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 ui for r > m,

there exists a subset T ⊂ {1, . . . , r} with |T| ≤ m such that x |

i∈T ui.

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 ui for r > m,

there exists a subset T ⊂ {1, . . . , r} with |T| ≤ m such that x |

i∈T ui.

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 ui for r > m,

there exists a subset T ⊂ {1, . . . , r} with |T| ≤ m such that x |

i∈T ui.

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(p1 · · · pr) = r for any primes p1, . . . , pr ∈ 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 ui for r > m,

there exists a subset T ⊂ {1, . . . , r} with |T| ≤ m such that x |

i∈T ui.

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 ui for r > m,

there exists a subset T ⊂ {1, . . . , r} with |T| ≤ m such that x |

i∈T ui.

Definition

A bullet for x ∈ M is a product u1 · · · ur of irreducible elements such that (i) x divides u1 · · · ur, and (ii) x does not divide u1 · · · ur/ui 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 ui for r > m,

there exists a subset T ⊂ {1, . . . , r} with |T| ≤ m such that x |

i∈T ui.

Definition

A bullet for x ∈ M is a product u1 · · · ur of irreducible elements such that (i) x divides u1 · · · ur, and (ii) x does not divide u1 · · · ur/ui for each i ≤ r. The set of bullets of x is denoted bul(x).

Proposition

ωM(x) = max{r : u1 · · · ur ∈ 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 g1, . . . , gk.

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 g1, . . . , gk.

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 g1, . . . , gk.

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 g1, . . . , gk.

Example

McN = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, 21, . . .}. “McNugget Monoid” n ∈ McN ω(n) bullet 6 3 3 · 20 9 3 3 · 20 12 3 3 · 20 15 4 4 · 6 18 3 3 · 6 n ∈ McN ω(n) bullet 20 10 10 · 6 21 5 5 · 6 24 4 4 · 6 26 11 11 · 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: Bullets in S: b1g1 + · · · + bkgk ← → b = (b1, . . . , bk) ∈ Nk.

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: Bullets in S: b1g1 + · · · + bkgk ← → b = (b1, . . . , bk) ∈ Nk. For each i ≤ k, we have ci ei ∈ bul(n) for some ci > 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: Bullets in S: b1g1 + · · · + bkgk ← → b = (b1, . . . , bk) ∈ Nk. For each i ≤ k, we have ci ei ∈ bul(n) for some ci > 0. bul(n) ⊂ k

i=1[0, ci].

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: Bullets in S: b1g1 + · · · + bkgk ← → b = (b1, . . . , bk) ∈ Nk. For each i ≤ k, we have ci ei ∈ bul(n) for some ci > 0. bul(n) ⊂ k

i=1[0, ci].

Algorithm

Search k

i=1[0, ci] 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: Bullets in S: b1g1 + · · · + bkgk ← → b = (b1, . . . , bk) ∈ Nk. For each i ≤ k, we have ci ei ∈ bul(n) for some ci > 0. bul(n) ⊂ k

i=1[0, ci].

Algorithm

Search k

i=1[0, ci] 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

Quasilinearity for numerical monoids

Theorem ((O.–Pelayo, 2013), (Garc´ ıa-Garc´ ıa et.al., 2013))

ωS(n) = 1

g1 n + a0(n) for n ≫ 0, where a0(n) periodic with period g1.

5 10 15 20 25 30 35 5 10 15 20 40 60 80 100 5 10 15 20

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))

ωS(n) = 1

g1 n + a0(n) for n ≫ 0, where a0(n) periodic with period g1.

5 10 15 20 25 30 35 5 10 15 20 40 60 80 100 5 10 15 20

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 N0 such that ω(n) is quasilinear for n > N0.

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 6 / 12

Quasilinearity for numerical monoids

Dissonance point: minimum N0 such that ω(n) is quasilinear for n > N0.

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 N0 such that ω(n) is quasilinear for n > N0.

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

Quasilinearity for numerical monoids

Dissonance point: minimum N0 such that ω(n) is quasilinear for n > N0.

Question (O.-Pelayo, 2013)

The upper bound for dissonance point is large. Can we do better?

Roadblock

Existing algorithms are slow for large n.

