How do you measure primality? Christopher ONeill Texas A&M - PowerPoint PPT Presentation

How do you measure primality? Christopher O’Neill Texas A&M University coneill@math.tamu.edu Joint with Thomas Barron and Roberto Pelayo September 26, 2014 Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 1 / 12

ω -primality Definition ( ω -primality) Fix a cancellative, commutative, atomic monoid M . For x ∈ M , ω ( x ) is the smallest positive integer m such that whenever x | � r i =1 u i for r > m , there exists a subset T ⊂ { 1 , . . . , r } with | T | ≤ m such that x | � i ∈ T u i . Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 2 / 12

ω -primality Definition ( ω -primality) Fix a cancellative, commutative, atomic monoid M . For x ∈ M , ω ( x ) is the smallest positive integer m such that whenever x | � r i =1 u i for r > m , there exists a subset T ⊂ { 1 , . . . , r } with | T | ≤ m such that x | � i ∈ T u i . Fact ω ( x ) = 1 if and only if x is prime (i.e. x | ab implies x | a or x | b ). Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 2 / 12

ω -primality Definition ( ω -primality) Fix a cancellative, commutative, atomic monoid M . For x ∈ M , ω ( x ) is the smallest positive integer m such that whenever x | � r i =1 u i for r > m , there exists a subset T ⊂ { 1 , . . . , r } with | T | ≤ m such that x | � i ∈ T u i . Fact ω ( x ) = 1 if and only if x is prime (i.e. x | ab implies x | a or x | b ). Fact M is factorial if and only if every irreducible element u ∈ M is prime. Moreover, ω ( p 1 · · · p r ) = r for any primes p 1 , . . . , p r ∈ M. Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 2 / 12

ω -primality Definition ( ω -primality) Fix a cancellative, commutative, atomic monoid M . For x ∈ M , ω ( x ) is the smallest positive integer m such that whenever x | � r i =1 u i for r > m , there exists a subset T ⊂ { 1 , . . . , r } with | T | ≤ m such that x | � i ∈ T u i . Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 2 / 12

ω -primality Definition ( ω -primality) Fix a cancellative, commutative, atomic monoid M . For x ∈ M , ω ( x ) is the smallest positive integer m such that whenever x | � r i =1 u i for r > m , there exists a subset T ⊂ { 1 , . . . , r } with | T | ≤ m such that x | � i ∈ T u i . Definition A bullet for x ∈ M is a product u 1 · · · u r of irreducible elements such that (i) x divides u 1 · · · u r , and (ii) x does not divide u 1 · · · u r / u i for each i ≤ r . The set of bullets of x is denoted bul( x ). Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 2 / 12

ω -primality Definition ( ω -primality) Fix a cancellative, commutative, atomic monoid M . For x ∈ M , ω ( x ) is the smallest positive integer m such that whenever x | � r i =1 u i for r > m , there exists a subset T ⊂ { 1 , . . . , r } with | T | ≤ m such that x | � i ∈ T u i . Definition A bullet for x ∈ M is a product u 1 · · · u r of irreducible elements such that (i) x divides u 1 · · · u r , and (ii) x does not divide u 1 · · · u r / u i for each i ≤ r . The set of bullets of x is denoted bul( x ). Proposition ω M ( x ) = max { r : u 1 · · · u r ∈ bul( x ) } . Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 2 / 12

Numerical monoids Definition A numerical monoid S is an additive submonoid of N with | N \ S | < ∞ . Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 3 / 12

Numerical monoids Definition A numerical monoid S is an additive submonoid of N with | N \ S | < ∞ . Fact Any numerical monoid S has a unique minimal generating set g 1 , . . . , g k . Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 3 / 12

Numerical monoids Definition A numerical monoid S is an additive submonoid of N with | N \ S | < ∞ . Fact Any numerical monoid S has a unique minimal generating set g 1 , . . . , g k . Example McN = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 3 / 12

Numerical monoids Definition A numerical monoid S is an additive submonoid of N with | N \ S | < ∞ . Fact Any numerical monoid S has a unique minimal generating set g 1 , . . . , g k . Example McN = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . “McNugget Monoid” Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 3 / 12

