Complete intersections¶

Let $S$ be a numerical semigroup minimally generated by $\{n_1,\dots, n_e\}$ with $m=\operatorname{m}(S)$ . The cardinality of a (any) minimal presentation is lowed bounded by $e-1$ and upper bounded by $m(m-1)/2$ .

Numerical semigroups attaining the upper bound are precisely those of maximal embedding dimension.

A numerical semigroup is a complete intersection if its number of minimal relations is $e-1$ .

LoadPackage("num");

true

s:=NumericalSemigroup(4,6,9);;

MinimalPresentation(s);

[ [ [ 0, 0, 2 ], [ 0, 3, 0 ] ], [ [ 0, 2, 0 ], [ 3, 0, 0 ] ] ]

IsCompleteIntersection(s);

true

Let $S_1$ and $S_2$ be numerical semigroups, and let $a_1\in S_2$ and $a_2\in S_1$ (none of them a minimal generator). Then $S=a_1 S_1+a_2 S_2$ is again a numerical semigroup, and if $A_1$ and $A_2$ are the minimal generating systems of $S_1$ and $S_2$ , respectively, $a_1A_1\cup a_2A_2$ is a minimal generating system of $S$ .

In this case, we say that $S$ is a gluing of $S_1$ and $S_2$ .

AsGluingOfNumericalSemigroups(s);

[ [ [ 4 ], [ 6, 9 ] ], [ [ 4, 6 ], [ 9 ] ] ]

JupyterSplashDot(DotTreeOfGluingsOfNumericalSemigroup(s));

Loading...

It can be shown taht $S$ is a complete intersection if it is either $\mathbb{N}$ or is a gluing of two complete intersection numerical semigroups.

A numerical semigroup $S$ is a gluing $S=a_1S_1+a_2S_2$ if and only if it admits a minimal presentation of the form $\sigma=\sigma_1\cup \sigma_2\cup \{(a,b)\}$ , where

$\sigma_1$ only “moves” generators in $a_1A_1$ (it is a presentation of $S_1$ ),
$\sigma_2$ only “moves” generators in $a_2A_2$ (it is a presentation of $S_2$ ),
$a$ is a factorization with support in $a_1A_1$ , and $b$ is a factorization with support in $a_2A_2$ .

\operatorname{Betti}(S)=a_1\operatorname{Betti}(S_1)\cup a_2\operatorname{Betti}(S_2) \cup \{a_1a_2\}.

(1)

Free numerical semigroups¶

A numerical semigroup $S$ minimally generated by $\{n_1,\dots,n_e\}$ is free for this arrangment of its minimal generators if either $\mathbb{N}$ or $S=d\langle n_1/d,\dots,n_{e-1}/d\rangle+ n_e\mathbb{N}$ is a gluing with $d=\gcd\{n_1,\dots,n_{e-1}\}$ and $\langle n_1/d,\dots,n_{e-1}/d\rangle$ is free.

Minimal free numerical semigroups have stair-like minimal presentations.

A numerical semigroup might be free for different arrangments of its minimal generators, but does not have to be free for all the arrangments of its minimal generators.

s:=NumericalSemigroup(4,6,9);;
IsFree(s);

true

IsUniversallyFree(s);

false

s:=NumericalSemigroup(6,10,15);;
IsUniversallyFree(s);

true

Telescopic numerical semigroups¶

A numerical semigroup $S$ minimally generated by $\{n_1<\dots<n_e\}$ is telescopic if it is free for the arrangment $(n_1,\dots,n_e)$ .

s:=NumericalSemigroup(4,6,9);;
IsTelescopic(s);

true

Suppose that $S$ is telescopic and define $d_i=\gcd\{n_1,\dots,n_{i-1}\}$ and $e_i=d_i/d_{i+1}$ . Then $S$ is the semigroup associated to a plane curve singularity if and only if $e_in_i<n_{i+1}$ for all $i$ .

