Computing the delta set of an affine semigroup: a status report - PowerPoint PPT Presentation

Affine semigroups (a geometric viewpoint) Fix an affine semigroup S = � n 1 , . . . , n k � ⊂ Z d ≥ 0 . � � a ∈ Z k Z( n ) = ≥ 0 : n = a 1 n 1 + · · · + a k n k A geometric viewpoint: non-negative integer solutions to linear equations. Example: S = � 6 , 9 , 20 � , n = 60. Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 5 / 27

Affine semigroups (a geometric viewpoint) Fix an affine semigroup S = � n 1 , . . . , n k � ⊂ Z d ≥ 0 . � � a ∈ Z k Z( n ) = ≥ 0 : n = a 1 n 1 + · · · + a k n k A geometric viewpoint: non-negative integer solutions to linear equations. Example: S = � 6 , 9 , 20 � , n = 60. � � a ∈ Z k Z(60) = ≥ 0 : 60 = 6 a 1 + 9 a 2 + 20 a 3 Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 5 / 27

l l l a t e x i t > < a < t e x i t s h a / 4 _ E 7 m p S t O 3 n M f 3 + B 8 / g D F G I 1 b d = N 5 W 8 a 4 X f g " D > A A A B 7 X i c T V a 0 s e 6 4 = " e y i m 8 t y A g E r z 9 Q z 5 V M 7 s r Y Q O q K U b M U d G G 4 C 2 e v C F z P w + K M k k w I d j K S U 9 o z l C O L E 8 V S j D L T T A B w Y n S u E V 3 p x H 5 8 6 D W u v W 6 6 t 1 d l O 1 C P I 0 C H M M J V M b B V 2 g j 1 S c x j 1 Q x x p p x J 6 h t m O q R 0 S u b W 9 k 5 x t 7 3 d f 3 B 4 V D 4 + e O m C H A Y C j y S L G M G T S p 1 q s 9 a o u h D i d m I R c t o J J 7 z D v / N E l W a x f x 3 N n R a j e C l 4 8 V B m t o V 1 K N s 2 2 7 i U 7 S w M x E J 2 t X V Q + V T 1 6 C R a h o l 5 A K 0 P B h 7 v z T z 3 L 0 w 4 0 8 Z 1 v o 3 2 H S b J C W f o j v h W Q 8 e r / 8 e a / M m g e r o p X / C i w c V / F 4 e b / 4 b 0 3 Y Y Z 2 7 W 8 O K x g m s L V X K y a b Y N z W b P v P u v b B a 2 i t s 7 3 q v 7 p Y P D B 5 N 1 r p w C 4 P H e D D P z l s Q K g 6 7 7 7 a y V B m I 6 4 = " Z / A 6 K c s s L S c O h e 2 5 e a n t t e x i t > < l a e b x i t s h a 1 _ 2 b b q x E J 3 1 s 9 a v k w c v w S J U k L J M S q > X Q G P b 7 4 = " A N A A B 7 n i c b V B k n s h y o a 0 z 9 u W K p N z E 4 x n + 0 7 I T s V R J q k x b c 9 N M h M T j Y J J P i l 2 q V l 4 Y F Z 9 K b i f 1 7 x w + g q G A u V Z x U 6 P J E q 0 / Q r J T 3 J d M a a x M a M 4 t C 7 G z Q 8 m Y 4 v J n 6 S u e u j U j U P Y 5 N T H L f J T L R r Z A a o q X i P g q U v J V S s i 4 D G S Z e D k p Q 5 v G t / T V 6 S U s u q S t 0 r 0 Q k G E U r S d q e e 2 7 f W b d U Z D / h x X m / X H H r 7 x z o l X g 5 q U C O a t r 8 H s x E 0 i D G Q E 1 i R r C x U r v q V / k k 4 X R a 6 q W a S z U G e E i 7 l g o c Y W 8 I 1 R s k J I 2 p M R Y j r b u e K 0 2 Q i V x u d b 6 e w t r 6 x V H X c L u 3 s 7 u 0 H t l A f m L Z 7 0 N Y H 4 O / 3 Z p i Z F 0 r f A p L T j t S U R y H n b Y D 8 e 3 M b z 9 R Z o + k P H n W S K k J 9 v + B E d U K s K W e C U 1 P 6 n A C p 1 A F D 6 g 0 C X f Q A h 8 I j i I E v 4 K 3 / P I q 8 R 1 v m 7 p 3 f 1 l p N O Z l = 4 A W V S N w A = < f / l a t e x i t > P g X 5 u H N k c 6 L 8 + 8 / L F o L T j 5 z D H G I k M r P U k D m 1 n j 1 h I L 3 s z 8 T + v w s 5 x 8 3 O n 6 M w q A Q n l y p Y w a K 7 + m r E L U o M n 1 i C i W 2 A V k R G W G F i b K G o i O s i Y k K m h g w 9 W R S l H J k G z 3 P e s J h k i w X R S l H k q b z 3 9 G Q K U o m J n + n w G a s V 7 2 5 J M / X S 0 1 0 H W R M 1 G f 6 8 R v 1 m 7 p 3 1 v V p N f I 0 i n A I + q W C i W L 2 V k T G G 3 F i b E I l G 4 K 2 C 5 8 R Q a Q j H W v c N k z F B h p V h h N 1 Y Z l W t u w u g d e L p 8 A I 5 2 o P y V 3 N q p o r D F E U K 1 v S I Q X 6 e y L D Q u u u o Z a 9 q m m A y w S P s d 1 R i Q X W Q L c 6 G 1 / x i c b V B N S w M E 7 J 2 t X 7 V + V T X B 6 1 / + b C z C l p m 9 A b v u A = " > A A 1 C w L b 0 G w 2 J F m h P 2 0 R X j y o e P X b m R v a h Q i m 7 V V B B l S 8 e K 7 i 2 0 C 4 5 N A T L 8 + 5 8 L F s L j M 5 z C n / g f P 4 6 J Y g F D 5 r Q g j t o w m 8 E J v A M r / D A o V 4 h a 1 _ b a s e 6 = " V Y r v 2 A K B s t y t N x g = = < / l a e i x i t > < l a t e x a c < < / l a t e x i t > l W a t e x i t s h < D 1 e q 2 P x W j B y n O J 4 Q + s z x + Y u a _ x c f c e u / a M w O = M " > A A A B 7 X i d O b H a s e 6 4 = " A J I t f e Q q l a P r 5 3 q b S p m s i I y w x E b m n k q m B G f 5 y k g v 0 j g U J B z p C M b g Q A M m K d F 8 a 6 E H D I I m 3 E I L X C w c B M / w C m / W x u U v X a h d 1 5 y 7 e q Z h y N s o w g m c Q c V K 1 x q g j 1 S c x j Q d 2 x p p x J 6 h t R e O t C x u b W 9 k 5 x 7 + S 3 f 3 B 4 V D 4 m O 5 G T h A Y C j y S L M D H G = S p 1 q s 9 D f 0 J m i m I R c t o J 7 x d z v / N E l W a 3 v B V 7 p a D e C l 4 8 n i B t o V 1 K N s 2 m Q o X N S w M x E J 2 t 7 h V + V T 1 6 C R a 2 U 1 z 3 o K 0 P B h 7 v T W A z L 0 w 4 0 8 Z 3 M l v 2 S b J C W f o j H / h Q 8 e r / 8 e a s o s / x Y M a r / 4 Z b 4 / b l 9 K D g p N s h t p o J l a I 0 J D t s U h N 2 2 3 d 3 Z K Q M l U c S z u 7 e / s H 5 O W j B x U l k l C X 3 h 5 W L 2 9 2 / J g z p 3 b 7 2 y q s r W 9 s j s D U b + J o U 7 j 1 f 2 H 6 i 0 H + I v 0 A s " E s < l a t e x i t h = a 1 _ b a s e 6 4 l I 4 c L 8 Q v 1 K O X j Q z E L 6 Q 1 R P F G F H W 3 M = " > A A A B 8 i F c b V B N T 8 J A R y 5 g i s a Y j P G Q d 0 o V O K T K S 7 P Q l u 3 4 V G j C s V J d x 6 V 1 l 2 K p G e F 0 M R K 7 M g u 5 7 M 3 F / u p o o O G l z I R J m v m C l A E K I m m e 0 h G T f 2 + k O F R q E p S h + G b u t y d U K a v J e z 2 N q R f i + t W m H R 8 r y p m g r a T a 0 0 4 s K Q 5 9 o A I F a 3 Y G t E q c n Q y g R 6 t f / u o N x 1 B I I 1 g b y e u F W 9 v U g B 6 r l / Z 5 a x u u S u 2 q V 7 u s e H f 9 c s W t u 3 O G M V D v / 1 E l W a J e A G T S Y M Y D w W L g e c h 6 r j R B h p V h N e N p q Z d q K j E 5 t L V l p A I 5 W v 3 y 2 O + Q k D S m w h C z 7 4 5 X i g 5 0 8 Z 1 v 3 F C 2 v r G 5 l Z x m 2 7 9 4 t X / 4 8 Z / Y r m u Q V s f D D z e u S x O p p w J 6 h t m O 1 m I R X E c c t o O x Q z t u 7 d / U D 4 8 e R 1 J q g j 1 S c I T Z y G K e / U W l 2 c j T M + I J n E I V P L i r b J b Z Y Y W J s Q m V A U j e 8 s u r x G / C t x 8 t O P n M M f y B / W k D X 3 S N x A = g s x e B C 3 w g M I Z n I z U 3 R z o v z r v E W H s C o L n 6 e y L D d F a T O L S d M T Y v K v z t W i p w T H W Q c U + d o j O r D F C j e Z p c m Q T N f k c D i K g x f G I J J o r U F z 6 N x P + 8 b m q i y Y B j Q q a G C r J g 1 P X R M J q m h k i w R H S l H J k b z 3 9 W 0 Q s u p C G 2 n w G a V 1 7 2 5 + J / X S 0 G K Q 1 K 3 + v I 6 8 R v m G 7 p 3 e 6 y 0 G n 4 l U 2 o M n 1 q C i W L V I k T G W G F i b E u D a H 8 Y k 1 R Q a Q j W V v c 8 N z F B h p 3 y h g e D c s W t u w u d P e L l p A I 5 2 o V h L U 6 d o Q u r D F E K L 1 v S o I X 6 e y c Q N y N Z q Z 9 q m m A w W S P a s 1 R i Q X k 3 R V A A A B 7 X i c b B " N S w M x E J 2 t > = 7 2 6 D v o / v T K k W o w x V L 7 x G N t X V Y 7 Q r u U b J p t Q P D Z k G S F s v R H a O + i V T 1 6 C R a h Q m r 7 R f y 4 F b x 4 G a j 8 D m J M 6 L 8 + 5 L r F s O T j 5 z C n / M g Q i D c 6 i C B 1 f g A j t o g w 8 E J v / L f _ e x i t s P a 1 b a a s e 6 4 = " 4 / t h l = 4 A < Q C N x Q = Y < / l a t e x i t > Affine semigroups (a geometric viewpoint) Fix an affine semigroup S = � n 1 , . . . , n k � ⊂ Z d ≥ 0 . � � a ∈ Z k Z( n ) = ≥ 0 : n = a 1 n 1 + · · · + a k n k A geometric viewpoint: non-negative integer solutions to linear equations. Example: S = � 6 , 9 , 20 � , n = 60. � � a ∈ Z k Z(60) = ≥ 0 : 60 = 6 a 1 + 9 a 2 + 20 a 3 Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 5 / 27

