Counting numerical semigroups of a given genus by using - - PowerPoint PPT Presentation

Counting numerical semigroups of a given genus by using γ-hyperelliptic semigroups

Matheus Bernardini1 University of Campinas / Federal Institute of S˜ ao Paulo

(Joint work in progress with Fernando Torres1)

International Meeting on Numerical Semigroups with Applications Levico Terme, July 5 2016

1Both authors are partially supported by CNPq Matheus Bernardini IMNS 2016 1 / 34

1 Introduction 2 A brief survey about counting numerical semigroups by genus 3 Our approach 4 A related problem Matheus Bernardini IMNS 2016 2 / 34

Introduction

Let S be a numerical semigroup. G(S) := N0 \ S - set of gaps of S; g(S) := #G(S) - genus of S; ng := #{S : g(S) = g}.

Examples

n0 = 1 − N0 n1 = 1 − N0 \ {1} n2 = 2 − N0 \ {1, 2} and N0 \ {1, 3} n3 = 4 − N0 \ {1, 2, 3}, N0 \ {1, 2, 4}, N0 \ {1, 2, 5} and N0 \ {1, 3, 5}

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Interest

Studying the behavior of ng.

Main Goal (but still not solved)

ng ≤ ng+1, for all g.

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Interest

Studying the behavior of ng.

Main Goal (but still not solved)

ng ≤ ng+1, for all g.

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A brief survey

First Bound

If g(S) = g, then 2g + N0 ⊂ S. Hence, ng ≤ 2g − 1 g

• .
• M. Bras-Amor´
• s and A. de Mier - 2007

ng ≤ Cg = 1 g + 1 2g g

• .

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A brief survey

First Bound

If g(S) = g, then 2g + N0 ⊂ S. Hence, ng ≤ 2g − 1 g

• .
• M. Bras-Amor´
• s and A. de Mier - 2007

ng ≤ Cg = 1 g + 1 2g g

• .

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From now on, let ϕ = 1+

√ 5 2

.

• M. Bras-Amor´
• s - 2006/2008 (Conjecture)

1

lim

g→∞

ng+1 ng = ϕ; lim

g→∞

ng+1 + ng ng+2 = 1.

2 ng + ng+1 ≤ ng+2, for all g. Matheus Bernardini IMNS 2016 6 / 34

• M. Bras-Amor´
• s - 2009

Let (Fn)n≥0 = (1, 1, 2, 3, 5, 8, 13, . . .) be the Fibonacci sequence. Then 2Fg ≤ ng ≤ 1 + 3 · 2g−3, ∀g ≥ 3.

• S. Elizalde - 2010

ag ≤ ng ≤ cg, ∀g ≥ 1 where ag and cg are coefficients of some explicit generating functions.

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• M. Bras-Amor´
• s - 2009

Let (Fn)n≥0 = (1, 1, 2, 3, 5, 8, 13, . . .) be the Fibonacci sequence. Then 2Fg ≤ ng ≤ 1 + 3 · 2g−3, ∀g ≥ 3.

• S. Elizalde - 2010

ag ≤ ng ≤ cg, ∀g ≥ 1 where ag and cg are coefficients of some explicit generating functions.

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• A. Zhai - 2011/2013

1

lim

g→∞

ng+1 ng = ϕ;

2

lim

g→∞

ng+1 + ng ng+2 = 1.

Remark

Zhai’s first item implies that ng < ng+1, for g ≫ 0. Checking if ng ≤ ng+1 for all g is still an open problem (weaker conjecture).

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• A. Zhai - 2011/2013

1

lim

g→∞

ng+1 ng = ϕ;

2

lim

g→∞

ng+1 + ng ng+2 = 1.

Remark

Zhai’s first item implies that ng < ng+1, for g ≫ 0. Checking if ng ≤ ng+1 for all g is still an open problem (weaker conjecture).

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m(S) := min{s ∈ S : s = 0} - multiplicity of S; N(m, g) := #{S : g(S) = g and m(S) = m}.

• N. Kaplan - 2012

If 2g < 3m, then N(m, g) = N(m − 1, g − 1) + N(m − 1, g − 2).

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Part of M. Bras-Amor´

• s’ presentation in last IMNS;

r(S) - ordinarization number of S; ng,r := #{S : g(S) = g and r(S) = r}.

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Part of M. Bras-Amor´

• s’ presentation in last IMNS;

r(S) - ordinarization number of S; ng,r := #{S : g(S) = g and r(S) = r}.

Matheus Bernardini IMNS 2016 11 / 34

• M. Bras-Amor´
• s - 2012

If r > max{g

3 + 1,

• g+1

2

• − 14}, then ng,r ≤ ng+1,r.
• M. Bras-Amor´
• s - 2012 (Conjecture)

If r > g

3, then ng,r ≤ ng+1,r.

