Some remarks of Wilf’s conjecture Shalom Eliahou Université du Littoral Côte d’Opale International Meeting on Numerical Semigroups 8 - 12 September, 2014 Cortona, September 12 Shalom Eliahou (ULCO) Wilf’s conjecture IMNS, September 12 1 / 23
Introduction Here, [ x , y [ means integer interval : all n ∈ Z such that x ≤ n < y . Let S ⊆ N be a numerical semigroup 1 . Notation S ∗ = S \{ 0 } m = min S ∗ , its multiplicity c its conductor , i.e. [ c , ∞ [ ⊆ S and c minimal L = S ∩ [ 0 , c [ , the left part of S P = the set of primitive elements , i.e. minimal generators of S 1 That is: 0 ∈ S , S + S ⊆ S , N \ S finite. Shalom Eliahou (ULCO) Wilf’s conjecture IMNS, September 12 2 / 23
Conjecture (Wilf, 1978) Let S ⊆ N be a numerical semigroup. The density of L = S ∩ [ 0 , c [ inside [ 0 , c [ should be bounded below as follows: | L | 1 ≥ | P | , c i.e. by the inverse of the embedding dimension of S. Equivalently, | P || L | ≥ c . Notation W ( S ) = | P || L |− c . Wilf’s conjecture states: W ( S ) ≥ 0 . Shalom Eliahou (ULCO) Wilf’s conjecture IMNS, September 12 3 / 23
Some results Wilf’s conjecture holds in various cases, including: for | P | = 2 [Sylvester 1884] for | P | = 3 [Fröberg et al. 1987] for genus = c −| L | ≤ 50 [Bras-Amorós 2008] for m ≥ c / 2 [Kaplan 2012] for | P | ≥ m / 2 [Sammartano 2012] for m ≤ 8 [Sammartano 2012] Of particular interest here: for | L | ≤ 4 [Dobbs and Matthews 2006] Shalom Eliahou (ULCO) Wilf’s conjecture IMNS, September 12 4 / 23
Main result Theorem (E., work in progress) Wilf’s conjecture holds for | L | ≤ 10 . The proof rests on suitably slicing the integers, some general statements independent of | L | , some case-by-case analysis specific to | L | ≤ 10. Shalom Eliahou (ULCO) Wilf’s conjecture IMNS, September 12 5 / 23
The parameters q , ρ Notation Let S be a numerical semigroup with m , c as above. Denote � c � q = m and ρ = qm − c . Thus, qm ≥ c, and c = qm − ρ with ρ ∈ [ 0 , m [ . Example q = 1 ⇐ ⇒ m = c ⇐ ⇒ S = { 0 }∪ [ c , ∞ [ . Shalom Eliahou (ULCO) Wilf’s conjecture IMNS, September 12 6 / 23
Slices Notation Let I q = [ c , c + m [ . More generally, for j ∈ Z , let I j be the translate of I q by ( j − q ) m. Thus, = [ c − m , c [ I q − 1 = [ c − 2 m , c − m [ I q − 2 . . . = [ c − qm , c − qm + m [ = [ ρ , ρ + m [ I 1 = [ ρ − m , ρ [ I 0 Shalom Eliahou (ULCO) Wilf’s conjecture IMNS, September 12 7 / 23
Remark Why consider these slices I j rather than the more obvious [ jm , jm + m [ ? Because the interval I q = [ c , c + m [ seems to play a key role in Wilf’s conjecture. (See below.) Notation For all j ≥ 0 , let S j = S ∩ I j . Example S 0 = { 0 } m ∈ S 1 , 2 m ∈ S 2 , . . . , jm ∈ S j for all j S q − 1 � I q − 1 (as c − 1 / ∈ S q − 1 ) S q = I q Shalom Eliahou (ULCO) Wilf’s conjecture IMNS, September 12 8 / 23