Delta set Fix an affine semigroup S = � n 1 , . . . , n k � ⊂ Z d ≥ 0 . � � a ∈ Z k Z( n ) = ≥ 0 : n = a 1 n 1 + · · · + a k n k Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 6 / 27

Delta set Fix an affine semigroup S = � n 1 , . . . , n k � ⊂ Z d ≥ 0 . � � a ∈ Z k Z( n ) = ≥ 0 : n = a 1 n 1 + · · · + a k n k a = ( a 1 , . . . , a k ) ∈ Z k | a | = a 1 + · · · + a k � ≥ 0 Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 6 / 27

Delta set Fix an affine semigroup S = � n 1 , . . . , n k � ⊂ Z d ≥ 0 . � � a ∈ Z k Z( n ) = ≥ 0 : n = a 1 n 1 + · · · + a k n k a = ( a 1 , . . . , a k ) ∈ Z k | a | = a 1 + · · · + a k � ≥ 0 L( n ) = {| a | : a ∈ Z( n ) } = { ℓ 1 < · · · < ℓ r } Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 6 / 27

Delta set Fix an affine semigroup S = � n 1 , . . . , n k � ⊂ Z d ≥ 0 . � � a ∈ Z k Z( n ) = ≥ 0 : n = a 1 n 1 + · · · + a k n k a = ( a 1 , . . . , a k ) ∈ Z k | a | = a 1 + · · · + a k � ≥ 0 L( n ) = {| a | : a ∈ Z( n ) } = { ℓ 1 < · · · < ℓ r } Definition The delta set of n ∈ S is ∆( n ) = { ℓ 2 − ℓ 1 , . . . , ℓ r − ℓ r − 1 } Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 6 / 27

Delta set Fix an affine semigroup S = � n 1 , . . . , n k � ⊂ Z d ≥ 0 . � � a ∈ Z k Z( n ) = ≥ 0 : n = a 1 n 1 + · · · + a k n k a = ( a 1 , . . . , a k ) ∈ Z k | a | = a 1 + · · · + a k � ≥ 0 L( n ) = {| a | : a ∈ Z( n ) } = { ℓ 1 < · · · < ℓ r } Definition The delta set of n ∈ S is ∆( n ) = { ℓ 2 − ℓ 1 , . . . , ℓ r − ℓ r − 1 } Example S = � 6 , 9 , 20 � : Z(60) = { (10 , 0 , 0) , (7 , 2 , 0) , (4 , 4 , 0) , (1 , 6 , 0) , (0 , 0 , 3) } Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 6 / 27

Delta set Fix an affine semigroup S = � n 1 , . . . , n k � ⊂ Z d ≥ 0 . � � a ∈ Z k Z( n ) = ≥ 0 : n = a 1 n 1 + · · · + a k n k a = ( a 1 , . . . , a k ) ∈ Z k | a | = a 1 + · · · + a k � ≥ 0 L( n ) = {| a | : a ∈ Z( n ) } = { ℓ 1 < · · · < ℓ r } Definition The delta set of n ∈ S is ∆( n ) = { ℓ 2 − ℓ 1 , . . . , ℓ r − ℓ r − 1 } Example S = � 6 , 9 , 20 � : Z(60) = { (10 , 0 , 0) , (7 , 2 , 0) , (4 , 4 , 0) , (1 , 6 , 0) , (0 , 0 , 3) } L(60) = { 3 , 7 , 8 , 9 , 10 } ∆(60) = { 1 , 4 } Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 6 / 27

Delta set Fix an affine semigroup S = � n 1 , . . . , n k � ⊂ Z d ≥ 0 . � � a ∈ Z k Z( n ) = ≥ 0 : n = a 1 n 1 + · · · + a k n k a = ( a 1 , . . . , a k ) ∈ Z k | a | = a 1 + · · · + a k � ≥ 0 L( n ) = {| a | : a ∈ Z( n ) } = { ℓ 1 < · · · < ℓ r } Definition The delta set of n ∈ S is ∆( n ) = { ℓ 2 − ℓ 1 , . . . , ℓ r − ℓ r − 1 } Example S = � 6 , 9 , 20 � : Z(60) = { (10 , 0 , 0) , (7 , 2 , 0) , (4 , 4 , 0) , (1 , 6 , 0) , (0 , 0 , 3) } L(60) = { 3 , 7 , 8 , 9 , 10 } ∆(60) = { 1 , 4 } L(142) = { 10 , 11 , 12 , 14 , 15 , 16 , 17 , 18 , 19 } ∆(142) = { 1 , 2 } Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 6 / 27

Delta set Fix an affine semigroup S = � n 1 , . . . , n k � ⊂ Z d ≥ 0 . � � a ∈ Z k Z( n ) = ≥ 0 : n = a 1 n 1 + · · · + a k n k a = ( a 1 , . . . , a k ) ∈ Z k | a | = a 1 + · · · + a k � ≥ 0 L( n ) = { ℓ 1 < · · · < ℓ r } ∆( n ) = { ℓ i − ℓ i − 1 } Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 7 / 27

Delta set Fix an affine semigroup S = � n 1 , . . . , n k � ⊂ Z d ≥ 0 . � � a ∈ Z k Z( n ) = ≥ 0 : n = a 1 n 1 + · · · + a k n k a = ( a 1 , . . . , a k ) ∈ Z k | a | = a 1 + · · · + a k � ≥ 0 L( n ) = { ℓ 1 < · · · < ℓ r } ∆( n ) = { ℓ i − ℓ i − 1 } Facts for large n ∈ S min ∆( n ) is as small as possible for S Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 7 / 27

Delta set Fix an affine semigroup S = � n 1 , . . . , n k � ⊂ Z d ≥ 0 . � � a ∈ Z k Z( n ) = ≥ 0 : n = a 1 n 1 + · · · + a k n k a = ( a 1 , . . . , a k ) ∈ Z k | a | = a 1 + · · · + a k � ≥ 0 L( n ) = { ℓ 1 < · · · < ℓ r } ∆( n ) = { ℓ i − ℓ i − 1 } Facts for large n ∈ S min ∆( n ) is as small as possible for S S = � 3 , 5 , 7 � : L(110) = { 16 , 18 , . . . , 34 , 36 } , ∆(110) = { 2 } Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 7 / 27

Delta set Fix an affine semigroup S = � n 1 , . . . , n k � ⊂ Z d ≥ 0 . � � a ∈ Z k Z( n ) = ≥ 0 : n = a 1 n 1 + · · · + a k n k a = ( a 1 , . . . , a k ) ∈ Z k | a | = a 1 + · · · + a k � ≥ 0 L( n ) = { ℓ 1 < · · · < ℓ r } ∆( n ) = { ℓ i − ℓ i − 1 } Facts for large n ∈ S min ∆( n ) is as small as possible for S S = � 3 , 5 , 7 � : L(110) = { 16 , 18 , . . . , 34 , 36 } , ∆(110) = { 2 } L( n ) is an arithmetic sequence with a few values removed near the ends Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 7 / 27

Delta set Fix an affine semigroup S = � n 1 , . . . , n k � ⊂ Z d ≥ 0 . � � a ∈ Z k Z( n ) = ≥ 0 : n = a 1 n 1 + · · · + a k n k a = ( a 1 , . . . , a k ) ∈ Z k | a | = a 1 + · · · + a k � ≥ 0 L( n ) = { ℓ 1 < · · · < ℓ r } ∆( n ) = { ℓ i − ℓ i − 1 } Facts for large n ∈ S min ∆( n ) is as small as possible for S S = � 3 , 5 , 7 � : L(110) = { 16 , 18 , . . . , 34 , 36 } , ∆(110) = { 2 } L( n ) is an arithmetic sequence with a few values removed near the ends S = � 42 , 86 , 245 , 285 , 365 , 463 � : L(3023) = { 7 , 9 , 11 , 12 , . . . , 46 , 47 , 58 , 62 , 64 } , ∆(3023) = { 1 , 2 , 4 , 9 } Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 7 / 27

Delta set (a geometric viewpoint) Fix an affine semigroup S = � n 1 , . . . , n k � ⊂ Z d ≥ 0 . � � a ∈ Z k Z( n ) = ≥ 0 : n = a 1 n 1 + · · · + a k n k a = ( a 1 , . . . , a k ) ∈ Z k | a | = a 1 + · · · + a k � ≥ 0 L( n ) = { ℓ 1 < · · · < ℓ r } ∆( n ) = { ℓ i − ℓ i − 1 } Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 8 / 27

Delta set (a geometric viewpoint) Fix an affine semigroup S = � n 1 , . . . , n k � ⊂ Z d ≥ 0 . � � a ∈ Z k Z( n ) = ≥ 0 : n = a 1 n 1 + · · · + a k n k a = ( a 1 , . . . , a k ) ∈ Z k | a | = a 1 + · · · + a k � ≥ 0 L( n ) = { ℓ 1 < · · · < ℓ r } ∆( n ) = { ℓ i − ℓ i − 1 } A geometric viewpoint: lattice width Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 8 / 27

Delta set (a geometric viewpoint) Fix an affine semigroup S = � n 1 , . . . , n k � ⊂ Z d ≥ 0 . � � a ∈ Z k Z( n ) = ≥ 0 : n = a 1 n 1 + · · · + a k n k a = ( a 1 , . . . , a k ) ∈ Z k | a | = a 1 + · · · + a k � ≥ 0 L( n ) = { ℓ 1 < · · · < ℓ r } ∆( n ) = { ℓ i − ℓ i − 1 } A geometric viewpoint: lattice width Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 8 / 27

Delta set (a geometric viewpoint) Fix an affine semigroup S = � n 1 , . . . , n k � ⊂ Z d ≥ 0 . � � a ∈ Z k Z( n ) = ≥ 0 : n = a 1 n 1 + · · · + a k n k a = ( a 1 , . . . , a k ) ∈ Z k | a | = a 1 + · · · + a k � ≥ 0 L( n ) = { ℓ 1 < · · · < ℓ r } ∆( n ) = { ℓ i − ℓ i − 1 } A geometric viewpoint: lattice width | a | = � a � 1 (the ℓ 1 -norm) Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 8 / 27

