On a question of Eliahou and a conjecture of Wilf
Manuel Delgado
www.fc.up.pt/cmup/mdelgado/
International meeting on numerical semigroups with applications
Levico Terme 2016
Research partially supported by:
CMUP — UID/MAT/00144/2013
Semigrupos Numéricos y Afines — MTM2014-55367-P
Motivation
Every integer is the Eliahou number of infinitely many NS
References
Outline
1
Motivation
A pictorial view of a numerical semigroup
Terminology, notation and a conjecture of Wilf
Some more notation and a problem of Eliahou
Fromentin-Eliahou examples
2
Every integer is the Eliahou number of infinitely many NS
Defining the numerical semigroup S(p)
A visual way to look to S(p)
The S(p) satisfy Wilf’s conjecture
Defining the numerical semigroup S(p, τ )
A visual way to look to S(p, τ )
Defining a sequence of numerical semigroups S (i) (p, τ )
A visual way to look to S (i) (p, τ )
3
References
M. Delgado
Question of Eliahou — Conjecture of Wilf
Levico Terme, 04/07/2016
2 / 21
Motivation
Every integer is the Eliahou number of infinitely many NS
References
Motivation
This work is motivated by a recent paper of Eliahou:
S. Eliahou, Wilf’s conjecture and Macaulay’s theorem, preprint.
It was presented in a special session on “Commutative Monoids” of the
AMS-EMS-SPM joint meeting that held in Porto, in 2015.
Many talks in former editions of the IMNS series of conferences on numerical
semigroups referred Wilf’s conjecture...
M. Delgado
Question of Eliahou — Conjecture of Wilf
Levico Terme, 04/07/2016
3 / 21
Motivation
Every integer is the Eliahou number of infinitely many NS
References
A pictorial view of a numerical semigroup
We represent a numerical semigroup with multiplicity m and conductor c by
highlighting some entries in a (dc/me + 1) × m table of consecutive integers.
The uppermost row starts in c. The lowest row contains 0.
52
41
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19
8
-3
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-2
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10
-1
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0
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1
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Pictorial representation of the numerical semigroup h11, 13, 21, 62i
Among the elements of the numerical semigroup, some are highlighted.
M. Delgado
Question of Eliahou — Conjecture of Wilf
Levico Terme, 04/07/2016
4 / 21
Motivation
Every integer is the Eliahou number of infinitely many NS
References
Terminology, notation and a conjecture
of Wilf
Let S be a numerical semigroup.
Primitive elements of S (also known as minimal generators) . . . . . . . . . . P(S)
Conductor of S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c(S)
the smallest integer from which all the integers belong to S
Lower elements of S (also known as left elements) . . . . . . . . . . . . . . . . . . . . L(S)
the elements of S that are smaller than c(S)
Wilf number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W(S) = |P||L| − c
Conjecture (Wilf)
If S is a numerical semigroup, then W(S) ≥ 0.
M. Delgado
Question of Eliahou — Conjecture of Wilf
Levico Terme, 04/07/2016
5 / 21
Motivation
Every integer is the Eliahou number of infinitely many NS
References
Some more notation
Multiplicity of S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m(S)
the smallest positive element of S
q-number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . q(S) = dc / me
The non primitive elements are called decomposable.
Decomposable elements at level q . . . . . . . . . . . . . . . .Dq (S) = [c, c + m[\P.
A kind of a remainder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ρ(S) = q · m − c.
Eliahou number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E(S) = |P ∩ L||L| − q|Dq | + ρ
Proposition (Eliahou)
If E(S) ≥ 0, then W(S) ≥ 0.
Theorem (Eliahou)
Numerical semigroups S satisfying q(S) = 3 have positive Eliahou number.
M. Delgado
Question of Eliahou — Conjecture of Wilf
Levico Terme, 04/07/2016
6 / 21
Motivation
Every integer is the Eliahou number of infinitely many NS
References
A problem of Eliahou
Most numerical semigroups satisfy q(S) ≤ 3... A natural question is:
“Is there any numerical semigroup S such that E(S) < 0?”
The answer is “yes”. In fact, there are infinitely many, but...
For instance, the probability of a numerical semigroup taken at random from
the set of numerical semigroups of genus up to 60 is ' 10513 .
So, without computational tools, we would possibly never find a single
example.
Problem (Eliahou)
Characterize the numerical semigroups having negative Eliahou number.
hm, g1 , g2 , . . . , gr ic is the numerical semigroup containing {m, g1 , g2 , . . . , gr }
and all the integers greater than or equal to c.