Question (O.–Pelayo, 2014)

Can we dynamically (inductively) compute several ω-values at once?

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 6 / 12

Quasilinearity for numerical monoids

Dissonance point: minimum N0 such that ω(n) is quasilinear for n > N0.

Question (O.-Pelayo, 2013)

The upper bound for dissonance point is large. Can we do better?

Roadblock

Existing algorithms are slow for large n.

Question (O.–Pelayo, 2014)

Can we dynamically (inductively) compute several ω-values at once?

Answer (Barron-O.-Pelayo, 2014)

Yes!

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 6 / 12

Toward a dynamic algorithm. . . the inductive step

For n ∈ S, let Z(n) = { a ∈ Nk : k

i=1 aigi = n}.

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 7 / 12

Toward a dynamic algorithm. . . the inductive step

For n ∈ S, let Z(n) = { a ∈ Nk : k

i=1 aigi = n}.

For each i ≤ k,

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 7 / 12

Toward a dynamic algorithm. . . the inductive step

For n ∈ S, let Z(n) = { a ∈ Nk : k

i=1 aigi = n}.

For each i ≤ k, φi : Z(n − gi) − → Z(n)

• a

− →

• a +

ei.

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 7 / 12

Toward a dynamic algorithm. . . the inductive step

For n ∈ S, let Z(n) = { a ∈ Nk : k

i=1 aigi = n}.

For each i ≤ k, φi : Z(n − gi) − → Z(n)

• a

− →

• a +

ei. In particular, Z(n) =

• i≤k

φi(Z(n − gi))

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 7 / 12

Toward a dynamic algorithm. . . the inductive step

For n ∈ S, let Z(n) = { a ∈ Nk : k

i=1 aigi = n}.

For each i ≤ k, φi : Z(n − gi) − → Z(n)

• a

− →

• a +

ei. In particular, Z(n) =

• i≤k

φi(Z(n − gi))

Definition/Proposition (Cover morphisms)

Fix n ∈ S and i ≤ k. The i-th cover morphism for n is the map ψi : bul(n − gi) − → bul(n)

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 7 / 12

Toward a dynamic algorithm. . . the inductive step

For n ∈ S, let Z(n) = { a ∈ Nk : k

i=1 aigi = n}.

For each i ≤ k, φi : Z(n − gi) − → Z(n)

• a

− →

• a +

ei. In particular, Z(n) =

• i≤k

φi(Z(n − gi))

Definition/Proposition (Cover morphisms)

Fix n ∈ S and i ≤ k. The i-th cover morphism for n is the map ψi : bul(n − gi) − → bul(n) given by

• b −

→ b + ei k

j=1 bjgj − n − gi /

∈ S

• b

k

j=1 bjgj − n − gi ∈ S

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 7 / 12

Toward a dynamic algorithm. . . the inductive step

For n ∈ S, let Z(n) = { a ∈ Nk : k

i=1 aigi = n}.

For each i ≤ k, φi : Z(n − gi) − → Z(n)

• a

− →

• a +

ei. In particular, Z(n) =

• i≤k

φi(Z(n − gi))

Definition/Proposition (Cover morphisms)

Fix n ∈ S and i ≤ k. The i-th cover morphism for n is the map ψi : bul(n − gi) − → bul(n) given by

• b −

→ b + ei k

j=1 bjgj − n − gi /

∈ S

• b

k

j=1 bjgj − n − gi ∈ S

Moreover, bul(n) =

i≤k ψi(bul(n − gi)).**

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 7 / 12

Toward a dynamic algorithm. . . the base case

Definition (ω-primality in numerical monoids)

Fix a numerical monoid S and n ∈ S.

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 8 / 12

Toward a dynamic algorithm. . . the base case

Definition (ω-primality in numerical monoids)

Fix a numerical monoid S and n ∈ S. ωS(n) is the minimal m such that whenever (r

i=1 gji) − n ∈ S for r > m,

there exists T ⊂ {1, . . . , r} with |T| ≤ m and (

i∈T gji) − n ∈ S.