Delta set (a geometric viewpoint) Fix an affine semigroup S = � n 1 , . . . , n k � ⊂ Z d ≥ 0 . � � a ∈ Z k Z( n ) = ≥ 0 : n = a 1 n 1 + · · · + a k n k a = ( a 1 , . . . , a k ) ∈ Z k | a | = a 1 + · · · + a k � ≥ 0 L( n ) = { ℓ 1 < · · · < ℓ r } ∆( n ) = { ℓ i − ℓ i − 1 } A geometric viewpoint: lattice width | a | = � a � 1 (the ℓ 1 -norm) min ∆( n ): min ℓ 1 width Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 8 / 27

Delta set (a geometric viewpoint) Fix an affine semigroup S = � n 1 , . . . , n k � ⊂ Z d ≥ 0 . � � a ∈ Z k Z( n ) = ≥ 0 : n = a 1 n 1 + · · · + a k n k a = ( a 1 , . . . , a k ) ∈ Z k | a | = a 1 + · · · + a k � ≥ 0 L( n ) = { ℓ 1 < · · · < ℓ r } ∆( n ) = { ℓ i − ℓ i − 1 } A geometric viewpoint: lattice width | a | = � a � 1 (the ℓ 1 -norm) min ∆( n ): min ℓ 1 width extremal lengths near vertices Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 8 / 27

Delta set of a semigroup Fix an affine semigroup S = � n 1 , . . . , n k � ⊂ Z d ≥ 0 . � � a ∈ Z k Z( n ) = ≥ 0 : n = a 1 n 1 + · · · + a k n k a = ( a 1 , . . . , a k ) ∈ Z k | a | = a 1 + · · · + a k � ≥ 0 L( n ) = { ℓ 1 < · · · < ℓ r } ∆( n ) = { ℓ i − ℓ i − 1 } Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 9 / 27

Delta set of a semigroup Fix an affine semigroup S = � n 1 , . . . , n k � ⊂ Z d ≥ 0 . � � a ∈ Z k Z( n ) = ≥ 0 : n = a 1 n 1 + · · · + a k n k a = ( a 1 , . . . , a k ) ∈ Z k | a | = a 1 + · · · + a k � ≥ 0 L( n ) = { ℓ 1 < · · · < ℓ r } ∆( n ) = { ℓ i − ℓ i − 1 } Definition The delta set of S is the union ∆( S ) = � n ∈ S ∆( n ) Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 9 / 27

Delta set of a semigroup Fix an affine semigroup S = � n 1 , . . . , n k � ⊂ Z d ≥ 0 . � � a ∈ Z k Z( n ) = ≥ 0 : n = a 1 n 1 + · · · + a k n k a = ( a 1 , . . . , a k ) ∈ Z k | a | = a 1 + · · · + a k � ≥ 0 L( n ) = { ℓ 1 < · · · < ℓ r } ∆( n ) = { ℓ i − ℓ i − 1 } Definition The delta set of S is the union ∆( S ) = � n ∈ S ∆( n ) Goal Given n 1 , . . . , n k as input, compute ∆( S ). Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 9 / 27

Delta set of a semigroup Fix an affine semigroup S = � n 1 , . . . , n k � ⊂ Z d ≥ 0 . � � a ∈ Z k Z( n ) = ≥ 0 : n = a 1 n 1 + · · · + a k n k a = ( a 1 , . . . , a k ) ∈ Z k | a | = a 1 + · · · + a k � ≥ 0 L( n ) = { ℓ 1 < · · · < ℓ r } ∆( n ) = { ℓ i − ℓ i − 1 } Definition The delta set of S is the union ∆( S ) = � n ∈ S ∆( n ) Goal Given n 1 , . . . , n k as input, compute ∆( S ). The primary difficulty: this is an infinite union! Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 9 / 27

Delta set of a semigroup Fix an affine semigroup S = � n 1 , . . . , n k � ⊂ Z d ≥ 0 . � � a ∈ Z k Z( n ) = ≥ 0 : n = a 1 n 1 + · · · + a k n k a = ( a 1 , . . . , a k ) ∈ Z k | a | = a 1 + · · · + a k � ≥ 0 L( n ) = { ℓ 1 < · · · < ℓ r } ∆( n ) = { ℓ i − ℓ i − 1 } Definition The delta set of S is the union ∆( S ) = � n ∈ S ∆( n ) Goal Given n 1 , . . . , n k as input, compute ∆( S ). The primary difficulty: this is an infinite union! Example S = � 17 , 33 , 53 , 71 � , ∆( S ) = { 2 , 4 , 6 } . Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 9 / 27

Delta set of a semigroup Fix an affine semigroup S = � n 1 , . . . , n k � ⊂ Z d ≥ 0 . � � a ∈ Z k Z( n ) = ≥ 0 : n = a 1 n 1 + · · · + a k n k a = ( a 1 , . . . , a k ) ∈ Z k | a | = a 1 + · · · + a k � ≥ 0 L( n ) = { ℓ 1 < · · · < ℓ r } ∆( n ) = { ℓ i − ℓ i − 1 } Definition The delta set of S is the union ∆( S ) = � n ∈ S ∆( n ) Goal Given n 1 , . . . , n k as input, compute ∆( S ). The primary difficulty: this is an infinite union! Example S = � 17 , 33 , 53 , 71 � , ∆( S ) = { 2 , 4 , 6 } . 6 ∈ ∆(266) , ∆(283) , ∆(300). Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 9 / 27

4 3 2 1 20 40 60 80 100 120 140 The first breakthrough: numerical semigroups Fix a numerical semigroup S = � n 1 , . . . , n k � ⊂ Z ≥ 0 . Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 10 / 27

4 3 2 1 20 40 60 80 100 120 140 The first breakthrough: numerical semigroups Fix a numerical semigroup S = � n 1 , . . . , n k � ⊂ Z ≥ 0 . Theorem (Chapman–Hoyer–Kaplan, 2009) For n ≥ 2 kn 2 n 2 k , ∆( n ) = ∆( n + n 1 n k ) . Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 10 / 27

The first breakthrough: numerical semigroups Fix a numerical semigroup S = � n 1 , . . . , n k � ⊂ Z ≥ 0 . Theorem (Chapman–Hoyer–Kaplan, 2009) For n ≥ 2 kn 2 n 2 k , ∆( n ) = ∆( n + n 1 n k ) . Example: S = � 6 , 9 , 20 � : 2 kn 2 n 2 k = 21600 4 3 2 1 20 40 60 80 100 120 140 ( n , d ) d ∈ ∆( n ) � Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 10 / 27

The first breakthrough: numerical semigroups Fix a numerical semigroup S = � n 1 , . . . , n k � ⊂ Z ≥ 0 . Theorem (Chapman–Hoyer–Kaplan, 2009) For n ≥ 2 kn 2 n 2 k , ∆( n ) = ∆( n + n 1 n k ) . Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 11 / 27

The first breakthrough: numerical semigroups Fix a numerical semigroup S = � n 1 , . . . , n k � ⊂ Z ≥ 0 . Theorem (Chapman–Hoyer–Kaplan, 2009) For n ≥ 2 kn 2 n 2 k , ∆( n ) = ∆( n + n 1 n k ) . For n ∈ S with 0 ≤ n ≤ 2 kn 2 n 2 k + n 1 n k , Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 11 / 27

The first breakthrough: numerical semigroups Fix a numerical semigroup S = � n 1 , . . . , n k � ⊂ Z ≥ 0 . Theorem (Chapman–Hoyer–Kaplan, 2009) For n ≥ 2 kn 2 n 2 k , ∆( n ) = ∆( n + n 1 n k ) . For n ∈ S with 0 ≤ n ≤ 2 kn 2 n 2 k + n 1 n k , compute: Z( n ) = { a ∈ Z k ≥ 0 : n = a 1 n 1 + · · · + a k n k } Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 11 / 27

The first breakthrough: numerical semigroups Fix a numerical semigroup S = � n 1 , . . . , n k � ⊂ Z ≥ 0 . Theorem (Chapman–Hoyer–Kaplan, 2009) For n ≥ 2 kn 2 n 2 k , ∆( n ) = ∆( n + n 1 n k ) . For n ∈ S with 0 ≤ n ≤ 2 kn 2 n 2 k + n 1 n k , compute: Z( n ) = { a ∈ Z k ≥ 0 : n = a 1 n 1 + · · · + a k n k } Z( n ) � L( n ) = { a 1 + · · · + a k : a ∈ Z( n ) } Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 11 / 27

The first breakthrough: numerical semigroups Fix a numerical semigroup S = � n 1 , . . . , n k � ⊂ Z ≥ 0 . Theorem (Chapman–Hoyer–Kaplan, 2009) For n ≥ 2 kn 2 n 2 k , ∆( n ) = ∆( n + n 1 n k ) . For n ∈ S with 0 ≤ n ≤ 2 kn 2 n 2 k + n 1 n k , compute: Z( n ) = { a ∈ Z k ≥ 0 : n = a 1 n 1 + · · · + a k n k } Z( n ) � L( n ) = { a 1 + · · · + a k : a ∈ Z( n ) } L( n ) = { ℓ 1 < . . . < ℓ r } � ∆( n ) = { ℓ i − ℓ i − 1 } Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 11 / 27

The first breakthrough: numerical semigroups Fix a numerical semigroup S = � n 1 , . . . , n k � ⊂ Z ≥ 0 . Theorem (Chapman–Hoyer–Kaplan, 2009) For n ≥ 2 kn 2 n 2 k , ∆( n ) = ∆( n + n 1 n k ) . For n ∈ S with 0 ≤ n ≤ 2 kn 2 n 2 k + n 1 n k , compute: Z( n ) = { a ∈ Z k ≥ 0 : n = a 1 n 1 + · · · + a k n k } Z( n ) � L( n ) = { a 1 + · · · + a k : a ∈ Z( n ) } L( n ) = { ℓ 1 < . . . < ℓ r } � ∆( n ) = { ℓ i − ℓ i − 1 } Compute ∆( S ) = � n ∆( n ). Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 11 / 27

The first breakthrough: numerical semigroups Fix a numerical semigroup S = � n 1 , . . . , n k � ⊂ Z ≥ 0 . Theorem (Chapman–Hoyer–Kaplan, 2009) For n ≥ 2 kn 2 n 2 k , ∆( n ) = ∆( n + n 1 n k ) . For n ∈ S with 0 ≤ n ≤ 2 kn 2 n 2 k + n 1 n k , compute: → Z( n ) = { a ∈ Z k ≥ 0 : n = a 1 n 1 + · · · + a k n k } ← Z( n ) � L( n ) = { a 1 + · · · + a k : a ∈ Z( n ) } L( n ) = { ℓ 1 < . . . < ℓ r } � ∆( n ) = { ℓ i − ℓ i − 1 } Compute ∆( S ) = � n ∆( n ). Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 11 / 27

The first breakthrough: numerical semigroups Fix a numerical semigroup S = � n 1 , . . . , n k � ⊂ Z ≥ 0 . Theorem (Chapman–Hoyer–Kaplan, 2009) For n ≥ 2 kn 2 n 2 k , ∆( n ) = ∆( n + n 1 n k ) . For n ∈ S with 0 ≤ n ≤ 2 kn 2 n 2 k + n 1 n k , compute: → Z( n ) = { a ∈ Z k ≥ 0 : n = a 1 n 1 + · · · + a k n k } ← Z( n ) � L( n ) = { a 1 + · · · + a k : a ∈ Z( n ) } L( n ) = { ℓ 1 < . . . < ℓ r } � ∆( n ) = { ℓ i − ℓ i − 1 } Compute ∆( S ) = � n ∆( n ). | Z( n ) | ≈ n k − 1 Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 11 / 27