M. Delgado
Question of Eliahou — Conjecture of Wilf
Levico Terme, 04/07/2016
7 / 21
Motivation
Every integer is the Eliahou number of infinitely many NS
References
Fromentin-Eliahou examples . . . . . . . . . . . . . . . . . . . . . notation . . . “generators”
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M. Delgado
. . . . . . . . . . . . . . . . . . . S(4) . . . h14, 22, 23i56
. . . . . . . . . . S (1) (4, 0) . . . h16, 25, 26i64
. . . . . . . . . . . . . . . . . . . . h17, 26, 28i68
. . . . . . . . . . . . . . . . . . . . h17, 27, 28i68
89
71
53
35
17
. . . . . S (2) (4, 0) . . . h18, 28, 29i72
Question of Eliahou — Conjecture of Wilf
Levico Terme, 04/07/2016
8 / 21
Motivation
Every integer is the Eliahou number of infinitely many NS
References
The Fromentin-Eliahou examples have Eliahou number equal to −1 and are
the only examples of numerical semigroups of genus not greater than 60
having negative Eliahou number.
M. Delgado
Question of Eliahou — Conjecture of Wilf
Levico Terme, 04/07/2016
9 / 21
Motivation
Every integer is the Eliahou number of infinitely many NS
References
Defining the numerical semigroup S(p)
A visual way to look to S(p)
We construct numerical semigroups whith pictorial representations similar to
those of some of the Fromentin-Eliahou examples.
One can partition these tables into several blocks (where p is an even positive
integer):
a block C consisting of the 2 leftmost columns;
blocks Bi , 1 ≤ i ≤ p2 , consisting of p2 + 4 consecutive columns.
M. Delgado
Question of Eliahou — Conjecture of Wilf
Levico Terme, 04/07/2016
10 / 21
Motivation
Every integer is the Eliahou number of infinitely many NS
References
The definition of S(p)
Let p be an even positive integer.
Consider the following integers (which depend on p):
p2
p p
+ 2p + 2 =
+ 4 + 2;
4
2 2
p
+4 .
γ(p) = 2µ(p) −
2
µ(p) =
Definition
The numerical semigroup hµ, γ, γ + 1ipµ , which contains {µ, γ, γ + 1} and all
the integers greater than or equal to pµ, is denoted by S(p).
After some technical results one gets:
Corollary
The conductor of S(p) is pµ.
M. Delgado
Question of Eliahou — Conjecture of Wilf
Levico Terme, 04/07/2016
11 / 21
Motivation
Every integer is the Eliahou number of infinitely many NS
References
After some more technical results one gets:
Corollary
|L| =
p3
24
+ 38 p 2 +
13
12 p.
And also:
Corollary
|Dp | =
p2
4
+ 52 p + 6 /2.
Now we can state the main result:
Theorem
E(S(p)) =
p
p
1−
4
2
Corollary
There exist numerical semigroups with arbitrarily large negative Eliahou
number.
M. Delgado
Question of Eliahou — Conjecture of Wilf
Levico Terme, 04/07/2016
12 / 21
Motivation
Every integer is the Eliahou number of infinitely many NS
References
The S(p) satisfy Wilf’s conjecture
From the previously stated results one gets:
Corollary
|P(S)| =
p2
8
+ 34 p + 2.
And gets that the Wilf’s conjecture holds for all numerical semigroups S(p).
Proposition
Let p be an even positive integer. Then
W(S(p)) =
p5
5p 4
p3
7p 2
p
+
+
−
+ > 0.
192
64
4
16
6
In particular, W(S(p)) > 0.
Remark
When p grows, E(S(p)) becomes large negative, while W(S(p)) becomes
large positive.
M. Delgado
Question of Eliahou — Conjecture of Wilf
Levico Terme, 04/07/2016
13 / 21
Motivation
Every integer is the Eliahou number of infinitely many NS
References
Defining the semigroup S(p, τ )
A visual way to look to S(p, τ )
Let τ be a non-negative integer τ .
For each of the figures representing one S(p) do the following slight
modifications:
add τ columns to the right of each of the Bi blocks, for 1 ≤ i ≤
p
2
− 1;
concerning the block B p2 , instead of adding columns to the right, τ
columns are put at the left of the table. Block B is therefore split into
two parts.
Block C now consists of the columns τ + 1 and τ + 2.
M. Delgado
Question of Eliahou — Conjecture of Wilf
Levico Terme, 04/07/2016
14 / 21
Motivation
Every integer is the Eliahou number of infinitely many NS
References
It is straightforward to observe that the former images represent the
semigroups S(p, τ ) defined below.
Consider the integers:
m = µ + τ p2 =
p
2
p
2
+4+τ +2=
p2
4
+ 2p + 2 + τ p2 ;
g = γ + τ (p − 1) = 2m − p2 + 4 + τ = 2m − (τ + 1) −
c = pµ + τ p2 − 1 = pm − τ .
p
2
+2+1 ;
Definition
The numerical semigroup hm, g , g + 1ic is denoted by S(p, τ ).