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 8 / 12

Toward a dynamic algorithm. . . the base case

Definition (ω-primality in numerical monoids)

Fix a numerical monoid S and n ∈ S. ωS(n) is the minimal m such that whenever (r

i=1 gji) − n ∈ S for r > m,

there exists T ⊂ {1, . . . , r} with |T| ≤ m and (

i∈T gji) − n ∈ S.

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 8 / 12

Toward a dynamic algorithm. . . the base case

Definition (ω-primality in numerical monoids)

Fix a numerical monoid S and n ∈ Z = q(S). ωS(n) is the minimal m such that whenever (r

i=1 gji) − n ∈ S for r > m,

there exists T ⊂ {1, . . . , r} with |T| ≤ m and (

i∈T gji) − n ∈ S.

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 8 / 12

Toward a dynamic algorithm. . . the base case

Definition (ω-primality in numerical monoids)

Fix a numerical monoid S and n ∈ Z = q(S). ωS(n) is the minimal m such that whenever (r

i=1 gji) − n ∈ S for r > m,

there exists T ⊂ {1, . . . , r} with |T| ≤ m and (

i∈T gji) − n ∈ S.

Remark

All properties of ω extend from S to Z.

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 8 / 12

Toward a dynamic algorithm. . . the base case

Definition (ω-primality in numerical monoids)

Fix a numerical monoid S and n ∈ Z = q(S). ωS(n) is the minimal m such that whenever (r

i=1 gji) − n ∈ S for r > m,

there exists T ⊂ {1, . . . , r} with |T| ≤ m and (

i∈T gji) − n ∈ S.

Remark

All properties of ω extend from S to Z.

Proposition

For n ∈ Z, the following are equivalent: (i) ω(n) = 0, (ii) bul(n) = { 0}, (iii) −n ∈ S.

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 8 / 12

A dynamic algorithm!

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 9 / 12

A dynamic algorithm!

Example

McN = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, 21, . . .}.

n ∈ Z ω(n) bul(n) n ∈ Z ω(n) bul(n)

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 9 / 12

A dynamic algorithm!

Example

McN = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, 21, . . .}.

n ∈ Z ω(n) bul(n) ≤ −44 { 0} n ∈ Z ω(n) bul(n)

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 9 / 12

A dynamic algorithm!

Example

McN = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, 21, . . .}.

n ∈ Z ω(n) bul(n) ≤ −44 { 0} −43 1 { e1, e2, e3} n ∈ Z ω(n) bul(n)

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 9 / 12

A dynamic algorithm!

Example

McN = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, 21, . . .}.

n ∈ Z ω(n) bul(n) ≤ −44 { 0} −43 1 { e1, e2, e3} −42 { 0} n ∈ Z ω(n) bul(n)

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 9 / 12

A dynamic algorithm!

Example

McN = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, 21, . . .}.

n ∈ Z ω(n) bul(n) ≤ −44 { 0} −43 1 { e1, e2, e3} −42 { 0} . . . . . . . . . −38 { 0} n ∈ Z ω(n) bul(n)

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 9 / 12

A dynamic algorithm!

Example

McN = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, 21, . . .}.

n ∈ Z ω(n) bul(n) ≤ −44 { 0} −43 1 { e1, e2, e3} −42 { 0} . . . . . . . . . −38 { 0} −37 2 {2 e1, e2, e3} n ∈ Z ω(n) bul(n)

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 9 / 12

A dynamic algorithm!

Example

McN = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, 21, . . .}.

n ∈ Z ω(n) bul(n) ≤ −44 { 0} −43 1 { e1, e2, e3} −42 { 0} . . . . . . . . . −38 { 0} −37 2 {2 e1, e2, e3} −36 { 0} −35 { 0} n ∈ Z ω(n) bul(n)

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 9 / 12

A dynamic algorithm!

Example

McN = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, 21, . . .}.

n ∈ Z ω(n) bul(n) ≤ −44 { 0} −43 1 { e1, e2, e3} −42 { 0} . . . . . . . . . −38 { 0} −37 2 {2 e1, e2, e3} −36 { 0} −35 { 0} −34 2 { e1, 2 e2, e3} n ∈ Z ω(n) bul(n)

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 9 / 12

A dynamic algorithm!