An improvement: the first usable algorithm Fix a numerical semigroup S = � n 1 , . . . , n k � ⊂ Z ≥ 0 . Theorem (Chapman–Hoyer–Kaplan, 2009) For n ≥ 2 kn 2 n 2 k , ∆( n ) = ∆( n + n 1 n k ) . Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 12 / 27

An improvement: the first usable algorithm Fix a numerical semigroup S = � n 1 , . . . , n k � ⊂ Z ≥ 0 . Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014) For n ≥ N S , ∆( n ) = ∆( n + lcm( n 1 , n k )) . Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 12 / 27

An improvement: the first usable algorithm Fix a numerical semigroup S = � n 1 , . . . , n k � ⊂ Z ≥ 0 . Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014) For n ≥ N S , ∆( n ) = ∆( n + lcm( n 1 , n k )) . Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 12 / 27

An improvement: the first usable algorithm Fix a numerical semigroup S = � n 1 , . . . , n k � ⊂ Z ≥ 0 . Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014) For n ≥ N S , ∆( n ) = ∆( n + lcm( n 1 , n k )) . S = � 6 , 9 , 20 � : 2 kn 2 n 2 k = 21600, N S = 144, actual = 91 Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 12 / 27

An improvement: the first usable algorithm Fix a numerical semigroup S = � n 1 , . . . , n k � ⊂ Z ≥ 0 . Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014) For n ≥ N S , ∆( n ) = ∆( n + lcm( n 1 , n k )) . S = � 6 , 9 , 20 � : 2 kn 2 n 2 k = 21600, N S = 144, actual = 91 2 kn 2 n 2 S N S ∆( S ) Runtime k � 7 , 15 , 17 , 18 , 20 � 60000 1935 { 1 , 2 , 3 } 1m 28s � 11 , 53 , 73 , 87 � 3209256 14381 { 2 , 4 , 6 , 8 , 10 , 22 } 0m 49s � 31 , 73 , 77 , 87 , 91 � 6045130 31364 { 2 , 4 , 6 } 400m 12s 9 · 10 7 � 100 , 121 , 142 , 163 , 284 � 24850 ——— ——— � 1001 , 1211 , 1421 , 1631 , 2841 � 6 · 10 15 2063141 ——— ——— Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 12 / 27

An improvement: the first usable algorithm Fix a numerical semigroup S = � n 1 , . . . , n k � ⊂ Z ≥ 0 . Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014) For n ≥ N S , ∆( n ) = ∆( n + lcm( n 1 , n k )) . S = � 6 , 9 , 20 � : 2 kn 2 n 2 k = 21600, N S = 144, actual = 91 2 kn 2 n 2 S N S ∆( S ) Runtime k � 7 , 15 , 17 , 18 , 20 � 60000 1935 { 1 , 2 , 3 } 1m 28s � 11 , 53 , 73 , 87 � 3209256 14381 { 2 , 4 , 6 , 8 , 10 , 22 } 0m 49s � 31 , 73 , 77 , 87 , 91 � 6045130 31364 { 2 , 4 , 6 } 400m 12s 9 · 10 7 � 100 , 121 , 142 , 163 , 284 � 24850 ——— ——— � 1001 , 1211 , 1421 , 1631 , 2841 � 6 · 10 15 2063141 ——— ——— GAP Numerical Semigroups Package, available at http://www.gap-system.org/Packages/numericalsgps.html . Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 12 / 27

An improvement: the first usable algorithm Fix a numerical semigroup S = � n 1 , . . . , n k � ⊂ Z ≥ 0 . Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014) For n ≥ N S , ∆( n ) = ∆( n + lcm( n 1 , n k )) . Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 13 / 27

An improvement: the first usable algorithm Fix a numerical semigroup S = � n 1 , . . . , n k � ⊂ Z ≥ 0 . Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014) For n ≥ N S , ∆( n ) = ∆( n + lcm( n 1 , n k )) . For n ∈ S with 0 ≤ n ≤ 2 kn 2 n 2 k + n 1 n k , compute: Z( n ) = { a ∈ Z k ≥ 0 : n = a 1 n 1 + · · · + a k n k } Z( n ) � L( n ) = { a 1 + · · · + a k : a ∈ Z( n ) } L( n ) = { ℓ 1 < . . . < ℓ r } � ∆( n ) = { ℓ i − ℓ i − 1 } Compute ∆( S ) = � n ∆( n ). Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 13 / 27

An improvement: the first usable algorithm Fix a numerical semigroup S = � n 1 , . . . , n k � ⊂ Z ≥ 0 . Theorem (Garc´ ıa-Garc´ ıa–Moreno-Fr´ ıas–Vigneron-Tenorio, 2014) For n ≥ N S , ∆( n ) = ∆( n + lcm( n 1 , n k )) . For n ∈ S with 0 ≤ n ≤ 2 kn 2 n 2 k + n 1 n k , compute: → Z( n ) = { a ∈ Z k ≥ 0 : n = a 1 n 1 + · · · + a k n k } ← Z( n ) � L( n ) = { a 1 + · · · + a k : a ∈ Z( n ) } L( n ) = { ℓ 1 < . . . < ℓ r } � ∆( n ) = { ℓ i − ℓ i − 1 } Compute ∆( S ) = � n ∆( n ). Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 13 / 27

A faster solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

A faster solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

A faster solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) a �− → a + e i Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

A faster solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) a �− → a + e i S = � 6 , 9 , 20 � : Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

7 U h m P A K H j O p o e a m O E U c o g h 7 i E e G z 0 I N 9 J O W f j X / J u Z 0 z l p Z D o P J U D K c I x N / q K l F G o N C p S G 5 k J a V w O V k L 0 f R r 8 s X v c < P a 5 j h S 9 5 t o D c M S N s N p 9 R j a 9 q Y F O N q M Q i h Y Q 9 + / W d D e I h w E A h 2 G H C 1 5 M Z I K j J I O 5 1 R W X p e I X t b Z C y G A / 6 D s / Q c O s e a + j Y q f 3 r f B r e O L l m b k i x X 0 p Y y A A p S z Q O C H H T + P y o g m F H B M C a b E x 2 h T r V R 6 y L H / Z Y S q C i u J R E D L j I y a m Y A Q 2 B H z I N J t 6 X Q 8 L D L Z F x I g T L D V b c i X 9 D k A n 8 Z 2 Q H 0 m 5 0 h C K d m h E G E l 3 E S M O d H V H G B A 9 x 6 e s a b _ 1 a h s t i e = t a l < / l a t e x i t > 4 " A J > " = 8 N 0 I X C T R l Q l y h 3 R O 8 j C H O 6 e H o A B u 6 N d c g H X S i Y H n T X j C 2 e G K s y j d Q J v E N F r x J L 5 R k 1 K 4 M m r g H B o i t T O N u h K W p I 1 / v z c a M X h 8 p 7 + b z x J n a m E 4 a L W G j 9 X n F / / W M i G S O q L i W P c f F u z + 7 t x G f l H p v 2 q r s 3 u y b t b L F A faster solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) a �− → a + e i S = � 6 , 9 , 20 � : Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

f t t a l < > t i x e a x l / < = o U k r j e i P 6 1 v n Q m s " = 4 e t s a b _ 1 a h s 6 f S w z J E g A o 1 w A p A u A g D X Z M A 3 / K i 0 1 H C z R 7 V y Z z b / x W 9 K W 9 T a 9 s I P Y 7 A 2 v g x K O 8 Z K P C s S Z s Q t I V y 9 S P e K + w F G + 3 o a Y F Z W 6 W z N Y b B 6 v T > " = U v Y X 8 A e A P Z c y U K a B M A A h x q / q 1 r W 3 M E M B w S N B V b c i n 8 R M X w M q r R 7 h M S m 0 B Y 1 L C N 9 p I 0 A V S B p f q G R 4 8 K 9 k W M M 6 E i 6 p p S 6 8 E I x B k S m 4 l B 6 U 2 k l k F v v T X 7 p B a 0 E N p 2 p j 6 V 6 4 C / B K O S C x b X J e 7 8 y r q m C t M t E C I Y E y Z K C Y G e S z Y p I X q q 0 v U z u N n N w m i V r F T 8 + r F S J G k P W d n 7 v J v V y X p N 5 M A P p U B t 7 Y 7 0 I A r R 7 Q k M 3 o Q z D S q s G 1 E J X e R S x / / 8 C P o E m B G X 2 O G w k y J R U j P t S t 5 A p o 6 Y m y O e L w U o t 4 E Q P c Y 6 + u 6 X A Y x t V Q j I W G W f x g 8 x J E Y b K 4 V e / v 4 m V m b x U Y k d A P K T G y b G + R 5 8 5 J F 0 q l c / X e F f e A N j + X g G < > t i x e t a l / = f = w W P C 3 c A 4 P r p g f R O r S b e Y 3 U Z 1 d X r j T 8 6 r r x F H T F G O i a W M a 3 E E l b L G Z V c g + Z x / P S 5 h Y p D D 1 F 6 F w l N z r Z p G l j G R O Y R X P h 7 9 3 M 3 e I T e g E 1 p 0 s k 3 T e j 4 R y F Z i p Z 3 / A G H Y N 0 7 Z L m f S C 5 R H d 4 B 7 v 3 S 3 x D 7 t 5 G b s l y 2 k q 4 u J D 8 S s k n x k q G q w j R W q y H V z O Q o Y N C T p d G u O S 2 T C h n a + V 6 Z J D P W E o C s n d e r U A G U f k i 6 M y 9 J Y a l d 7 A I U U x J 3 O x b k G p u Q 5 G 0 y R m q r P H 5 r 8 s X v c 7 P a j f Y d A 6 5 R k 1 K R L M S Q 9 + 9 S 0 h C p G N 5 k J a V w O V k 4 m h n d c g H X S i Y H T 2 X j N C e G K s y r x r u g H B o L i T O N h t K W p I 1 / v z c Y i d K U D K c I x N / q l P o 9 E N h m P A K J D j j H B M f W o / W f X O / J u Z 0 z l p Z H O Q R D c M S N s N p 9 j t 5 a q Y F O N q M o 9 p g o e a m U O U c o h G 7 i E e G z 0 I F J j Q m c F x I g T L D V b i 2 X 9 B A A A > " = L D N B d H V H G h k Z D H L z I N J t 6 X Q 8 8 0 M h 4 6 e s a b _ 1 a s " t i x e t a l < = y I 3 X C T R l Q J l h R A O 8 j C H O 6 e H o O S J b p v 2 q r s 3 u y t f b L l O L S 8 p 7 H L b W v E N F a J M q i P G c f F u z + 7 t x + z E h 8 Z 2 Q H 0 m 5 0 C 9 K d n m E G E l 3 B u x G J n a m E 4 h a W j X 9 X n F / / W M i G F b g 4 8 2 8 4 + 7 6 x 1 C w N O s / T E y o 2 s y p T b r 0 Z S s X 9 0 z w u P 3 k T m E P U O O b L z I N J t 6 X Q 8 D B 2 L F x o g T L D H D 0 o R j D U m + I I 8 I Z i x l b J 6 S j B k 9 e b r U w C V p + x t m H 3 + e v t T 8 4 Q K d o 4 A k q H w v M k 0 2 r S R 8 F 8 Q A f T U k 0 V a 5 v 4 9 2 d n t 3 h S b K / N W f t V W B 4 0 c a a m p Q z U 5 V o W j K 7 h Q J R L J x V I c e J I O 5 1 R W X p l K t m b x 2 h T r V j I 6 w O L e X i I e I h E M A h 2 G H C 1 5 D R y r S x X 0 p Y y A A p z C Q k O H H T + P y i C L J H / Z E Y q C i u R a E D L j I y a m S e Z B m M J a a D u C z w a F 8 " = 4 f e s a b U O 1 0 X 9 B A A A > " = U d s h j V i 0 w D I _ 6 a O G A / 6 h s / Q c b C s a + j Y q f 3 r y D Z < s t i F e t a l x > a t Q / l < t e x i A faster solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) a �− → a + e i S = � 6 , 9 , 20 � : Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