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• M. Bras-Amor´
• s - 2012

If r > max{g

3 + 1,

• g+1

2

• − 14}, then ng,r ≤ ng+1,r.
• M. Bras-Amor´
• s - 2012 (Conjecture)

If r > g

3, then ng,r ≤ ng+1,r.

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Our approach

γ(S): number of even gaps of S - #[G(S) ∩ 2Z]; γ-hyperelliptic semigroup: numerical semigroup with γ even gaps; Nγ(g) := #{S : g(S) = g and γ(S) = γ}. ng =

g

• γ=0

Nγ(g)

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Our approach

γ(S): number of even gaps of S - #[G(S) ∩ 2Z]; γ-hyperelliptic semigroup: numerical semigroup with γ even gaps; Nγ(g) := #{S : g(S) = g and γ(S) = γ}. ng =

g

• γ=0

Nγ(g)

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Examples

n0 = 1 (N0) and Nγ(0) =

• 1, if γ = 0

0, if γ ≥ 1. n1 = 1 (N0 \ {1}) and Nγ(1) =

• 1, if γ = 0

0, if γ ≥ 1. n2 = 2 (N0 \ {1, 2} and N0 \ {1, 3}) and Nγ(2) =      1, if γ = 0 1, if γ = 1 0, if γ ≥ 2.

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• F. Torres - 1997

If γ and g are the number of even gaps and the genus of a numerical semigroup S, respectively, then 3γ ≤ 2g.

Remark

If γ is even, then N0 \ ({2, 4, . . . , 2γ} ∪ {1, 3, . . . , γ − 1}) is a numerical semigroup with genus g = 3γ

2 .

ng = ⌊ 2g

3 ⌋

• γ=0

Nγ(g)

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• F. Torres - 1997

If γ and g are the number of even gaps and the genus of a numerical semigroup S, respectively, then 3γ ≤ 2g.

Remark

If γ is even, then N0 \ ({2, 4, . . . , 2γ} ∪ {1, 3, . . . , γ − 1}) is a numerical semigroup with genus g = 3γ

2 .

ng = ⌊ 2g

3 ⌋

• γ=0

Nγ(g)

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Theorem 1

Let γ be a nonnegative integer and g ≥ 3γ. Then Nγ(g) = Nγ(3γ). Thus, Nγ(g) = Nγ(g + 1), for all g ≥ 3γ.

Theorem 2

Let γ be a nonnegative integer and g < 3γ. Then Nγ(g) < Nγ(3γ).

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Theorem 1

Let γ be a nonnegative integer and g ≥ 3γ. Then Nγ(g) = Nγ(3γ). Thus, Nγ(g) = Nγ(g + 1), for all g ≥ 3γ.

Theorem 2

Let γ be a nonnegative integer and g < 3γ. Then Nγ(g) < Nγ(3γ).

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g / γ 1 2 3 4 5 6 7 8 9 10 ng 1 1 1 1 1 2 1 1 2 3 1 2 1 4 4 1 2 4 7 5 1 2 6 3 12 6 1 2 7 12 1 23 7 1 2 7 19 10 39 8 1 2 7 21 32 4 67 9 1 2 7 23 51 33 1 118 10 1 2 7 23 62 91 18 204 11 1 2 7 23 65 142 98 5 343 12 1 2 7 23 68 174 257 59 1 592 13 1 2 7 23 68 192 412 271 25 1001 14 1 2 7 23 68 197 514 678 197 6 1693 15 1 2 7 23 68 200 570 1100 793 92 1 2857

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Notice that ng = ⌊ g

3⌋

• γ=0

Nγ(g) + ⌊ 2g

3 ⌋

• γ=⌊ g

3⌋+1

Nγ(g). ng+1 = ⌊ g

3⌋

• γ=0

Nγ(g + 1) +

2(g+1)

3

• γ=⌊ g

3⌋+1

Nγ(g + 1). Theorem 1 states that Nγ(g) = Nγ(g + 1), for γ ≤ g

3.

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Notice that ng = ⌊ g

3⌋

• γ=0

Nγ(g) + ⌊ 2g

3 ⌋

• γ=⌊ g

3⌋+1

Nγ(g). ng+1 = ⌊ g

3⌋

• γ=0

Nγ(g + 1) +

2(g+1)

3

• γ=⌊ g

3⌋+1

Nγ(g + 1). Theorem 1 states that Nγ(g) = Nγ(g + 1), for γ ≤ g

3.

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Notice that ng = ⌊ g

3⌋

• γ=0

Nγ(g) + ⌊ 2g

3 ⌋

• γ=⌊ g

3⌋+1

Nγ(g). ng+1 = ⌊ g

3⌋

• γ=0

Nγ(g + 1) +

2(g+1)

3

• γ=⌊ g

3⌋+1

Nγ(g + 1). Theorem 1 states that Nγ(g) = Nγ(g + 1), for γ ≤ g

3.