Example

McN = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, 21, . . .}.

n ∈ Z ω(n) bul(n) ≤ −44 { 0} −43 1 { e1, e2, e3} −42 { 0} . . . . . . . . . −38 { 0} −37 2 {2 e1, e2, e3} −36 { 0} −35 { 0} −34 2 { e1, 2 e2, e3} −33 { 0} −32 { 0} n ∈ Z ω(n) bul(n)

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 9 / 12

A dynamic algorithm!

Example

McN = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, 21, . . .}.

n ∈ Z ω(n) bul(n) ≤ −44 { 0} −43 1 { e1, e2, e3} −42 { 0} . . . . . . . . . −38 { 0} −37 2 {2 e1, e2, e3} −36 { 0} −35 { 0} −34 2 { e1, 2 e2, e3} −33 { 0} −32 { 0} −31 3 {3 e1, e2, e3} n ∈ Z ω(n) bul(n)

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 9 / 12

A dynamic algorithm!

Example

McN = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, 21, . . .}.

n ∈ Z ω(n) bul(n) ≤ −44 { 0} −43 1 { e1, e2, e3} −42 { 0} . . . . . . . . . −38 { 0} −37 2 {2 e1, e2, e3} −36 { 0} −35 { 0} −34 2 { e1, 2 e2, e3} −33 { 0} −32 { 0} −31 3 {3 e1, e2, e3} . . . . . . . . . n ∈ Z ω(n) bul(n)

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 9 / 12

A dynamic algorithm!

Example

McN = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, 21, . . .}.

n ∈ Z ω(n) bul(n) ≤ −44 { 0} −43 1 { e1, e2, e3} −42 { 0} . . . . . . . . . −38 { 0} −37 2 {2 e1, e2, e3} −36 { 0} −35 { 0} −34 2 { e1, 2 e2, e3} −33 { 0} −32 { 0} −31 3 {3 e1, e2, e3} . . . . . . . . . n ∈ Z ω(n) bul(n) 1 5 {5 e1, (2, 1, 0), . . .}

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 9 / 12

A dynamic algorithm!

Example

McN = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, 21, . . .}.

n ∈ Z ω(n) bul(n) ≤ −44 { 0} −43 1 { e1, e2, e3} −42 { 0} . . . . . . . . . −38 { 0} −37 2 {2 e1, e2, e3} −36 { 0} −35 { 0} −34 2 { e1, 2 e2, e3} −33 { 0} −32 { 0} −31 3 {3 e1, e2, e3} . . . . . . . . . n ∈ Z ω(n) bul(n) 1 5 {5 e1, (2, 1, 0), . . .} 2 7 {7 e1, 6 e2, . . .}

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 9 / 12

A dynamic algorithm!

Example

McN = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, 21, . . .}.

n ∈ Z ω(n) bul(n) ≤ −44 { 0} −43 1 { e1, e2, e3} −42 { 0} . . . . . . . . . −38 { 0} −37 2 {2 e1, e2, e3} −36 { 0} −35 { 0} −34 2 { e1, 2 e2, e3} −33 { 0} −32 { 0} −31 3 {3 e1, e2, e3} . . . . . . . . . n ∈ Z ω(n) bul(n) 1 5 {5 e1, (2, 1, 0), . . .} 2 7 {7 e1, 6 e2, . . .} 3 3 {3 e3, 2 e2, . . .} 4 4 {4 e1, 4 e2, . . .} 5 9 {9 e1, (6, 1, 0), . . .} 6 3 {3 e3, 2 e2, . . .} 7 6 {6 e1, (3, 1, 0), . . .} 8 8 {8 e1, (5, 2, 0), . . . , } 9 3 {3 e1, 3 e3, . . .} . . . . . . . . .

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 9 / 12

A dynamic algorithm!