Remark m + S j ⊆ S j + 1 for all j . It follows that 1 ≤ | S 1 | ≤ ··· ≤ | S q − 1 | . Of course, | L | = 1 + | S 1 | + ··· + | S q − 1 | . Remark In general, we only have a weak grading: S 1 + S j ⊆ S 1 + j ∪ S 1 + j + 1 for j ≥ 1 S i + S j ⊆ S i + j − 1 ∪ S i + j ∪ S i + j + 1 for i , j ≥ 2 . Shalom Eliahou (ULCO) Wilf’s conjecture IMNS, September 12 9 / 23
When, for a given S , this weak grading is actually a true grading up to degree q − 1, things are simpler! Theorem (E., work in progress) Let S ⊂ N satisfy S i + S j ⊆ S i + j for i + j ≤ q − 1 , and P ∩ L = P 1 . Then S satisfies Wilf’s conjecture. Proof. Involved, using a theorem of Macaulay (1927) on the growth of Hilbert functions of standard graded algebras. Example Let m = 1000, c = 4000. Assume that all left minimal generators of S (those less than c ) are contained in [ 1000 , 1333 [ . Then S satisfies Wilf. Shalom Eliahou (ULCO) Wilf’s conjecture IMNS, September 12 10 / 23
Primitives and decomposables Definition Let x ∈ S ∗ . We say that x is decomposable if x = x 1 + x 2 with x 1 , x 2 ∈ S ∗ primitive otherwise. Notation P = set of primitive elements = set of minimal generators of S D = set of decomposable elements. Thus, S ∗ = P ⊔ D . Note that P ⊆ [ 0 , c + m [ . Indeed, [ c + m , ∞ [ ⊆ m + S ∗ . Shalom Eliahou (ULCO) Wilf’s conjecture IMNS, September 12 11 / 23
Notation For j ∈ N , let = P ∩ S j , = | P j | , P j p j = D ∩ S j , = | D j | . D j d j Note that p 1 ≥ 1 since m ∈ P 1 . Also S 1 = P 1 , as x ∈ D = ⇒ x ≥ 2 m . Definition The profile of S is ( p 1 ,..., p q − 1 ) ∈ N q − 1 . Any ( p 1 ,..., p q − 1 ) ∈ N q − 1 with p 1 ≥ 1 is the profile of a suitable S . Shalom Eliahou (ULCO) Wilf’s conjecture IMNS, September 12 12 / 23
Discarding the right primitives • The number p q of right primitives, i.e. those in I q = [ c , c + m [ , is the out-degree of the vertex S in the tree of numerical semigroups. • By definition, it is not included in the profile ( p 1 ,..., p q − 1 ) . • Now, p q is involved in two terms in W ( S ) : W ( S ) = | P || L |− c = | P || L |− qm + ρ . Indeed, | P | = | P ∩ L | + p q = p q + d q , m since m = | [ c , c + m [ | = | I q | = p q + d q . Shalom Eliahou (ULCO) Wilf’s conjecture IMNS, September 12 13 / 23
The constant W 0 ( S ) Factoring out p q from the expression of W ( S ) = | P || L |− qm + ρ , we obtain: Definition W 0 ( S ) = | P ∩ L || L |− qd q + ρ . By construction, we have W ( S ) = p q ( | L |− q )+ W 0 ( S ) . Shalom Eliahou (ULCO) Wilf’s conjecture IMNS, September 12 14 / 23
Proposition W ( S ) ≥ W 0 ( S ) . Proof. W ( S ) = p q ( | L |− q )+ W 0 ( S ) , and | L | ≥ q since L ⊇ { 0 , m ,..., ( q − 1 ) m } . Corollary If S satisfies W 0 ( S ) ≥ 0 then S satisfies Wilf. ✷ We shall use it a lot below! Shalom Eliahou (ULCO) Wilf’s conjecture IMNS, September 12 15 / 23
The case q = 2 Then W ( S ) ≥ 0 in this case [Kaplan 2012]. Stronger yet simpler: Proposition For S with q = 2 , we have W 0 ( S ) ≥ ρ ≥ 0 . Proof. Let k = p 1 . Then | L | = 1 + k since L = S 0 ⊔ S 1 = { 0 }⊔ P 1 here. Now W 0 ( S ) − ρ = | P ∩ L || L |− 2 d 2 = k ( 1 + k ) − 2 d 2 . But d 2 ≤ k ( k + 1 ) / 2 , as any decomposable in I 2 = [ c , c + m [ is a sum of two primitives in P 1 . Therefore W 0 ( S ) ≥ ρ ≥ 0. Shalom Eliahou (ULCO) Wilf’s conjecture IMNS, September 12 16 / 23