R n 4 q b p a 3 t d W 0 9 e / + g p N f J 0 Z 4 5 f s y Z o t 7 z b R V W V a E u 2 e W F I c + p 1 a / f D 3 z 2 4 S N E x t o k E Y 9 k 8 P e K c i Z o U z 5 P U L N y U p i i 6 b t l x W s A 9 s Q o 5 p S s N A F L 2 r B 1 p f U Z d u Y l 3 T n 9 x 5 9 8 4 b i P Q 1 g M D h 1 j M M J + 3 Q y S T T C J Q S + h 9 u X 9 K D 6 5 a V Z 6 i k M x c o A B Z E 0 S h 2 j O i 3 1 E k V T C M Z 4 S L u G T i x p n 9 e N 9 H B Z m c y E S e a C 4 R U q q b 8 3 U h w N P Q l 9 M 5 m l X w I h q X Y j 1 S A b v a 1 M 6 d 0 7 5 E 6 d R a J O z 2 J q f b i o W A B I 1 g d q 1 p Q i o 0 4 V i r F u v E 2 k u x J h p y O B r R M 3 J U B I U e j b 3 / 1 L V I 4 Y U x m b V m v d / e V 4 K Y c k A P u V t x Y A X 6 + P J j E b G T K P R Q w O Q E 4 t o U L G e y t m Y 6 o j U J B y k w G O 2 X S x m E o P C 8 / / 6 I A R P z V H y q W j W w G J q k x n o 7 s A 0 G 5 Q u p k b I x O 3 J x U U k S W G d p T C N u / x u x 8 g x f W G G O 8 Z D O q Q T D J 6 S V + a n h C 2 Y p 5 b h t e x i t s a l 1 _ b a s e 6 a < = < 3 C P W w = = / > l a t e x i t 4 " A > 5 H I K A = " A r A A B 9 X i c Y R H W B j d o I I v 9 z G J V e 5 s y v c 4 t E S r O R x 1 F Z e + g c V Z G L b Y l 6 S y Y b g r r 8 3 T j r X d Z f U b E P J X c + l q 0 F 5 F 8 5 R X / g f e f T G 3 a M W a i O F e H p G r + j N A j / m 6 O q Q T D J Z V 8 + a n h C 2 Y D S O j z V H y q W R w s G J q k x n k S u o + t x Y A X 6 u J 6 j E b G T K P V Q G G d p T C N u / x I x 8 g x f W G W P W d R M 3 9 7 h P X q Y j 4 E a Y J 9 3 e M 3 Y h 5 G w R I k T s 0 p 1 E g e y 6 7 b G 5 Q u p k A x y O 3 J x U U I 0 R i e k U o G A U r d 2 n s C l d J z a A k D F a M W a i O G H T p G r + j N A 3 E f Z r O R x 1 F E + l g c V Z G L b e F b / 6 P W w = = < l c a t e x i t > 2 A e J X c + l q 0 F 5 4 8 5 R X / g f P S e Y m w G O 2 X B S E y o P C 8 / / x k J G v U x m b V m 4 / R e V 4 K Y c P P U j Y T Y b g r r 6 8 j S r X d Z f U 3 y t O L Q E 4 t o U w e 5 y t m Y 6 o p A p D C X 5 z i H 8 g f 5 V A 6 R c k U c j L < 1 w D K / w Z j Z f L 9 a 7 9 T E = / J " b a s e 6 4 = t 1 2 v n x A 8 1 _ a l < a t e x i t > l h a t e x i t s D E s M K Z r I i M s R J E m 6 J K p g p I 3 k Q c 6 Q j N K A I D J i n R f G R 8 m c w B M d Q A R u o o A 4 3 0 I A p O c k v L p F W r u 7 y V v T 0 r 1 6 / 9 / 1 3 0 N Y H A 4 / Z Z p i Z F y R S 7 L C K S 3 + G F w + e m P X X e P P f G D Y Z e O T j l G p r H z N l F S 6 F 5 d k t 2 y l s b G 5 7 4 x R 3 S 3 v 7 B o F T B B 8 n i c b V N A S w M x E M 3 A A r N F J R r r V a D > Z P E + H 0 = " W 1 Z y g v 2 A 7 V K a K b Y N z W 6 W x O q t / q h 6 9 B I Q 8 s Z S s C P Z Y y F q L C q Y E Z / H y u 6 R V r T h 2 i J b S 0 X x i C C a m R a y I j L D E x m Y I G 2 h A E w h e C I Z X e L O e H 1 t 5 l e p X W R 1 O 5 I J j K I M D p y B q h N E 9 o l K / P N x I c K D U m A P U 7 h g o c U O m K a e o p O j H J D A H m g o y P + T H H O k Q z S p A F B O J Z p l z 0 Z u / M X j f W / o W b r f E Q x F L 2 D L 8 X g 6 t J N I z H I T D A U 0 = " > A A B L 9 X i c b V D B Z h T O p O 2 o y E / R s O N G 4 1 x 0 j k x B j S 6 J b l i D I o 8 I I + m U s j r a C Z F Q < / l t G e x i t > < l y A t s 3 f q Y j + a b / O c Q / s D 6 a e V U C u D a a J M F w O d I D w 0 i z m x _ i t s h a 1 b a a s e 6 4 = " 8 i e 7 0 E m n d K C h 5 E m 0 H Q 2 Z 8 G l 9 G z H B D Z k h H 3 V H d O M S E B u N 8 J x z b + 7 p S a L O l L b t b n m X n G i M W / / F X E 9 j G W a h 4 I J u 6 = " y A o H e O 6 H C j 8 O R 3 4 e l t < l a t e x i s s h a 1 _ b a h J t F V D L T g I x L c 2 D L 8 Q X 6 b i Q N l R T C X I 0 8 X = " > A A A B 9 y 3 G w R f L N k V O V 8 a J k 5 G S p r s h d K 1 k R 5 6 A Y X j 5 a P 7 c v C 0 M o N s N S M c D t 9 9 G J F I 0 z p R S Q 9 + 9 Q Y h i M j q N O F Y q a 5 4 m s E i q M J a F N v P J Q d j y s K W c e L r q 2 v p H f G f x t 7 + z u F G C r N 1 I p W K h u O v T i L o B H g / z N S j X T n H Y i X c H g c d r 2 x t 6 C + e a K U y c Z P T M A 8 X Y v U = B v > " b a s e 6 4 = s I m Q n v 1 s S " A 1 I q / q h 6 9 B t r Q s Z S s C P 1 W A V A B 8 n i c b B 3 N S w M x E M _ a Y a J z / A K b 9 T g 9 W K 9 W x / E A 0 D P R 3 4 M Z X g o A u p w A w 1 z Z h l t e x i t > < a l t e x i t s a / y i V 7 R z C H 1 f < P 6 j r k U o = Z 8 y R D p Y h 5 G w I 1 3 k s 0 p 1 E D F e G H 5 e O T j l p 6 Z r z N l F S g T 4 r i k U o G A U e M d n s C l f d 6 y e P Y 3 M 3 9 7 h X 9 R q j 4 E a Y J d B O w Y o S 3 + G F + Z K e P X X e P F W f V K x g v 2 A 7 K 6 y a b Y N z W P m 7 5 k 2 y l s b G t C 7 x R 3 S 3 v D G L 4 Z 7 0 N Y H A / S 3 Z p i Z F y R M A q Q 3 o S U T f A 8 U F 8 R r A 2 0 d w M + Q 4 8 T t v e H C r m t x + p V k v k B 6 B l 4 m S k B x k a q e J X b U 2 w r H q k f H R f p l 7 X T v v F k K 4 C 9 X s S Z 0 r b b 0 e V W V t f W 9 T / U 8 2 8 s w y w P z E m T k 3 P u N K 0 5 h 7 K j o c V U J z Q p m a a 0 Q R b 9 h a 3 t n d 2 4 L v 5 B S 4 W x J x / S A 7 M k Q 7 R r I Q 0 Y J I S z S 3 z G 7 7 n d W O t X o D J E 1 G s q S e Y J N 8 T F r V i m X r n N u z U v 0 + F C E K Z y Y r I C t w M t C m q Y E p v P v w 9 N C L 1 Y 0 m I S M V 7 R r q p 0 A 6 K B C E 4 6 V j 8 p 2 p N 0 E S M h B p p 7 I 8 V B U P q A M 5 N p y k f G G E W 6 R p p 6 i S 6 9 M 8 K 4 B k M A faster solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) a �− → a + e i S = � 6 , 9 , 20 � : Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