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Corollary

ng ≤ ng+1 if, and only if, ⌊ 2g

3 ⌋

• γ=⌊ g

3⌋+1

Nγ(g) ≤

2(g+1)

3

• γ=⌊ g

3⌋+1

Nγ(g + 1).

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g / γ 1 2 3 4 5 6 7 8 9 10 ng 1 1 1 1 1 2 1 1 2 3 1 2 1 4 4 1 2 4 7 5 1 2 6 3 12 6 1 2 7 12 1 23 7 1 2 7 19 10 39 8 1 2 7 21 32 4 67 9 1 2 7 23 51 33 1 118 10 1 2 7 23 62 91 18 204 11 1 2 7 23 65 142 98 5 343 12 1 2 7 23 68 174 257 59 1 592 13 1 2 7 23 68 192 412 271 25 1001 14 1 2 7 23 68 197 514 678 197 6 1693 15 1 2 7 23 68 200 570 1100 793 92 1 2857

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Conjecture

Let γ be a non-negative integer. Then Nγ(g) ≤ Nγ(g + 1), ∀g.

Remark

If this Conjecture holds, then ng ≤ ng+1 for all g.

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Conjecture

Let γ be a non-negative integer. Then Nγ(g) ≤ Nγ(g + 1), ∀g.

Remark

If this Conjecture holds, then ng ≤ ng+1 for all g.

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Construction of a γ-hyperelliptic semigroup of genus g

γ - positive integer T = N0 \ {q1, . . . , qγ} numerical semigroup S = 2 · T ∪ (2 · N0 + 1) \ {“suitable choice”of g − γ odd numbers}

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“suitable choice”ensures that the final set is closed under addition. S is a γ-hyperelliptic semigroup: even gaps and even non-gaps are determined by T. S has genus g if, and only if, the number of green points choosen as gaps is g − γ − k.

Lemma

Let S be a γ-hyperelliptic semigroup of genus g and O the first odd number in S. Then 2g − 4γ + 1 ≤ O ≤ 2g − 2γ + 1.

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“suitable choice”ensures that the final set is closed under addition. S is a γ-hyperelliptic semigroup: even gaps and even non-gaps are determined by T. S has genus g if, and only if, the number of green points choosen as gaps is g − γ − k.

Lemma

Let S be a γ-hyperelliptic semigroup of genus g and O the first odd number in S. Then 2g − 4γ + 1 ≤ O ≤ 2g − 2γ + 1.

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Proof (Thm 1)

Numbers qi must be gaps! (otherwise, S ∋ qi + qi = 2qi / ∈ S) g ≥ 3γ ⇒

• O≥2g−4γ+1

O ≥ 2γ + 1. Hence,

O > qγ ≥ qi, for all i sum of odd elements of S is greater than 4γ + 2.

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Proof (Thm 1)

Numbers qi must be gaps! (otherwise, S ∋ qi + qi = 2qi / ∈ S) g ≥ 3γ ⇒

• O≥2g−4γ+1

O ≥ 2γ + 1. Hence,

O > qγ ≥ qi, for all i sum of odd elements of S is greater than 4γ + 2.

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Proof (Thm 1)

Sγ(g) := {S : g(S) = g and γ(S) = γ} For a fixed g ≥ 3γ, we find a bijection between Sγ(g) and Sγ(3γ)

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Proof (Thm 1)

Given S(g) ∈ Sγ(g), let O(g) be the first odd number of S(g) Let M := g − 3γ. Making a translation by −2M only on the odd numbers higher than or equal to O(g), we obtain a NS S, such that O(3γ) = O(g) − 2M ≥ 2γ + 1 (and this is the first odd number of S) The even gaps of S(g) and S are the same, as the odd gaps of S(g) and S lower than O(3γ). The odd gaps of S(g) and S higher than O(3γ) are translated by −2M Under this construction, we have g(S) = g − M = 3γ. Hence, S ∈ Sγ(3γ) and #Sγ(g) ≤ #Sγ(3γ) Similarly (by making a translation by +2M), we can verify the other inequality and the result follows.

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A related problem

γ non-negative integer For g ≥ 3γ, the sequence Nγ(g) is constant and equal to Nγ(3γ) A natural task is about the behavior of fγ := Nγ(3γ)

Lemma

Let γ be a non-negative integer and Mγ := 2γ γ

2 + 1

• − 1. Then

Mγ + (nγ − γ) · (γ + 1) ≤ fγ ≤ Mγ + (nγ − γ) · 2γ.