Example

McN = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, 21, . . .}.

n ∈ Z ω(n) bul(n) ≤ −44 { 0} −43 1 { e1, e2, e3} −42 { 0} . . . . . . . . . −38 { 0} −37 2 {2 e1, e2, e3} −36 { 0} −35 { 0} −34 2 { e1, 2 e2, e3} −33 { 0} −32 { 0} −31 3 {3 e1, e2, e3} . . . . . . . . . n ∈ Z ω(n) bul(n) 1 5 {5 e1, (2, 1, 0), . . .} 2 7 {7 e1, 6 e2, . . .} 3 3 {3 e3, 2 e2, . . .} 4 4 {4 e1, 4 e2, . . .} 5 9 {9 e1, (6, 1, 0), . . .} 6 3 {3 e3, 2 e2, . . .} 7 6 {6 e1, (3, 1, 0), . . .} 8 8 {8 e1, (5, 2, 0), . . . , } 9 3 {3 e1, 3 e3, . . .} . . . . . . . . . 148 28 {28 e1, . . .}

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 9 / 12

A dynamic algorithm!

Example

McN = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, 21, . . .}.

n ∈ Z ω(n) bul(n) ≤ −44 { 0} −43 1 { e1, e2, e3} −42 { 0} . . . . . . . . . −38 { 0} −37 2 {2 e1, e2, e3} −36 { 0} −35 { 0} −34 2 { e1, 2 e2, e3} −33 { 0} −32 { 0} −31 3 {3 e1, e2, e3} . . . . . . . . . n ∈ Z ω(n) bul(n) 1 5 {5 e1, (2, 1, 0), . . .} 2 7 {7 e1, 6 e2, . . .} 3 3 {3 e3, 2 e2, . . .} 4 4 {4 e1, 4 e2, . . .} 5 9 {9 e1, (6, 1, 0), . . .} 6 3 {3 e3, 2 e2, . . .} 7 6 {6 e1, (3, 1, 0), . . .} 8 8 {8 e1, (5, 2, 0), . . . , } 9 3 {3 e1, 3 e3, . . .} . . . . . . . . . 148 28 {28 e1, . . .} 149 33 {33 e1, . . .}

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 9 / 12

A dynamic algorithm!

Example

McN = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, 21, . . .}.

n ∈ Z ω(n) bul(n) ≤ −44 { 0} −43 1 { e1, e2, e3} −42 { 0} . . . . . . . . . −38 { 0} −37 2 {2 e1, e2, e3} −36 { 0} −35 { 0} −34 2 { e1, 2 e2, e3} −33 { 0} −32 { 0} −31 3 {3 e1, e2, e3} . . . . . . . . . n ∈ Z ω(n) bul(n) 1 5 {5 e1, (2, 1, 0), . . .} 2 7 {7 e1, 6 e2, . . .} 3 3 {3 e3, 2 e2, . . .} 4 4 {4 e1, 4 e2, . . .} 5 9 {9 e1, (6, 1, 0), . . .} 6 3 {3 e3, 2 e2, . . .} 7 6 {6 e1, (3, 1, 0), . . .} 8 8 {8 e1, (5, 2, 0), . . . , } 9 3 {3 e1, 3 e3, . . .} . . . . . . . . . 148 28 {28 e1, . . .} 149 33 {33 e1, . . .} 150 25 {25 e1, . . .}

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 9 / 12

A dynamic algorithm!

Example

McN = 6, 9, 20 = {0, 6, 9, 12, 15, 18, 20, 21, . . .}.

n ∈ Z ω(n) bul(n) n ∈ Z ω(n) bul(n) 6 3 {3 e3, 2 e2, . . .} 9 3 {3 e1, 3 e3, . . .} . . . . . . . . . 148 28 {28 e1, . . .} 149 33 {33 e1, . . .} 150 25 {25 e1, . . .}

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 9 / 12

Runtime comparison

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 10 / 12

Runtime comparison

S n ∈ S ωS(n) Existing Dynamic 6, 9, 20 1000 170 0m 67s 0.4s 6, 9, 20 2000 340 17m 20s 3.1s 31, 39, 45, 52 1000 40 0m 39s 0.3s 31, 39, 45, 52 2000 71 24m 31s 2.0s 54, 67, 69, 73, 75 1000 23 2m 02s 0.7s 54, 67, 69, 73, 75 1500 33 22m 50s 2.3s 54, 67, 69, 73, 75 3000 61 ——— 44.3s 54, 67, 69, 73, 75 5000 98 ——— 8m 59.0s