v J Q 6 V t x Y A X 6 u + j W E b G T K P A d k Y U I G m p a n h C 2 Y S O u G d T W C N u / G x x 8 g x f x b V o / / x U G j O Q E 4 t U C w L e y t m Y 6 o p A 8 P V P m 4 v / e V 4 K Y c P R o J y k w G O 2 X B S m E + 6 t M X R q j 4 E a Y J 9 y 6 h i k U o G A U r e d n P 7 C k D D p Y h 5 G w R I 3 s 9 0 p 1 E g e T e Y 3 M 3 s l Z x H y q W R j w G J q k n z k s S 8 D O q Q T D J V P d p J z a 2 R y 0 G 5 Q u k o A b x O 3 J x U U I 7 W 5 S F 9 1 r 6 0 = " > A A A B X m i c b V D L T g I x F G b 2 A e 6 4 = " i N d x g B 2 3 G B 6 W e P D 9 u c w L D a Q m n d K C h 0 5 m 0 H 2 G Z 8 B 9 u X G i M W / E E L D 8 Q X 6 t J N I z H B Z l k B E 3 V H d O M S E 3 s b y a g c V Z G L b l E T 3 M Z W a i O G F H p G r + + E N d Y b g r r 6 8 T j r X Z F f U 3 Y e b S r O R x 1 j A _ < = < / l a t e x i t > l w a t e x i t s h a 1 = W e 5 f F e X c + l q 0 F J 8 P 5 R X / g f P 4 A c 2 6 1 6 F K D J i n R f G I I J p Z k r I i M s M R E m 6 J A K p e e N 9 H B p Z c y E S a N C j I / F C Q c 6 Q j K g n m A R c u o A 4 3 0 I A E d J D w D K / w Z j 1 Z Q M R 7 3 8 c v L p F W r u k V B v T 0 r 1 6 / y O o p w 9 4 9 u F J Q i o 0 4 V i p r v B E 2 k u x 1 I x w O i h 1 1 B M 6 d 0 7 5 d d q p O r / R M 3 J y U I U e j b 3 3 E m w 0 T 2 i U q b 8 3 U h q Z N Q l 9 M 5 m l X P R E B k h V j T M Z 4 S L u G C x A S 5 a V Z 6 i k 6 M c o L a S F W 6 W Z F Y o S 3 + G w N + K e P X X e P P f m z Y Z Y B I t Q s Z S s C P Z 8 b O K x g v 2 A 7 V K y a L 7 6 e R 3 S 3 v 7 B 4 d H 5 O 7 T j l G p Z r z N l F x t 0 F N Y H A 4 / 3 Z p i Z y 5 R S G C D k 2 y l s b G 9 h 7 i e x i t > < l a t e x t a s h a 1 _ b a s e 6 t l = 8 9 T E f L V j 5 z i H g / f X 5 A 6 R c k U c = < 4 " q N A A A B 8 n i c b V B S " w M x E M 3 W r 1 q / > = t F 2 v n x A 8 1 9 s / T J 0 R r r V a N D Z P E + H / n q h E g e T e Y 3 M 3 9 7 P p X R q j 4 E a Y J 9 y 1 0 6 1 G p Z r z N l F S 6 F D s D p Y h 5 G w R I 3 k M i j z O 3 J x U U I 7 W o P V b H y q W R j w G J q k x A k l U o G A U r e d n s C d k J j d T c y 0 G 5 Q u p l T n z v 2 A 7 V K y a b Y N W x 6 W Z F Y o S 3 + G F g K + 9 E M 3 W r 1 q / q h 6 B O I t Q s Z S s C P Z Y 8 w K O x D k 2 y l s b G 5 t 7 R G 3 S 3 v 7 B 4 d H 5 e C S e 0 P X X e P P f m L Z 7 N R Y H A 4 / 3 Z p i Z F y x k M Z Y e b S r O R x 1 F E + U g c V Z G L b l E T 3 3 f M y y t m Y 6 o p A 5 t S Y Z b g r r 6 8 T j r X d a W L w / g f P 4 A c 2 y P W = R = < / l a t e x i t > X 5 a A i O G F H p G r + j N e 8 f F e X c + l q 0 F J 5 e w s I / G x x 8 g x f W G W Q N 6 V t x Y A X 6 u + J u C E V S 8 D O q Q T D J Z 6 + T a n h C 2 Y S O u G d p j b U 8 G O 2 X B S m E o P C / k / x U G j O Q E 4 t o w y G V T K P A d k Y U x m b m J 4 v / e V 4 K Y c P P R x w 9 e y c j J c j Q 6 B e / o t O Q x A E V m n C s V E I e c R 5 6 A d Y j 5 a P 7 r Q V 2 e d o v l u y K n N x 1 P K T N P U U n R h l g x q Q m i c 0 S t X f G w H O I e + 0 m W G p G O J 0 W r U G i E S Z j P K R d Q w k K O t s r q 2 v p H f L G x 7 u + z u F f c P W i q M 3 y a x j G W a h 4 E m a n J z b b + 7 p 8 S L O l L b t J F 4 u t c z v / 1 I p W K h N 2 O T i L o B H g r m M x r N C E v J Q d j y s K G e N d j X T n H Y i S X H g c k l S t A G u / Z F O < / l a e 6 x i t > < l a t e x i / D r e I Z X e L O e r B f 3 s f q Y j + a s b O c Q / t s h > D a K v I s 9 L 0 = " A 3 A A B 8 n i c b V B N m f h A a 1 _ b a s e 6 4 = " 2 V J j 5 i 4 H O s n V Q L I w J W Z H / J j j n S I Z h g Q A Z O U a D 4 x B B P W T T / o E n p l M U y 5 6 M E i / r x t r / 8 J N m I h J F E j l e p X W R 1 5 O I J K y I M D 5 1 C H G 2 h A e t Z O E R l h i o k 1 R B V C 4 s / j l Z d K q V h y 7 1 X b B 6 t J N I z H B D Z k j Q S 6 J b l x i I o 8 I X 8 + c 0 = " > A A A B 9 X i b L V D L T g I x F L 2 D I m s z y w U P E m T k 3 P u T s 0 9 X s S Z 0 r b 9 b w 8 U T D j R 0 O p O 2 o y E / 2 s O N C 4 1 x 6 7 + 4 8 U h V b B f r 3 f q Y j + a s O e c Q / s D 6 / A G y C r O F G I J j K I M D 5 1 C H 2 L h A E w h I e I Z X e Z Q j a = " 8 a m w z C u D a J 6 M U F O d I D w 0 i V 4 e < a / l a t e x i t > < l t s e x i t s h a 1 _ b a e W 5 1 0 E S 8 0 I p 9 N C L Y p 0 w m S M h 7 R r q M N 2 B x S k B B k a / e J X b C p S O K B C E 4 6 V 6 j A V 4 7 A M 5 N p y k V v J v n p d W P t X 7 o A E 1 G P U W B 6 S p p 6 i E 6 M M k 9 B K 8 4 R G q f p 7 I 8 G m l V p h 7 K j o c V 5 U z Q m J a a 0 0 4 k K Q 4 8 T Q R v n t f W N / K b h a 3 t d L 2 9 4 v 5 B S 4 W x J t e B p M v w H q k f H R f r 7 0 X T v v F k l 2 x U 6 k 2 + d H r m t x + p V C w U 3 A o S U T f A Q 8 F 8 R r O 1 q l J x z b + 7 p 8 S L O L a b t b y u 3 s r q 2 v n m H W Q 2 Z 8 B 9 u X G i M / E / F n X 9 j G W a h 4 p f 0 X e C N j X T n H Y i S H K g c d r 2 x t c z v / G s L W G x t 7 + z u F f c P i y q M J a F N E v J Q d j H m I Q O H C j 8 O R 3 h l J l e R T C X I 0 N 8 = " > 6 H A a < l a t e x i t s h 1 o _ b a s e 6 4 = " y A A A 5 E Z k h G H V H d O M S 3 B l E G E m n d K C h 0 D H B x 9 X i c b V D L T g I F z L 2 D L 8 Q X 6 t J N I 1 p R b Z u J / X j f W / o W M z B H F m g o y P + T H 0 l O / A P m h N E 9 o l K q N p x I c K D U J P D O Z H k H r q Y E Z / H L y 6 R V T i h 2 x b m t l e p X W C u Q i z S p A A y Y p 0 X x C J C a S m a y I j L D E R K j W k 7 c v X s 8 r R f L N V a O w V a J k 5 G S p C P 5 0 g K h u N O T i L o B H r j m M 4 K 1 k R 5 6 A d Y h S O 7 9 G J F I 0 z G e E i h o g o c U O U m a e o p t D 9 F + 9 Q Y h i Q M q N O Y c q a 5 j R 9 p N s N S M s J S + M W a i O G F H p G r j 3 N A e f F e X c + l q a T F 1 f U 3 Y e b S r O R x F E E Z + g c V Z G L b l 0 J d 1 l a t e x i t s h a _ > b a s e 6 4 = " H B j < t 5 3 8 5 R X / g f P 4 A c C i P W w = = < / l a t e x Z X o x E b G T K P A d k Y U m J b V m 4 v / e V 4 K Y j + P W C N u / D x x 8 g x f G u W I Q 6 V t x Y A X 6 c P r 5 w L e y t m Y 6 o p A t o S y Y b g r r 6 8 T j U t R E J y k w G O 2 X B S m o 4 P C 8 / / x U G j O Q E d I p J t n d 0 9 e / + g p a E a E t o k E Y 9 k x 8 e 3 p c 7 u 5 Z 4 5 f s y Z 0 o z b b R V W V t f W N 4 q K i 3 B O z 2 J q R f i o W A I a 1 g b 6 a E X Y j 1 S J h Z c o U z P N a S e W F I + K p 2 1 / f D 3 z 2 4 9 U P h I L i c b V D L S s N A F 2 9 p r 1 p f U Z d u B o X B Q z v W 9 G J V e 5 s y v R A r Y 5 H I K A = " > A A t N D 9 S + h 9 u X C j i 1 n x J 5 9 8 4 T b P Q 1 g M Q T y A U p i i 6 L b l x W s 9 M s Y 5 l M J + 3 Q y S T T G d " h a 1 _ b a s e 6 4 = s m Q n v 1 s S I v M B s t K a P 6 j r k U o = < / l t i e x i t > < l a t e x a U i 6 x E M 3 W r 1 q / q h 9 w B I t Q s Z G s C P Z M S y = c Z P e T A 8 X Y v U " N > A A A B 8 n i c b V B f 1 8 p I C E t M t C m q Y E w Y F r + 8 T F r V i m N r y n I z Q 3 7 M k Q 7 R r A 0 Z Y J I S z S e G Y C K X N H 9 z / A K b 9 a T 9 W K W E x / z 0 Z y V 7 R z C J g u A z U v 0 q q y M P R 3 M A Z X D g A u p w A w 1 o Y S O s i k U o G A U r e d n C M l f d G q m R y 0 G 5 6 y u 9 p 1 E g K T e Y 3 M 3 7 9 h P X R q j 4 E a Y J Q p s Z k s S 8 D O q Q T D J 6 x V + a n h C 2 Y S O u n k k W A b x O 3 J x U U I 7 o q P z V H y q W R j w G J 0 e k Z K e P X X e P P f m L 7 w 0 N Y H A 4 3 3 Z p i + F F Y x g v 2 A 7 V K y a b N G z W 6 W Z F Y o S 3 + Z / y 6 j l G p R r z N l F S F O 1 D D p Y h 5 G w R I T Z e 5 S G C D k 5 y l s b G 2 t 7 x R 3 S 3 v 7 B 4 d H A faster solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) a �− → a + e i S = � 6 , 9 , 20 � : Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

