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 Bernardini 1 University of Campinas / Federal Institute of S˜ ao Paulo (Joint work in progress with Fernando Torres 1 ) International Meeting on Numerical Semigroups with Applications Levico Terme, July 5 2016 1 Both 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 ) := N 0 \ S - set of gaps of S ; g ( S ) := # G ( S ) - genus of S ; n g := # { S : g ( S ) = g } . Examples n 0 = 1 − N 0 n 1 = 1 − N 0 \ { 1 } n 2 = 2 − N 0 \ { 1 , 2 } and N 0 \ { 1 , 3 } n 3 = 4 − N 0 \ { 1 , 2 , 3 } , N 0 \ { 1 , 2 , 4 } , N 0 \ { 1 , 2 , 5 } and N 0 \ { 1 , 3 , 5 } Matheus Bernardini IMNS 2016 3 / 34

Interest Studying the behavior of n g . Main Goal (but still not solved) n g ≤ n g +1 , for all g . Matheus Bernardini IMNS 2016 4 / 34

Interest Studying the behavior of n g . Main Goal (but still not solved) n g ≤ n g +1 , for all g . Matheus Bernardini IMNS 2016 4 / 34

A brief survey First Bound If g ( S ) = g , then 2 g + N 0 ⊂ S . Hence, � 2 g − 1 � n g ≤ . g M. Bras-Amor´ os and A. de Mier - 2007 � 2 g � 1 n g ≤ C g = . g + 1 g Matheus Bernardini IMNS 2016 5 / 34

A brief survey First Bound If g ( S ) = g , then 2 g + N 0 ⊂ S . Hence, � 2 g − 1 � n g ≤ . g M. Bras-Amor´ os and A. de Mier - 2007 � 2 g � 1 n g ≤ C g = . g + 1 g Matheus Bernardini IMNS 2016 5 / 34

√ From now on, let ϕ = 1+ 5 . 2 M. Bras-Amor´ os - 2006/2008 (Conjecture) n g +1 lim = ϕ ; 1 n g g →∞ n g +1 + n g lim = 1 . n g +2 g →∞ 2 n g + n g +1 ≤ n g +2 , for all g. Matheus Bernardini IMNS 2016 6 / 34

M. Bras-Amor´ os - 2009 Let ( F n ) n ≥ 0 = (1 , 1 , 2 , 3 , 5 , 8 , 13 , . . . ) be the Fibonacci sequence. Then 2 F g ≤ n g ≤ 1 + 3 · 2 g − 3 , ∀ g ≥ 3 . S. Elizalde - 2010 a g ≤ n g ≤ c g , ∀ g ≥ 1 where a g and c g are coefficients of some explicit generating functions. Matheus Bernardini IMNS 2016 7 / 34

M. Bras-Amor´ os - 2009 Let ( F n ) n ≥ 0 = (1 , 1 , 2 , 3 , 5 , 8 , 13 , . . . ) be the Fibonacci sequence. Then 2 F g ≤ n g ≤ 1 + 3 · 2 g − 3 , ∀ g ≥ 3 . S. Elizalde - 2010 a g ≤ n g ≤ c g , ∀ g ≥ 1 where a g and c g are coefficients of some explicit generating functions. Matheus Bernardini IMNS 2016 7 / 34

A. Zhai - 2011/2013 n g +1 lim = ϕ ; 1 n g g →∞ n g +1 + n g lim = 1 . 2 n g +2 g →∞ Remark Zhai’s first item implies that n g < n g +1 , for g ≫ 0. Checking if n g ≤ n g +1 for all g is still an open problem (weaker conjecture). Matheus Bernardini IMNS 2016 8 / 34

A. Zhai - 2011/2013 n g +1 lim = ϕ ; 1 n g g →∞ n g +1 + n g lim = 1 . 2 n g +2 g →∞ Remark Zhai’s first item implies that n g < n g +1 , for g ≫ 0. Checking if n g ≤ n g +1 for all g is still an open problem (weaker conjecture). Matheus Bernardini IMNS 2016 8 / 34