NumSgpsUseSingular();
x:=X(Rationals,"x");; y:=X(Rationals,"y");;
f:= y^4-2*x^3*y^2-4*x^5*y+x^6-x^7;;
s:=SemigroupOfValuesOfPlaneCurve(f);
MinimalGenerators(s);

true

<Numerical semigroup with 3 generators>

[ 4, 6, 13 ]

IsTelescopic(s);

true

IsNumericalSemigroupAssociatedIrreduciblePlanarCurveSingularity(s);

true

# LoadPackage("jupyterviz");
# is:=[0..60];;
# Plot([is,i->Length(CompleteIntersectionNumericalSemigroupsWithFrobeniusNumber(2*i+1))],
# [is, i->Length(FreeNumericalSemigroupsWithFrobeniusNumber(2*i+1))],
# [is, i->Length(TelescopicNumericalSemigroupsWithFrobeniusNumber(2*i+1))]);
Display(List([0..60],i->[i,Length(CompleteIntersectionNumericalSemigroupsWithFrobeniusNumber(2*i+1)),
                    Length(FreeNumericalSemigroupsWithFrobeniusNumber(2*i+1)),
                    Length(TelescopicNumericalSemigroupsWithFrobeniusNumber(2*i+1))]));

[ [    0,    1,    1,    1 ],
  [    1,    1,    1,    1 ],
  [    2,    2,    2,    2 ],
  [    3,    3,    3,    2 ],
  [    4,    2,    2,    2 ],
  [    5,    4,    4,    4 ],
  [    6,    5,    5,    3 ],
  [    7,    3,    3,    2 ],
  [    8,    7,    7,    5 ],
  [    9,    8,    8,    6 ],
  [   10,    5,    5,    4 ],
  [   11,   11,   11,    8 ],
  [   12,   11,   11,    8 ],
  [   13,    9,    9,    7 ],
  [   14,   14,   14,   10 ],
  [   15,   17,   17,    9 ],
  [   16,   12,   12,    8 ],
  [   17,   18,   18,   12 ],
  [   18,   24,   24,   12 ],
  [   19,   16,   16,   11 ],
  [   20,   27,   27,   18 ],
  [   21,   31,   31,   19 ],
  [   22,   21,   21,   13 ],
  [   23,   36,   35,   20 ],
  [   24,   38,   38,   22 ],
  [   25,   27,   27,   16 ],
  [   26,   46,   46,   24 ],
  [   27,   45,   45,   25 ],
  [   28,   34,   33,   20 ],
  [   29,   57,   57,   32 ],
  [   30,   62,   62,   31 ],
  [   31,   43,   43,   25 ],
  [   32,   65,   65,   37 ],
  [   33,   77,   76,   39 ],
  [   34,   53,   52,   29 ],
  [   35,   84,   83,   43 ],
  [   36,   90,   90,   47 ],
  [   37,   61,   61,   37 ],
  [   38,  100,  100,   52 ],
  [   39,  110,  109,   54 ],
  [   40,   80,   79,   47 ],
  [   41,  122,  120,   61 ],
  [   42,  120,  120,   60 ],
  [   43,   94,   94,   48 ],
  [   44,  143,  142,   73 ],
  [   45,  151,  149,   72 ],
  [   46,  108,  106,   57 ],
  [   47,  158,  157,   75 ],
  [   48,  179,  179,   84 ],
  [   49,  128,  128,   68 ],
  [   50,  197,  194,   86 ],
  [   51,  209,  207,   89 ],
  [   52,  142,  142,   76 ],
  [   53,  229,  227,  101 ],
  [   54,  238,  235,  104 ],
  [   55,  172,  169,   83 ],
  [   56,  264,  260,  122 ],
  [   57,  259,  258,  118 ],
  [   58,  202,  200,   96 ],
  [   59,  294,  291,  128 ],
  [   60,  311,  310,  140 ] ]

Apéry sets and gluings¶

Assume that $S=a_1S_1+a_2S_2$ is a gluing of $S_1$ and $S_2$ . Then

\operatorname{Ap}(S,a_1a_2)=a_1\operatorname{Ap}(S_1,a_1)+a_2\operatorname{Ap}(S_2,a_2).

(2)

In particular,

\operatorname{F}(S)=a_1\operatorname{F}(S_1)+a_2\operatorname{F}(S_2)+a_1a_2,

(3)

and

\operatorname{t}(S)=\operatorname{t}(S_1)\operatorname{t}(S_2).

(4)

In particular, the gluing of two symmetric numerical semigroups is symmetric, and every complete intersection is a symmetric numerical semigroup.

Hilbert series (or generating functions) and gluings¶

Let $S$ be a numerical semigroup. The Hilbert series associated to $S$ is the formal series

H_S(x)= \sum_{s\in S} x^s.