q + I Q 6 V t x Y A X 6 u J G j E b G T K P A d k Y W W x d + a n h C 2 Y S O u G p f T C N u / G x x 8 g x U m 6 t 8 / / x U G j O Q E 4 o P U w L e y t m Y 6 o p C o b P V m 4 v / e V 4 K Y c P E R J y k w G O 2 X B S m V Z 5 y P X R q j 4 E a Y J 9 M 7 6 i k U o G A U r e d h 9 s 3 1 D D p Y h 5 G w R I k 3 s 0 p 1 E g e T e Y 3 M n C J k V H y q W R j w G J q x P n k s S 8 D O q Q T D z o l u d J z a 2 R y 0 G 5 Q p W k A b x O 3 J x U U I 7 A t 6 B G 1 r 6 0 = " > A A A 9 b X i c b V D L T g I x m 3 L B s e 6 4 = " i N d x g A w 2 G B 6 W e P D 9 u c F 2 b H E m n d K C h 0 5 m 0 Q E 2 Z 8 B 9 u X G i M W G l D B L 8 Q X 6 t J N I z H D 3 Z k B E 3 V H d O M S E a _ S 3 + g c V Z G L b l E T a E M W a i O G F H p G r Z F j X y Y b g r r 6 8 T j r d 1 Z f U 3 Y e b S r O R x + N 1 > = = < / l a t e x i t < W l a t e x i t s h a w P A J e f F e X c + l q 0 F 5 6 8 5 R X / g f P 4 A c 2 F S / p A D J i n R f G I I J K K Z r I i M s M R E m 6 k N K S 9 e N 9 H B p Z c y E e j a C j I / F C Q c 6 Q J p 4 A Q A R c u o A 4 3 0 I m M E J D w D K / w Z j 1 d B g k R 3 8 c v L p F W r u 7 w V v T 0 r 1 6 / y O o p n m L r h F J Q i o 0 4 V i p u 1 v E 2 k u x 1 I x w O B / 3 O 1 M 6 d 0 7 5 d d q p B 3 r R M 3 J y U I U e j b i 1 R h E 0 T 2 i U q b 8 3 U w B q N Q l 9 M 5 m l X P Z A E C k V j T M Z 4 S L u G h o x S 5 a V Z 6 i k 6 M c Z 9 F G z W 6 W Z F Y o S 3 + F Y w + K e P X X e P P f N b L Z 9 B I t Q s Z S s C P Y a 8 O K x g v 2 A 7 V K y m Z h 5 x R 3 S 3 v 7 B 4 d H e t O T j l G p Z r z N l 7 5 7 Z 0 N Y H A 4 / 3 Z p i F G y R S G C D k 2 y l s b 6 q a x t e x i t > < l a t e i l t s h a 1 _ b a s e a / 4 H 7 9 T E f L V j 5 z i 8 < g f X 5 A 6 R c k U c = 6 = / B > A A A B 8 n i c b V N = S w M x E M 3 W r 1 q " 0 " T t 2 v n x A 8 1 9 s / F H J R r r V a N D Z P E + / F b 7 1 E g e T e Y 3 M 3 9 h 0 P X R q j 4 E a Y J 9 p s M F l G p Z r z N l F S 6 1 k D D p Y h 5 G w R I 3 y 6 T P x O 3 J x U U I 7 W o z A V H y q W R j w G J q b k i C k U o G A U r e d n s l p d J j d T c y 0 G 5 Q u j O x N g v 2 A 7 V K y a b Y z K W 6 W Z F Y o S 3 + G x O w 6 x E M 3 W r 1 q / q h 9 8 B I t Q s Z S s C P Z Y F + e 7 C D k 2 y l s b G 5 t x S R 3 S 3 v 7 B 4 d H 5 G R K 7 e P X X e P P f m L Z 0 y N Y H A 4 / 3 Z p i Z F k n w Z Y e b S r O R x 1 F E + U g c V Z G L b l E T 3 3 f M S e y t m Y 6 o p A 5 t y Z Y b g r r 6 8 T j r X d a W w w / g f P 4 A c 2 y P W = R = < / l a t e x i t > X 5 a A i O G F H p G r + j N e 8 f F e X c + l q 0 F J 5 L U k W u / G x x 8 g x f W G I C Q 6 V t x Y A X 6 u + N T j 6 s S 8 D O q Q T D J Z V p + a n h C 2 Y S O u G d J E o C w G O 2 X B S m E o P 8 y / / x U G j O Q E 4 t k J b b G T K P A d k Y U x m V R m 4 v / e V 4 K Y c P P M S X / E y c j J c j Q 6 B e e I o O Q x A E V m n C s t Q N 7 k R 5 6 A d Y j 5 a P c n r V 2 e d o v l u y K V e K g H K T N P U U n R h l P O x Q m i c 0 S t X f G q U x W I + 0 m W G p G O J 0 e w r G i E S Z j P K R d Q 1 4 k x 3 s r q 2 v p H f L G t y 7 + z u F f c P W i q u b J J 9 j G W a h 4 E m a n x t z b + 7 p 8 S L O l L b M a M h x t c z v / 1 I p W K u r N O T i L o B H g r m 2 d F e N E v J Q d j y s K G C c N j X T n H Y i S X H g w O N a / A G u / Z F O < / l t D e x i t > < l a t e x 6 s t f I e I Z X e L O e r B r / 3 f q Y j + a s b O c Q i w " m D a K v I s 9 L 0 = > f A A A B 8 n i c b V B 3 V s " h a 1 _ b a s e 6 4 = A L 2 J j 5 i 4 H O s n V Q h E l h W Z H / J j j n S I Z W T g A Z O U a D 4 x B B Q i J M J o E n p l M U y 5 6 / h E / r x t r / 8 J N m I P T A J e l e p X W R 1 5 O I j t K I M D 5 1 C H G 2 h y 4 F V Z E R l h i o k 1 R B O 7 C s / j l Z d K q V h y v n A j t J N I z H B D Z k B S X 6 J b l x i I o 8 I I 6 Q m b = " > A A A B 9 X i c V 8 D L T g I x F L 2 D L + U U T w U P E m T k 3 P u z 0 w 9 X s S Z 0 r b 9 b e y s D / j R 0 O p O 2 o y E T s 8 O N C 4 1 x 6 7 + 4 8 2 0 s W O f r 3 f q Y j + a s b c r Q / s D 6 / A G y C Z B e Q 2 J j K I M D 5 1 C H G h O A E w h I e I Z X e L F < h J " 8 a m w z C u D a a M 4 U F O d I D w 0 i V j = 6 / t l a t e x i t > < l a e e x i t s h a 1 _ b a s V V O Y E S 8 0 I p 9 N C L 1 0 N w m S M h 7 R r q M A 0 p V C k B B k a / e J X b x S 2 O K B C E 4 6 V 6 j p B W m n M 5 N p y k V v J v 7 d P W P t X 7 S o J E 1 G A p 6 9 S p p 6 i E 6 M M k B K U 8 4 R G q f p 7 I 8 G B S 4 t m 7 K j o c V 5 U z Q p a Q a 0 0 4 k K Q 4 8 T t h J e d f W N / K b h a 3 t n 2 R 9 4 v 5 B S 4 W x J L v + l 7 v w H q k f H R f r p X k T v v F k l 2 x U 6 B M 0 H 3 r m t x + p V C w U d o 2 S U T f A Q 8 F 8 R r A I 5 q l J x z b + 7 p 8 S L O L a b t b y u 3 s r q 2 v n m H W Q 2 Z 8 B 9 u X G i M / E / F n X 9 j G W a h 4 p f 0 X e C N j X T n H Y i S H K g c d r 2 x t c z v / G s L W G x t 7 + z u F f c P i y q M J a F N E v J Q d j H m I Q O H C j 8 O R 3 h l J l e R T C X I 0 N 8 = " > 6 H A a < l a t e x i t s h 1 o _ b a s e 6 4 = " y A A A 5 E Z k h G H V H d O M S 3 B l E G E m n d K C h 0 D H B x 9 X i c b V D L T g I F z L 2 D L 8 Q X 6 t J N I 1 p 1 b Z u J / X j f W / o W M z B H F m g o y P + T H 0 l O / A P m h N E 9 o l K q N p x I c K D U J P D O Z H k H r q Y E Z / H L y 6 R V T i h 2 b m t l e p X W R C u Q i z S p A A y Y p 0 X x C J C a S m a y I j L D E R K j W k 7 c v X s 8 r R f L N V a O w V a J k 5 G S p C P 5 0 g K h u N O T i L o B H r j m M 4 K 1 k R 5 6 A d Y h S O 7 9 G J F I 0 z G e E i h o g o c U O U m a e o p t D 9 F + 9 Q Y h i Q M q N O Y c q a 5 j R 9 p N s N S M s x S + M W a i O G F H p G r j 3 N A e f F e X c + l q a T F 1 f U 3 Y e b S r O R x F E E Z + g c V Z G L b l 0 J d 1 l a t e x i t s h a _ > b a s e 6 4 = " H B j < t 5 3 8 5 R X / g f P 4 A c C i P W w = = < / l a t e x Z X o x E b G T K P A d k Y U m J b V m 4 v / e V 4 K Y j + P W C N u / G D x 8 g x f G u W I Q 6 V t x Y A X 6 c P r 5 w L e y t m Y 6 o p A t o S y Y b g r r 6 8 T j U t R E J y k w G O 2 X B S m o 4 P C 8 / / x U G j O Q E d I p a 3 t n d 0 9 e / + g p J p E E t o k E Y 9 k x 8 a b K o P u 5 Z 4 5 f s y Z 0 7 q z b R V W V t f W N 4 e c h A J O z 2 J q R f i o W B h I 1 g b 6 a E X Y j 1 a K i I Z o U z P N a S e W F c U + p 2 1 / f D 3 z 2 4 9 3 D I F X i c b V D L S s N A L B 2 p r 1 p f U Z d u B 9 A t z v W 9 G J V e 5 s y v R A r Y 5 H I K A = " > A o Q M n Q S + h 9 u X C j i 1 9 T x 5 9 8 4 T b P Q 1 g J M N s y U p i i 6 L b l x W A T 9 s Y 5 l M J + 3 Q y S T x d " h a 1 _ b a s e 6 4 = s m Q n v 1 s S I v M B s t K a P 6 j r k U o = < / l t i e x i t > < l a t e x a U i 6 x E M 3 W r 1 q / q h 9 w B I t Q s Z G s C P Z M S y = c Z P e T A 8 X Y v U " N > A A A B 8 n i c b V B f 1 8 p I C E t M t C m q Y E w Y F r + 8 T F r V i m N r y n I z Q 3 7 M k Q 7 R r A 0 Z Y J I S z S e G Y C K X N H 9 z / A K b 9 a T 9 W K W E x / z 0 Z y V 7 R z C J g u A z U v 0 q q y M P R 3 M A Z X D g A u p w A w 1 o Y S O s i k U o G A U r e d n C M l f d G q m R y 0 G 5 6 y u 9 p 1 E g e K e Y 3 M 3 7 9 h P X R q j 4 E a Y J Q p s Z k s S 8 D O q Q T D J 6 x V + a n h C 2 Y S O u n k k W A b x O 3 J x U U I 7 o q P z V H y q W R j w G J 0 T k Z K e P X X e P P f m L 7 w 0 N Y H A 3 / 3 Z p i + F F Y x g v 2 A 7 V K y a b N G z W 6 W Z F Y o S 3 + Z 4 y 6 j l G p Z R z N l F S F O 1 D D p Y h 5 G w R I T r e 5 S G C D k 5 y l s b G 2 t v 7 H 4 B 7 d 3 S 3 R x A faster solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) a �− → a + e i Z( n ) = � i ≤ k φ i (Z( n − n i )) S = � 6 , 9 , 20 � : Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