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 2 g < 3 m, then N ( m , g ) = N ( m − 1 , g − 1) + N ( m − 1 , g − 2) . Matheus Bernardini IMNS 2016 9 / 34

Matheus Bernardini IMNS 2016 10 / 34

Part of M. Bras-Amor´ os’ presentation in last IMNS; r ( S ) - ordinarization number of S ; n g , r := # { S : g ( S ) = g and r ( S ) = r } . Matheus Bernardini IMNS 2016 11 / 34

Part of M. Bras-Amor´ os’ presentation in last IMNS; r ( S ) - ordinarization number of S ; n g , r := # { S : g ( S ) = g and r ( S ) = r } . Matheus Bernardini IMNS 2016 11 / 34

M. Bras-Amor´ os - 2012 � � If r > max { g g +1 3 + 1 , − 14 } , then n g , r ≤ n g +1 , r . 2 M. Bras-Amor´ os - 2012 (Conjecture) If r > g 3 , then n g , r ≤ n g +1 , r . Matheus Bernardini IMNS 2016 12 / 34

M. Bras-Amor´ os - 2012 � � If r > max { g g +1 3 + 1 , − 14 } , then n g , r ≤ n g +1 , r . 2 M. Bras-Amor´ os - 2012 (Conjecture) If r > g 3 , then n g , r ≤ n g +1 , r . Matheus Bernardini IMNS 2016 12 / 34

Our approach γ ( S ): number of even gaps of S - #[ G ( S ) ∩ 2 Z ]; γ -hyperelliptic semigroup: numerical semigroup with γ even gaps; N γ ( g ) := # { S : g ( S ) = g and γ ( S ) = γ } . g � n g = N γ ( g ) γ =0 Matheus Bernardini IMNS 2016 13 / 34

Our approach γ ( S ): number of even gaps of S - #[ G ( S ) ∩ 2 Z ]; γ -hyperelliptic semigroup: numerical semigroup with γ even gaps; N γ ( g ) := # { S : g ( S ) = g and γ ( S ) = γ } . g � n g = N γ ( g ) γ =0 Matheus Bernardini IMNS 2016 13 / 34

Examples n 0 = 1 ( N 0 ) and � 1 , if γ = 0 N γ (0) = 0 , if γ ≥ 1 . n 1 = 1 ( N 0 \ { 1 } ) and � 1 , if γ = 0 N γ (1) = 0 , if γ ≥ 1 . n 2 = 2 ( N 0 \ { 1 , 2 } and N 0 \ { 1 , 3 } ) and  1 , if γ = 0   N γ (2) = 1 , if γ = 1   0 , if γ ≥ 2 . Matheus Bernardini IMNS 2016 14 / 34

F. Torres - 1997 If γ and g are the number of even gaps and the genus of a numerical semigroup S, respectively, then 3 γ ≤ 2 g. Remark If γ is even, then N 0 \ ( { 2 , 4 , . . . , 2 γ } ∪ { 1 , 3 , . . . , γ − 1 } ) is a numerical semigroup with genus g = 3 γ 2 . ⌊ 2 g 3 ⌋ � n g = N γ ( g ) γ =0 Matheus Bernardini IMNS 2016 15 / 34

F. Torres - 1997 If γ and g are the number of even gaps and the genus of a numerical semigroup S, respectively, then 3 γ ≤ 2 g. Remark If γ is even, then N 0 \ ( { 2 , 4 , . . . , 2 γ } ∪ { 1 , 3 , . . . , γ − 1 } ) is a numerical semigroup with genus g = 3 γ 2 . ⌊ 2 g 3 ⌋ � n g = N γ ( g ) γ =0 Matheus Bernardini IMNS 2016 15 / 34