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 10 / 12

Runtime comparison

S n ∈ S ωS(n) Existing Dynamic 6, 9, 20 1000 170 0m 67s 0.4s 6, 9, 20 2000 340 17m 20s 3.1s 31, 39, 45, 52 1000 40 0m 39s 0.3s 31, 39, 45, 52 2000 71 24m 31s 2.0s 54, 67, 69, 73, 75 1000 23 2m 02s 0.7s 54, 67, 69, 73, 75 1500 33 22m 50s 2.3s 54, 67, 69, 73, 75 3000 61 ——— 44.3s 54, 67, 69, 73, 75 5000 98 ——— 8m 59.0s Sage: Open Source Mathematics Software, available at www.sagemath.org. GAP Numerical Semigroups Package, available at http://www.gap-system.org/Packages/numericalsgps.html.

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 10 / 12

Future directions

What about more general (finitely generated) monoids M?

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 11 / 12

Future directions

What about more general (finitely generated) monoids M? Characterization of ωM in terms of maximal length bullets?

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 11 / 12

Future directions

What about more general (finitely generated) monoids M? Characterization of ωM in terms of maximal length bullets?

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 11 / 12

Future directions

What about more general (finitely generated) monoids M? Characterization of ωM in terms of maximal length bullets? Extension of ωM to q(M)?

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 11 / 12

Future directions

What about more general (finitely generated) monoids M? Characterization of ωM in terms of maximal length bullets? Extension of ωM to q(M)?

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 11 / 12

Future directions

What about more general (finitely generated) monoids M? Characterization of ωM in terms of maximal length bullets? Extension of ωM to q(M)? Iterative construction of bullets from cover maps?

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 11 / 12

Future directions

What about more general (finitely generated) monoids M? Characterization of ωM in terms of maximal length bullets? Extension of ωM to q(M)? Iterative construction of bullets from cover maps?

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 11 / 12

Future directions

What about more general (finitely generated) monoids M? Characterization of ωM in terms of maximal length bullets? Extension of ωM to q(M)? Iterative construction of bullets from cover maps? Problem:

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 11 / 12

Future directions

What about more general (finitely generated) monoids M? Characterization of ωM in terms of maximal length bullets? Extension of ωM to q(M)? Iterative construction of bullets from cover maps? Problem: the base case!

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 11 / 12

Future directions

What about more general (finitely generated) monoids M? Characterization of ωM in terms of maximal length bullets? Extension of ωM to q(M)? Iterative construction of bullets from cover maps? Problem: the base case!

Problem

Find a dynamic algorithm to compute ω-primality in M.

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 11 / 12

Future directions

What about more general (finitely generated) monoids M? Characterization of ωM in terms of maximal length bullets? Extension of ωM to q(M)? Iterative construction of bullets from cover maps? Problem: the base case!

Problem

Find a dynamic algorithm to compute ω-primality in M.

Question

Are there dynamic algorithms for computing other factorization invariants?

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 11 / 12

References

Alfred Geroldinger (1997) Chains of factorizations in weakly Krull domains.

• Colloq. Math. 72 (1997) 53 – 81.

David Anderson, Scott Chapman, Nathan Kaplan, and Desmond Torkornoo (2011) An algorithm to compute ω-primality in a numerical monoid. Semigroup Forum 82 (2011), no. 1, 96 – 108. Christopher O’Neill, Roberto Pelayo (2014) How do you measure primality? American Mathematical Monthly, forthcoming. Thomas Barron, Christopher O’Neill, Roberto Pelayo (2014) On the computation of delta sets and ω-primality in numerical monoids. preprint.

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 12 / 12

References

Alfred Geroldinger (1997) Chains of factorizations in weakly Krull domains.

• Colloq. Math. 72 (1997) 53 – 81.

David Anderson, Scott Chapman, Nathan Kaplan, and Desmond Torkornoo (2011) An algorithm to compute ω-primality in a numerical monoid. Semigroup Forum 82 (2011), no. 1, 96 – 108. Christopher O’Neill, Roberto Pelayo (2014) How do you measure primality? American Mathematical Monthly, forthcoming. Thomas Barron, Christopher O’Neill, Roberto Pelayo (2014) On the computation of delta sets and ω-primality in numerical monoids. preprint. Thanks!

Christopher O’Neill (Texas A&M University) How do you measure primality? September 26, 2014 12 / 12