A faster solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) a �− → a + e i Z( n ) = � i ≤ k φ i (Z( n − n i )) Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

A faster solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) a �− → a + e i Z( n ) = � i ≤ k φ i (Z( n − n i )) n ∈ S = � 6 , 9 , 20 � Z( n ) L( n ) 0 { 0 } { 0 } Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

A faster solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) a �− → a + e i Z( n ) = � i ≤ k φ i (Z( n − n i )) n ∈ S = � 6 , 9 , 20 � Z( n ) L( n ) 0 { 0 } { 0 } 6 6 { e 1 } { 1 } 0 � e 1 Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

A faster solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) a �− → a + e i Z( n ) = � i ≤ k φ i (Z( n − n i )) n ∈ S = � 6 , 9 , 20 � Z( n ) L( n ) 0 { 0 } { 0 } 6 6 { e 1 } { 1 } 0 � e 1 9 9 0 � e 2 { e 2 } { 1 } Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

A faster solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) a �− → a + e i Z( n ) = � i ≤ k φ i (Z( n − n i )) n ∈ S = � 6 , 9 , 20 � Z( n ) L( n ) 0 { 0 } { 0 } 6 6 { e 1 } { 1 } 0 � e 1 9 9 0 � e 2 { e 2 } { 1 } 6 12 e 1 � 2 e 1 { 2 e 1 } { 2 } Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

A faster solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) a �− → a + e i Z( n ) = � i ≤ k φ i (Z( n − n i )) n ∈ S = � 6 , 9 , 20 � Z( n ) L( n ) 0 { 0 } { 0 } 6 6 { e 1 } { 1 } 0 � e 1 9 9 0 � e 2 { e 2 } { 1 } 6 12 e 1 � 2 e 1 { 2 e 1 } { 2 } 6 15 e 2 � (1 , 1 , 0) { (1 , 1 , 0) } { 2 } Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

A faster solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) a �− → a + e i Z( n ) = � i ≤ k φ i (Z( n − n i )) n ∈ S = � 6 , 9 , 20 � Z( n ) L( n ) 0 { 0 } { 0 } 6 6 { e 1 } { 1 } 0 � e 1 9 9 0 � e 2 { e 2 } { 1 } 6 12 e 1 � 2 e 1 { 2 e 1 } { 2 } 6 15 e 2 � (1 , 1 , 0) { (1 , 1 , 0) } { 2 } 9 e 1 � (1 , 1 , 0) Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

A faster solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) a �− → a + e i Z( n ) = � i ≤ k φ i (Z( n − n i )) n ∈ S = � 6 , 9 , 20 � Z( n ) L( n ) 0 { 0 } { 0 } 6 6 { e 1 } { 1 } 0 � e 1 9 9 0 � e 2 { e 2 } { 1 } 6 12 e 1 � 2 e 1 { 2 e 1 } { 2 } 6 15 e 2 � (1 , 1 , 0) { (1 , 1 , 0) } { 2 } 9 e 1 � (1 , 1 , 0) 6 18 2 e 1 � 3 e 1 { 3 e 1 , 2 e 2 } { 2 , 3 } Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

A faster solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) a �− → a + e i Z( n ) = � i ≤ k φ i (Z( n − n i )) n ∈ S = � 6 , 9 , 20 � Z( n ) L( n ) 0 { 0 } { 0 } 6 6 { e 1 } { 1 } 0 � e 1 9 9 0 � e 2 { e 2 } { 1 } 6 12 e 1 � 2 e 1 { 2 e 1 } { 2 } 6 15 e 2 � (1 , 1 , 0) { (1 , 1 , 0) } { 2 } 9 e 1 � (1 , 1 , 0) 6 18 2 e 1 � 3 e 1 { 3 e 1 , 2 e 2 } { 2 , 3 } 9 e 2 � 2 e 2 Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

A faster solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) a �− → a + e i Z( n ) = � i ≤ k φ i (Z( n − n i )) n ∈ S = � 6 , 9 , 20 � Z( n ) L( n ) 0 { 0 } { 0 } 6 6 { e 1 } { 1 } 0 � e 1 9 9 0 � e 2 { e 2 } { 1 } 6 12 e 1 � 2 e 1 { 2 e 1 } { 2 } 6 15 e 2 � (1 , 1 , 0) { (1 , 1 , 0) } { 2 } 9 e 1 � (1 , 1 , 0) 6 18 2 e 1 � 3 e 1 { 3 e 1 , 2 e 2 } { 2 , 3 } 9 e 2 � 2 e 2 20 20 0 � e 3 { e 3 } { 1 } Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

A faster solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) a �− → a + e i Z( n ) = � i ≤ k φ i (Z( n − n i )) n ∈ S = � 6 , 9 , 20 � Z( n ) L( n ) 0 { 0 } { 0 } 6 6 { e 1 } { 1 } 0 � e 1 9 9 0 � e 2 { e 2 } { 1 } 6 12 e 1 � 2 e 1 { 2 e 1 } { 2 } 6 15 e 2 � (1 , 1 , 0) { (1 , 1 , 0) } { 2 } 9 e 1 � (1 , 1 , 0) 6 18 2 e 1 � 3 e 1 { 3 e 1 , 2 e 2 } { 2 , 3 } 9 e 2 � 2 e 2 20 20 0 � e 3 { e 3 } { 1 } . . . . . . . . . Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

A faster solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) ψ i : L( n − n i ) − → L( n ) a �− → a + e i Z( n ) = � i ≤ k φ i (Z( n − n i )) n ∈ S = � 6 , 9 , 20 � Z( n ) L( n ) 0 { 0 } { 0 } 6 6 { e 1 } { 1 } 0 � e 1 9 9 0 � e 2 { e 2 } { 1 } 6 12 e 1 � 2 e 1 { 2 e 1 } { 2 } 6 15 e 2 � (1 , 1 , 0) { (1 , 1 , 0) } { 2 } 9 e 1 � (1 , 1 , 0) 6 18 2 e 1 � 3 e 1 { 3 e 1 , 2 e 2 } { 2 , 3 } 9 e 2 � 2 e 2 20 20 0 � e 3 { e 3 } { 1 } . . . . . . . . . Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

A faster solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) ψ i : L( n − n i ) − → L( n ) a �− → a + e i ℓ �− → ℓ + 1 Z( n ) = � i ≤ k φ i (Z( n − n i )) n ∈ S = � 6 , 9 , 20 � Z( n ) L( n ) 0 { 0 } { 0 } 6 6 { e 1 } { 1 } 0 � e 1 9 9 0 � e 2 { e 2 } { 1 } 6 12 e 1 � 2 e 1 { 2 e 1 } { 2 } 6 15 e 2 � (1 , 1 , 0) { (1 , 1 , 0) } { 2 } 9 e 1 � (1 , 1 , 0) 6 18 2 e 1 � 3 e 1 { 3 e 1 , 2 e 2 } { 2 , 3 } 9 e 2 � 2 e 2 20 20 0 � e 3 { e 3 } { 1 } . . . . . . . . . Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

A faster solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) ψ i : L( n − n i ) − → L( n ) a �− → a + e i ℓ �− → ℓ + 1 Z( n ) = � L( n ) = � i ≤ k φ i (Z( n − n i )) i ≤ k ψ i (L( n − n i )) n ∈ S = � 6 , 9 , 20 � Z( n ) L( n ) 0 { 0 } { 0 } 6 6 { e 1 } { 1 } 0 � e 1 9 9 0 � e 2 { e 2 } { 1 } 6 12 e 1 � 2 e 1 { 2 e 1 } { 2 } 6 15 e 2 � (1 , 1 , 0) { (1 , 1 , 0) } { 2 } 9 e 1 � (1 , 1 , 0) 6 18 2 e 1 � 3 e 1 { 3 e 1 , 2 e 2 } { 2 , 3 } 9 e 2 � 2 e 2 20 20 0 � e 3 { e 3 } { 1 } . . . . . . . . . Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

A faster solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) ψ i : L( n − n i ) − → L( n ) a �− → a + e i ℓ �− → ℓ + 1 Z( n ) = � L( n ) = � i ≤ k φ i (Z( n − n i )) i ≤ k ψ i (L( n − n i )) n ∈ S = � 6 , 9 , 20 � L( n ) 0 { 0 } 6 9 12 15 18 20 . . . Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

A faster solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) ψ i : L( n − n i ) − → L( n ) a �− → a + e i ℓ �− → ℓ + 1 Z( n ) = � L( n ) = � i ≤ k φ i (Z( n − n i )) i ≤ k ψ i (L( n − n i )) n ∈ S = � 6 , 9 , 20 � L( n ) 0 { 0 } 6 6 { 1 } 0 � 1 9 12 15 18 20 . . . Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

A faster solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) ψ i : L( n − n i ) − → L( n ) a �− → a + e i ℓ �− → ℓ + 1 Z( n ) = � L( n ) = � i ≤ k φ i (Z( n − n i )) i ≤ k ψ i (L( n − n i )) n ∈ S = � 6 , 9 , 20 � L( n ) 0 { 0 } 6 6 { 1 } 0 � 1 9 9 { 1 } 0 � 1 12 15 18 20 . . . Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

A faster solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) ψ i : L( n − n i ) − → L( n ) a �− → a + e i ℓ �− → ℓ + 1 Z( n ) = � L( n ) = � i ≤ k φ i (Z( n − n i )) i ≤ k ψ i (L( n − n i )) n ∈ S = � 6 , 9 , 20 � L( n ) 0 { 0 } 6 6 { 1 } 0 � 1 9 9 { 1 } 0 � 1 6 12 { 2 } 1 � 2 15 18 20 . . . Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27

A faster solution: dynamic programming Fix n ∈ S = � n 1 , . . . , n k � . For each i ≤ k , φ i : Z( n − n i ) − → Z( n ) ψ i : L( n − n i ) − → L( n ) a �− → a + e i ℓ �− → ℓ + 1 Z( n ) = � L( n ) = � i ≤ k φ i (Z( n − n i )) i ≤ k ψ i (L( n − n i )) n ∈ S = � 6 , 9 , 20 � L( n ) 0 { 0 } 6 6 { 1 } 0 � 1 9 9 { 1 } 0 � 1 6 12 { 2 } 1 � 2 6 15 { 2 } 1 � 2 18 20 . . . Christopher O’Neill (SDSU) Computing delta sets of affine semigroups Feb 10, 2020 14 / 27