Remark
1
|L(S(p))| = |L(S(p, τ ))|;
2
|Dp (S(p))| = |Dp (S(p, τ ))|;
M. Delgado
Question of Eliahou — Conjecture of Wilf
Levico Terme, 04/07/2016
15 / 21
Motivation
Every integer is the Eliahou number of infinitely many NS
References
Theorem
Let p be an even positive
integer and let τ be a non negative integer. Then
E(S(p, τ )) = p4 1 − p2 + τ .
Corollary
Every integer is the Eliahou number of some numerical semigroup of the form
S(p, τ ).
Remark
|P(S(p, τ ))| = |P(S(p))| + τ p2 ; c(S(p, τ )) = c(S(p)) + p p2 τ − τ .
Proposition
Let p be an even positive integer and let τ be a non negative integer. Then
W(S(p, τ )) > 0.
M. Delgado
Question of Eliahou — Conjecture of Wilf
Levico Terme, 04/07/2016
16 / 21
Motivation
Every integer is the Eliahou number of infinitely many NS
References
Defining the semigroups S (i) (p, τ )
A visual way to look to S (i) (p, τ )
Consider the image obtained by adding one column to the right of each of the
Bi blocks...
In the case of the block B p2 we add one column to its right part.
M. Delgado
Question of Eliahou — Conjecture of Wilf
Levico Terme, 04/07/2016
17 / 21
Motivation
Every integer is the Eliahou number of infinitely many NS
References
The definition of S (i) (p, τ )
For p and τ as above, we define a sequence of semigroups S (i) (p, τ ), i ≥ 0 as
follows. We take S (0) (p, τ ) = S(p, τ ) and give the following recursive
definition for the numbers involved.
m(i+1) = m(i) + p2 ;
g (i+1) = g (i) + p − 1;
c (i+1) = pm(i) +
p2
2
− τ = c (i) +
p2
2 .
Definition
Now, for i ∈ Z, we define the semigroup S (i) (p, τ ) as the semigroup
hm(i) , g (i) , g (i) + 1ic (i) .
Remark
The sequence S (i) (p, τ ) is infinite.
M. Delgado
Question of Eliahou — Conjecture of Wilf
Levico Terme, 04/07/2016
18 / 21
Motivation
Every integer is the Eliahou number of infinitely many NS
References
One can easily observe that the constants involved in the Eliahou number do
not change when we go from i to i + 1... Thus:
Theorem
Let p be an even positive integer and let τ be a non negative integer. For any
positive integer i, E(S(p, τ )) = E(S (i) (p, τ )).
The S (i) (p, τ ) satisfy Wilf’s conjecture:
Theorem
Let p be an even positive integer and let τ be a non negative integer. Then
W(S (i) (p, τ )) > 0, for every positive integer i.
The proof uses:
Lemma
W(S (i+1) (p, τ )) = W(S (i) (p, τ )) +
p 2 (p 2 +9p+2)
48
Corollary
Given integers n and N, there are infinitely many numerical semigroups S
such that E(S) = n and W(S) > N.
M. Delgado
Question of Eliahou — Conjecture of Wilf
Levico Terme, 04/07/2016
19 / 21
Motivation
Every integer is the Eliahou number of infinitely many NS
References
M. Delgado, “intpic”, a GAP package for drawing integers, Version 0.2.1;
2015. Available via http://www.gap-system.org/.
M. Delgado, On a question of Eliahou and a conjecture of Wilf,
http://arxiv.org/abs/1608.01353.
M. Delgado, P. A. Garcı́a-Sánchez and J. Morais, “NumericalSgps”, a
GAP package for numerical semigroups, Version 1.0.1; 2015. Available via
http://www.gap-system.org/.
The GAP Group, GAP – Groups, Algorithms, and Programming, Version
4.8.2; 2016. Available via http://www.gap-system.org/.
S. Eliahou, Wilf’s conjecture and Macaulay’s theorem, preprint. Available
at http:
//www.ugr.es/~imns2010/2016/preprints/eliahou-imns2016.pdf
J. C. Rosales and P. A. Garcı́a-Sánchez, “Numerical Semigroups”,
Developments in Maths. 20, Springer (2010).
M. Delgado
Question of Eliahou — Conjecture of Wilf
Levico Terme, 04/07/2016
20 / 21
Motivation
Every integer is the Eliahou number of infinitely many NS
References
Grazie
M. Delgado
Question of Eliahou — Conjecture of Wilf
Levico Terme, 04/07/2016
21 / 21