Numerical monoids Definition A numerical monoid S is an additive submonoid of N with | N \ S | < ∞ . Fact Any numerical monoid S has a unique minimal generating set g 1 , . . . , g k . Example McN = � 6 , 9 , 20 � = { 0 , 6 , 9 , 12 , 15 , 18 , 20 , 21 , . . . } . “McNugget Monoid” n ∈ McN ω ( n ) bullet n ∈ McN ω ( n ) bullet 6 3 3 · 20 20 10 10 · 6 9 3 3 · 20 21 5 5 · 6 12 3 3 · 20 24 4 4 · 6 15 4 4 · 6 26 11 11 · 6 18 3 3 · 6 27 6 6 · 6 Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 3 / 12

Algorithms to compute ω -primality ω -primality in a numerical monoid S : Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 4 / 12

Algorithms to compute ω -primality ω -primality in a numerical monoid S : → � b = ( b 1 , . . . , b k ) ∈ N k . Bullets in S : b 1 g 1 + · · · + b k g k ← Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 4 / 12

Algorithms to compute ω -primality ω -primality in a numerical monoid S : → � b = ( b 1 , . . . , b k ) ∈ N k . Bullets in S : b 1 g 1 + · · · + b k g k ← For each i ≤ k , we have c i � e i ∈ bul( n ) for some c i > 0. Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 4 / 12

Algorithms to compute ω -primality ω -primality in a numerical monoid S : → � b = ( b 1 , . . . , b k ) ∈ N k . Bullets in S : b 1 g 1 + · · · + b k g k ← For each i ≤ k , we have c i � e i ∈ bul( n ) for some c i > 0. bul( n ) ⊂ � k i =1 [0 , c i ]. Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 4 / 12

Algorithms to compute ω -primality ω -primality in a numerical monoid S : → � b = ( b 1 , . . . , b k ) ∈ N k . Bullets in S : b 1 g 1 + · · · + b k g k ← For each i ≤ k , we have c i � e i ∈ bul( n ) for some c i > 0. bul( n ) ⊂ � k i =1 [0 , c i ]. Algorithm Search � k i =1 [0 , c i ] for bullets, compute ω ( n ) = max {| � b | : � b ∈ bul( n ) } . Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 4 / 12

Algorithms to compute ω -primality ω -primality in a numerical monoid S : → � b = ( b 1 , . . . , b k ) ∈ N k . Bullets in S : b 1 g 1 + · · · + b k g k ← For each i ≤ k , we have c i � e i ∈ bul( n ) for some c i > 0. bul( n ) ⊂ � k i =1 [0 , c i ]. Algorithm Search � k i =1 [0 , c i ] for bullets, compute ω ( n ) = max {| � b | : � b ∈ bul( n ) } . Remark Several improvements on this algorithm exist. Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 4 / 12

15 20 15 10 10 5 5 5 10 15 20 25 30 35 20 40 60 80 100 Quasilinearity for numerical monoids Theorem ((O.–Pelayo, 2013), (Garc´ ıa-Garc´ ıa et.al., 2013)) ω S ( n ) = 1 g 1 n + a 0 ( n ) for n ≫ 0 , where a 0 ( n ) periodic with period g 1 . Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 5 / 12

Quasilinearity for numerical monoids Theorem ((O.–Pelayo, 2013), (Garc´ ıa-Garc´ ıa et.al., 2013)) ω S ( n ) = 1 g 1 n + a 0 ( n ) for n ≫ 0 , where a 0 ( n ) periodic with period g 1 . 15 20 15 10 10 5 5 20 40 60 80 100 5 10 15 20 25 30 35 S = � 3 , 7 � McN = � 6 , 9 , 20 � Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 5 / 12

Quasilinearity for numerical monoids Dissonance point : minimum N 0 such that ω ( n ) is quasilinear for n > N 0 . Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 6 / 12

Quasilinearity for numerical monoids Dissonance point : minimum N 0 such that ω ( n ) is quasilinear for n > N 0 . Question (O.-Pelayo, 2013) The upper bound for dissonance point is large. Can we do better? Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 6 / 12

Quasilinearity for numerical monoids Dissonance point : minimum N 0 such that ω ( n ) is quasilinear for n > N 0 . Question (O.-Pelayo, 2013) The upper bound for dissonance point is large. Can we do better? Roadblock Existing algorithms are slow for large n . Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 6 / 12