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A related problem

γ non-negative integer For g ≥ 3γ, the sequence Nγ(g) is constant and equal to Nγ(3γ) A natural task is about the behavior of fγ := Nγ(3γ)

Lemma

Let γ be a non-negative integer and Mγ := 2γ γ

2 + 1

• − 1. Then

Mγ + (nγ − γ) · (γ + 1) ≤ fγ ≤ Mγ + (nγ − γ) · 2γ.

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Theorem 3

Let ǫ > 0. Then lim

γ→∞

fγ (2ϕ + ǫ)γ = 0 and lim

γ→∞

fγ 2γ = ∞. It suggests that the asymptotic behavior of fγ is exponencial of order βγ, where 2 < β ≤ 2ϕ.

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Theorem 3

Let ǫ > 0. Then lim

γ→∞

fγ (2ϕ + ǫ)γ = 0 and lim

γ→∞

fγ 2γ = ∞. It suggests that the asymptotic behavior of fγ is exponencial of order βγ, where 2 < β ≤ 2ϕ.

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g / γ 1 2 3 4 5 6 7 8 9 10 ng 1 1 1 1 1 2 1 1 2 3 1 2 1 4 4 1 2 4 7 5 1 2 6 3 12 6 1 2 7 12 1 23 7 1 2 7 19 10 39 8 1 2 7 21 32 4 67 9 1 2 7 23 51 33 1 118 10 1 2 7 23 62 91 18 204 11 1 2 7 23 65 142 98 5 343 12 1 2 7 23 68 174 257 59 1 592 13 1 2 7 23 68 192 412 271 25 1001 14 1 2 7 23 68 197 514 678 197 6 1693 15 1 2 7 23 68 200 570 1100 793 92 1 2857

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γ fγ fγ/fγ−1 n2γ fγ/n2γ 1 1 1 1 2 2 2 1 2 7 3,5 7 1 3 23 3,285714 23 1 4 68 2,956522 67 1,015 5 200 2,941176 204 0,981 6 615 3,075 592 1,039 7 1764 2,868292 1693 1,042 8 5060 2,868480 4806 1,053 9 14626 2,890514 13467 1,086 10 41785 2,856899 37396 1,117 11 117573 2,813761 103246 1,139 12 332475 2,827818 282828 1,176 13 933891 2,808905 770832 1,212 14 2609832 2,794579 2091030 1,248

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Conjecture

lim

γ→∞

fγ fγ−1 = ϕ2 ≈ 2, 618 and lim

γ→∞

fγ n2γ = C, where C is a constant.

Remark

Firts item is a consequence of second item.

Remark

There is a relation between the sequence fγ and the conjecture proposed by M. Bras-Amor´

• s (12). In fact, if fγ is an increasing sequence, then the

conjecture is also true.

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Conjecture

lim

γ→∞

fγ fγ−1 = ϕ2 ≈ 2, 618 and lim

γ→∞

fγ n2γ = C, where C is a constant.

Remark

Firts item is a consequence of second item.

Remark

There is a relation between the sequence fγ and the conjecture proposed by M. Bras-Amor´

• s (12). In fact, if fγ is an increasing sequence, then the

conjecture is also true.

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References I

• M. Bernardini and F. Torres, Counting numerical semigroups of a

given genus via even gaps, In progress.

• M. Bras-Amor´
• s, Bounds on the number of numerical semigroups of a

given genus, Journal of Pure and Applied Algebra 213:6 (2009), 997–1001.

• M. Bras-Amor´
• s, Fibonacci-like behavior of the number of numerical

semigroups of a given genus, Semigroup Forum 76 (2008), 379–384.

• M. Bras-Amor´
• s, The ordinarization transform of a numerical

semigroup and semigroups with a large number of intervals, Journal of Pure and Applied Algebra 216 (2012), 2507–2518.

• M. Bras-Amor´
• s and A. de Mier, Representation of numerical

semigroups by Dyck Paths, Semigroup Forum 75 (2007), 676–681.

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References II

• S. Elizalde, Improved bounds on the number of numerical semigroups
• f a given genus, Journal of Pure and Applied Algebra 214 (2010),

1862–1873.

• N. Kaplan, Counting numerical semigroups by genus and some cases
• f a question of Wilf, Journal of Pure and Applied Algebra 216

(2012), 1016–1032.

• F. Torres, On γ-hyperelliptic numerical semigroups, Semigroup Forum

55 (1997), 364-379.

• F. Torres, On n-sheeted Covering of Curves and Semigroups which

Cannot Be Realized as Weierstrass Semigroups, Communications in Algebra 23 (11) (1995), 4211-4228.

• A. Zhai, Fibonacci-like growth of numerical semigroups of a given

genus, Semigroup Forum 86 (2013), 634–662.

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Thank You!

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