 = = 3, g 2 = = 4, g 3 = = 7) A. M. Robles-Pérez (UGR) Suitable sets of pseudo-frobenius numbers EACA / 25

9 General results: initial necessary condition Lemma If S is a numerical semigroup, then x Z \ S if and only if f x S for some f PF(S). Lemma Let PF(S) = PF = {g 1 < < g n }, with n 2. If i {2,...,n} and g PF \ {0} with g < g i, then there exists k {1,...,n} such that g k (g i g) S. Corollary g 1 g n g n 1. Example PF = {17,41} S(PF) =. A. M. Robles-Pérez (UGR) Suitable sets of pseudo-frobenius numbers EACA / 25

10 General results Lemma If S({g 1,g 2,g 3 = g 1 + g 2 }), then g 3 is odd. Proposition Let α,β,g be positive integers. Proposition 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}). If g 1 + g 2 3g and g 3 is odd, then S({g 1,g 2,g 3 }). A. M. Robles-Pérez (UGR) Suitable sets of pseudo-frobenius numbers EACA / 25

11 A simple idea to compute S(PF): by filtering S(max(PF)) gap> pf := [17,21];; gap> nsf21 := NumericalSemigroupsWithFrobeniusNumber(21);;time; Length(nsf21); gap> nspf1721 := Filtered(nsf21, s -> PseudoFrobeniusOfNumerical Semigroup(s) = pf);;time;length(nspf1721); gap> pf := [17,31];; gap> nsf31 := NumericalSemigroupsWithFrobeniusNumber(31);;time; Length(nsf31); gap> nspf1731 := Filtered(nsf31, s -> PseudoFrobeniusOfNumerical Semigroup(s) = pf);;time;length(nspf1731); A. M. Robles-Pérez (UGR) Suitable sets of pseudo-frobenius numbers EACA / 25

12 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 PF = {17,21} A. M. Robles-Pérez (UGR) Suitable sets of pseudo-frobenius numbers EACA / 25

13 Forced and free integers PF = {g 1,g 2,...,g n 1,g n } a given set ( N \ {0}). l GF (PF) if l is a gap of any S S(PF). l EF (PF) if l g n + 1 and it is an element of any S S(PF). Proposition S(PF) if and only if GF (PF) EF (PF) =. Free(PF) = {1,...,g n } \ (GF (PF) EF (PF)). Example PF = {17,21}. GF (PF) = {1,2,3,4,7,8,17,21}. EF (PF) = {0,13,14,18,19,20,22}. Free(PF) = {5,6,9,10,11,12,15,16}. A. M. Robles-Pérez (UGR) Suitable sets of pseudo-frobenius numbers EACA / 25

14 Starting forced gaps Lemma If S is a numerical semigroup, then m(s) t(s) + 1. Lemma Let PF(S) = {g 1 < g 2 < < g n 1 < g n }, with n > 1. Let i {2,...,t(S)} and g PF(S) with g < g i. Then g i g gaps(s) = N \ S. PF = {g 1 < g 2 < < g n 1 < g n } (n > 1). Let sfg(pf) be the set of positive divisors of Corollary PF {x N 1 x n} {g i g i {2,...,n},g PF,g i > g}. Let PF be a set of positive integers. Then sfg(pf) GF (PF). Example PF = {17,21} {17,21} {1,2} {4} sfg(pf) = {1,2,3,4,7,17,21}. A. M. Robles-Pérez (UGR) Suitable sets of pseudo-frobenius numbers EACA / 25

15 Forced elements: big forced elements Lemma If S is a numerical semigroup such that m(s) = m, then F(S) i S PF(S) for all i {1,...,m 1}. Consequence Let fg(pf) GF (PF). If m = min(n \ (fg(pf) {0})), then max(pf) i EF (PF) PF for all i {1,...,m 1}. Example PF = {17,21} fg(pf) = sfg(pf) = {1,2,3,4,7,17,21} bfe(pf) = {18,19,20}. PF = {4,9} fg(pf) = sfg(pf) = {1,2,3,4,5,9} bfe(pf) = {5,6,7,8} Contradiction! A. M. Robles-Pérez (UGR) Suitable sets of pseudo-frobenius numbers EACA / 25

16 Forced elements: elements forced by exclusion (1) Lemma If S is a numerical semigroup, then x Z \ S if and only if f x S for some f PF(S). Consequence Let x fg(pf) GF (PF). If there exists a unique f PF(S) such that ( f x > 0 and f x fg(pf) ), then f x efe 1 (PF) EF (PF). Example PF = {17,21} fg(pf) = sfg(pf) = {1,2,3,4,7,17,21} efe 1 (PF) = {13} (from x = 4). A. M. Robles-Pérez (UGR) Suitable sets of pseudo-frobenius numbers EACA / 25

17 Forced elements: elements forced by exclusion (2) Lemma If S is a numerical semigroup, then x S if and only if f x S for all f PF(S). Consequence Let x fg(pf) bfe(pf) efe 1 (PF) such that x max(pf) + 1. If ( f x < 0 or f x fg(pf) ) for all f PF, then x efe2 (PF) EF (PF). Example PF = {17,21} fg(pf) = sfg(pf) = {1,2,3,4,7,17,21} bfe(pf) efe 1 (PF) = {13,18,19,20} efe 2 (PF) = {14,22}. fe(pf) = bfe(pf) efe 1 (PF) efe 2 (PF) [1,max(PF)+1] EF (PF). fe({17,21}) = 13,14,18,19,20,22 [1,22] = {0,13,14,18,19,20,22} EF ({19,29}). A. M. Robles-Pérez (UGR) Suitable sets of pseudo-frobenius numbers EACA / 25