Some reductions Wilf’s conjecture is open for q ≥ 3. But reductions are available, e.g.: Proposition (work in progress) Assume q = 3 . Let S with profile ( p 1 , ∗ ) ∈ N 2 and p 1 ≤ 4 . Then W 0 ( S ) ≥ 0 . Proof. Factoring out p 2 in W 0 ( S ) , plus Macaulay, plus some ad-hoc computations for the reduced profile ( p 1 , 0 ) assuming p 1 ≤ 4. Shalom Eliahou (ULCO) Wilf’s conjecture IMNS, September 12 17 / 23
Yet another reduction: Proposition (work in progress) Assume q ≥ 3 . Let S with profile ( 1 , 0 ,..., 0 , ∗ ,..., ∗ ) ∈ N q − 1 where the leftmost ∗ occurs at index h with h ≥ q / 2 . Then W 0 ( S ) ≥ 0 . Proof. A lengthy computation – with a few tricks. Shalom Eliahou (ULCO) Wilf’s conjecture IMNS, September 12 18 / 23
Towards the case | L | ≤ 10 Lemma Let S with profile ( p 1 ,..., p q − 1 ) . Then | L | ≥ 1 + p q − 1 + 2 p q − 2 + ··· +( q − 1 ) p 1 . Proof. We have S 0 = { 0 } . For 1 ≤ i ≤ q − 1, S i contains � � � � P i ⊔ m + P i − 1 ⊔ ... ⊔ ( i − 1 ) m + P 1 . Hence | S i | ≥ p i + p i − 1 + ··· + p 1 . Example If S is of profile ( 3 , 1 , 0 ) then | L | ≥ 11. Shalom Eliahou (ULCO) Wilf’s conjecture IMNS, September 12 19 / 23
Proposition For | L | ≤ 10 , the only possible profiles with q ≥ 3 are: ( p 1 , ∗ ) with p 1 ≤ 4 ( 3 , 0 , 0 ) , ( 2 , 1 , ∗ ) , ( 2 , 0 , ∗ ) or ( 1 , ∗ , ∗ ) ( 2 , 0 , 0 , ∗ ) , ( 1 , 1 , 1 , 0 ) , ( 1 , 1 , 0 , ∗ ) , ( 1 , 0 , 1 , ∗ ) or ( 1 , 0 , 0 , ∗ ) ( 1 , 1 , 0 , 0 , 0 ) , ( 1 , 0 , 0 , 1 , 0 ) and ( 1 , 0 , 0 , 0 , ∗ ) ( 1 , 0 , 0 , 0 , 0 , ∗ ) In each case, one may show that W 0 ( S ) ≥ 0 by either using above results if applicable (e.g. reductions, or true grading) or else by a specific ad-hoc analysis. Theorem (work in progress) If | L | ≤ 10 then W 0 ( S ) ≥ 0 . Shalom Eliahou (ULCO) Wilf’s conjecture IMNS, September 12 20 / 23
A stronger conjecture? In view of the above results, can one conjecture W 0 ( S ) ≥ 0 always, implying Wilf? Not quite so! Assertion (Jean Fromentin, September 5, 2014) For genus g ≤ 42 , W 0 ( S ) ≥ 0 . For genus g ≤ 60 , there are exactly 5 counterexamples. The smallest one has genus g = 43 : S = � 14 , 22 , 23 �∪ [ 56 , ∞ [ . The other 4 occur in genus 51, 55, 55 and 59, respectively. All 5 cases satisfy W 0 ( S ) = − 1, q = 4, | L | = 13 and W ( S ) ≥ 35. Shalom Eliahou (ULCO) Wilf’s conjecture IMNS, September 12 21 / 23
Corollary Wilf’s conjecture holds up to genus 60. Announcement Assertion (Fromentin & Hivert, work in progress) There are exactly 377 866 907 506 273 numerical semigroups of genus g = 67 . Shalom Eliahou (ULCO) Wilf’s conjecture IMNS, September 12 22 / 23
Grazie mille! Shalom Eliahou (ULCO) Wilf’s conjecture IMNS, September 12 23 / 23
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