(5)

Also, for every $n\in S\setminus\{0\}$

H_S(x)= \frac{1}{1-x^n} \sum_{w\in \operatorname{Ap}(S,n)} x^w.

(6)

Thus, if $S=a_1S_1+a_2S_2$ is a gluing, then

H_S(x) = (1-x^{a_1a_2})H_{S_1}(x^{a_1})H_{S_2}(x^{a_2}).

(7)

Define the polynomial associated to $S$ as

P_S(x) = (1-x)H_S(x)= 1+(1-x)\sum_{g\in \mathbb{N}\setminus S} x^g.

(8)

s:=NumericalSemigroup(4,6,9);;
p:=NumericalSemigroupPolynomial(s,x);

x^12-x^11+x^8-x^7+x^6-x^5+x^4-x+1

Print(p);

x^12-x^11+x^8-x^7+x^6-x^5+x^4-x+1

If $S$ is a complete intersection, then

H_S(x)= \frac{\prod_{b\in \operatorname{Betti}(S)} (1-x^b)^{nrc(b)-1}}{\prod_{i=1}^d (1-x^{n_i})},

(9)

where $nrc(b)$ is the number of $R$ -classes of $b$ . Thus,

P_S(x)=\frac{(1-x)\prod_{b\in \operatorname{Betti}(S)} (1-x^b)^{nrc(b)-1}}{\prod_{i=1}^d (1-x^{n_i})}.

(10)

Since all the roots of $P_S(x)$ are in the unit circle, by Kronecker’s theorem $P_S$ is a product of cyclotomic polynomial.

A numerical semigroup is cyclotomic if $P_S(x)$ is a product of cyclotomic polynomials. Consequently, every complete intersection is cyclotomic.

IsCyclotomicNumericalSemigroup(s);

true

IsKroneckerPolynomial(p);

true

For every numerical semigroup $S$ , the polynomial $P_S(x)$ can be expressed in the following form (uniquely)

P_S(x)= \prod_{d\in \mathbb{N}\setminus\{0\}}(1-x^d)^{e_d}.

(11)

The sequence $(e_1,e_2,\dots)$ is known as the cyclotomic exponent sequence associated to $S$ .

CyclotomicExponentSequence(s,40);

[ 1, 0, 0, -1, 0, -1, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 1 ]

WittCoefficients(p,40);

[ 1, 0, 0, -1, 0, -1, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0,   0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]

The numerical semigroup $S$ is cyclotomic if and only if its cyclotomic exponent sequence has finite support.

s:=NumericalSemigroup(3,5,7);;
p:=NumericalSemigroupPolynomial(s,x);;
WittCoefficients(p,150);

[ 1, 0, -1, 0, -1, 0, -1, 0, 0, 1, 0, 1, 0, 1, 0, 0, -1, 0, -1, 0, 0, 1, 0,   1, 0, 1, -1, 0, -2, 0, -2, 1, -1, 3, 0, 3, -1, 3, -3, 1, -5, 1, -5, 3, -3,   7, -2, 8, -4, 7, -9, 4, -14, 6, -14, 12, -10, 22, -9, 25, -16, 23, -30, 17,   -42, 23, -43, 41, -36, 66, -37, 76, -60, 73, -100, 66, -133, 91, -139, 148,   -129, 219, -146, 252, -222, 252, -340, 255, -438, 346, -469, 524, -473,   731, -564, 846, -820, 887, -1183, 973, -1488, 1309, -1635, 1889, -1756,   2530, -2157, 2947, -3026, 3214, -4181, 3701, -5187, 4922, -5839, 6834,   -6563, 8905, -8200, 10467, -11195, 11807, -14992, 14052, -18463, 18510,   -21237, 24982, -24675, 31960, -31101, 37904, -41573, 43905, -54450, 53343,   -66840, 69606, -78312, 91968, -93176, 116272, -117909, 139142, -155059,   164573, -199918, 202659, -245305, 262345 ]

We still do not know if every cyclotomic polynomial is a complete intersection.

Gluings of numerical semigroups

Complete intersections¶

Free numerical semigroups¶

Telescopic numerical semigroups¶

Apéry sets and gluings¶

Hilbert series (or generating functions) and gluings¶