18 Further forced gaps Remark Let S be a numerical semigroup. If e S and f gaps(s), then f e < 0 or f e gaps(s). Consequence Let fg(pf) GF (PF) and fe(pf) EF (PF). Then = (fg(pf) e) N ffg(pf) GF (PF) for every e fe(pf). Example PF = {17,21} sfg(pf) = {1,2,3,4,7,17,21} fe(pf) = {0,13,14,18,19,20,22} = {8} (from e = 13). ffg(pf) = sfg(pf) {Positive divisors of } GF (PF). PF = {17,21} ffg(pf) = {1,2,3,4,7,17,21} {1,2,4,8} = {1,2,3,4,7,8,17,21}. A. M. Robles-Pérez (UGR) Suitable sets of pseudo-frobenius numbers EACA / 25

19 Algorithm PF PF sfg(pf) PF, sfg(pf) fe(pf) PF, sfg(pf), fe(pf) ffg 1 (PF) PF, ffg 1 (PF), fe(pf) ffe 1 (PF) PF, ffg 1 (PF), ffe 1 (PF) ffg 2 (PF) PF, ffg 2 (PF), ffe 1 (PF) ffe 2 (PF). Repeat until ffg k+1 (PF) = ffg k (PF) and ffe k+1 (PF) = ffe k (PF) ffg k (PF) GF (PF), ffe k (PF) EF (PF) [1,max(PF) + 1] \ (ffg k (PF) ffe k (PF)) Free(PF) A. M. Robles-Pérez (UGR) Suitable sets of pseudo-frobenius numbers EACA / 25

20 Example PF = {17,31} PF sfg(pf) = {1,2,7,14,17,31} PF, sfg(pf) fe(pf) = {0,3,6,9,12,15,18,21,24,27,29,30,32} PF, sfg(pf), fe(pf) ffg 1 (PF) = {1,2,4,5,7,8,10,11,13,14,16,17,19,22,25,28,31} PF, ffg 1 (PF), fe(pf) ffe 1 (PF) = {0,3,6,9,12,15,18,20,21,23,24,26,27,29,30,32} PF, ffg 1 (PF), ffe 1 (PF) ffg 2 (PF) = ffg 1 (PF) PF, ffg 2 (PF), ffe 1 (PF) ffe 2 (PF) = ffe 1 (PF) ffg 2 (PF) = GF (PF) = {1,2,4,5,7,8,10,11,13,14,16,17,19,22,25,28,31} ffe 2 (PF) = EF (PF) = {0,3,6,9,12,15,18,20,21,23,24,26,27,29,30,32} Free(PF) = A. M. Robles-Pérez (UGR) Suitable sets of pseudo-frobenius numbers EACA / 25

21 Example PF = {11,17}, GF (PF) = {1,2,3,4,6,7,11,12,17}, EF (PF) = {0,5,10,13,14,15,16,18}, Free(PF) = {8,9} Algorithm with ffg(pf) = {1,2,3,4,6,7,11,12,17} and ffe(pf) = {0,5,8,10,13,14,15,16,18} S = 5,8,14 = {0,5,8,10,13,14,15,16,18, } Algorithm with ffg(pf) = {1,2,3,4,6,7,8,11,12,17} and ffe(pf) = {0,5,9,10,13,14,15,16,18} S = 5,9,13,16 = {0,5,9,10,13,14,15,16,18, } A. M. Robles-Pérez (UGR) Suitable sets of pseudo-frobenius numbers EACA / 25

22 Example PF = {17,21}, {1,2,3,4,7,8,17,21} GF (PF), {0,13,14,18,19,20,22} EF (PF), Free(PF) {5,6,9,10,11,12,15,16} ,9,13 6,10,13, ,10,13,14,15,16 9,11,13,14,15,16,19 10,12,13,14,15,16,18,19 11,12,13,14,15,16,18,19,20 STOP Numerical semigroups with pseudo-frobenius numbers {17, 21} A. M. Robles-Pérez (UGR) Suitable sets of pseudo-frobenius numbers EACA / 25

23 Running times gap> pf:=[11,17];; gap> ans:=numericalsemigroupswithpseudofrobeniusnumbers(pf);; gap> time;length(ans); 0 2 gap> pf:=[17,21];; gap> ans:=numericalsemigroupswithpseudofrobeniusnumbers(pf);; gap> time;length(ans); 16 6 gap> pf:=[17,31];; gap> ans:=numericalsemigroupswithpseudofrobeniusnumbers(pf);; gap> time;length(ans); 16 1 A. M. Robles-Pérez (UGR) Suitable sets of pseudo-frobenius numbers EACA / 25

24 References V. Blanco and J.C. Rosales. The tree of irreducible numerical semigroups with fixed Frobenius number. Forum Math. 25 (2013), M. Delgado, P.A. García-Sánchez and J. Morais. NumericalSgps, a GAP package for numerical semigroups, Version 1.0.1; Available via M. Delgado, P.A. García-Sánchez, and A.M. Robles-Pérez. Numerical semigroups with a given set of pseudo-frobenius numbers. LMS J. Comput. Math. 19(1) (2016), A.M. Robles-Pérez and J.C. Rosales. The genus, the Frobenius number, and the pseudo-frobenius numbers of numerical semigroups with type two. To appear in Proc. Roy. Soc. Edinburgh Sect. A. J.C. Rosales, P.A. García-Sánchez, J.I. García-García, and J.A. Jiménez-Madrid. Fundamental gaps in numerical semigroups. J. Pure Appl. Algebra 189 (2004), A. M. Robles-Pérez (UGR) Suitable sets of pseudo-frobenius numbers EACA / 25

25 THANK YOU VERY MUCH FOR YOUR ATTENTION! A. M. Robles-Pérez (UGR) Suitable sets of pseudo-frobenius numbers EACA / 25