Theorem 1 Let γ be a nonnegative integer and g ≥ 3 γ . Then N γ ( g ) = N γ (3 γ ) . Thus, N γ ( g ) = N γ ( g + 1) , for all g ≥ 3 γ . Theorem 2 Let γ be a nonnegative integer and g < 3 γ . Then N γ ( g ) < N γ (3 γ ) . Matheus Bernardini IMNS 2016 16 / 34

Theorem 1 Let γ be a nonnegative integer and g ≥ 3 γ . Then N γ ( g ) = N γ (3 γ ) . Thus, N γ ( g ) = N γ ( g + 1) , for all g ≥ 3 γ . Theorem 2 Let γ be a nonnegative integer and g < 3 γ . Then N γ ( g ) < N γ (3 γ ) . Matheus Bernardini IMNS 2016 16 / 34

g / γ 0 1 2 3 4 5 6 7 8 9 10 n g 0 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 Matheus Bernardini IMNS 2016 17 / 34

Notice that ⌊ g ⌊ 2 g 3 ⌋ 3 ⌋ � � n g = N γ ( g ) + N γ ( g ) . γ =0 γ = ⌊ g 3 ⌋ +1 � 2( g +1) � ⌊ g 3 ⌋ 3 � � n g +1 = N γ ( g + 1) + N γ ( g + 1) . γ =0 γ = ⌊ g 3 ⌋ +1 Theorem 1 states that N γ ( g ) = N γ ( g + 1), for γ ≤ g 3 . Matheus Bernardini IMNS 2016 18 / 34

Notice that ⌊ g ⌊ 2 g 3 ⌋ 3 ⌋ � � n g = N γ ( g ) + N γ ( g ) . γ =0 γ = ⌊ g 3 ⌋ +1 � 2( g +1) � ⌊ g 3 ⌋ 3 � � n g +1 = N γ ( g + 1) + N γ ( g + 1) . γ =0 γ = ⌊ g 3 ⌋ +1 Theorem 1 states that N γ ( g ) = N γ ( g + 1), for γ ≤ g 3 . Matheus Bernardini IMNS 2016 18 / 34

Notice that ⌊ g ⌊ 2 g 3 ⌋ 3 ⌋ � � n g = N γ ( g ) + N γ ( g ) . γ =0 γ = ⌊ g 3 ⌋ +1 � 2( g +1) � ⌊ g 3 ⌋ 3 � � n g +1 = N γ ( g + 1) + N γ ( g + 1) . γ =0 γ = ⌊ g 3 ⌋ +1 Theorem 1 states that N γ ( g ) = N γ ( g + 1), for γ ≤ g 3 . Matheus Bernardini IMNS 2016 18 / 34

Corollary n g ≤ n g +1 if, and only if, � 2( g +1) � ⌊ 2 g 3 ⌋ 3 � � N γ ( g ) ≤ N γ ( g + 1) . γ = ⌊ g 3 ⌋ +1 γ = ⌊ g 3 ⌋ +1 Matheus Bernardini IMNS 2016 19 / 34

g / γ 0 1 2 3 4 5 6 7 8 9 10 n g 0 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 Matheus Bernardini IMNS 2016 20 / 34

Conjecture Let γ be a non-negative integer. Then N γ ( g ) ≤ N γ ( g + 1) , ∀ g . Remark If this Conjecture holds, then n g ≤ n g +1 for all g . Matheus Bernardini IMNS 2016 21 / 34

Conjecture Let γ be a non-negative integer. Then N γ ( g ) ≤ N γ ( g + 1) , ∀ g . Remark If this Conjecture holds, then n g ≤ n g +1 for all g . Matheus Bernardini IMNS 2016 21 / 34

Construction of a γ -hyperelliptic semigroup of genus g γ - positive integer T = N 0 \ { q 1 , . . . , q γ } numerical semigroup S = 2 · T ∪ (2 · N 0 + 1) \ { “suitable choice”of g − γ odd numbers } Matheus Bernardini IMNS 2016 22 / 34