Sample of Gröbner bases for toric ideals


Maintained by Olav Geil and Ali Sepas



Let $$a_1, \ldots , a_m \in {\mathbb{Z}}^+ $$ be relatively prime numbers and consider the corresponding numerical semigroup $$\Gamma = \langle a_1, \ldots , a_m\rangle =\{\gamma_1 a_1+ \cdots +\gamma_m a_m \mid \gamma_1, \ldots , \gamma_m \in {\mathbb{N}} \} \subseteq {\mathbb{N}}.$$ We shall assume that $a_1, \ldots , a_m$ are chosen to be the (unique) minimal set generating \( \Gamma \). Given a field \( {\mathbb{F}} \) denote by $$I_\Gamma \subseteq {\mathbb{F}}[X_1, \ldots , X_m]$$ the toric ideal associated with \( \Gamma \), i.e. the kernel of the homomorphism $$\varphi : {\mathbb{F}}[X_1, \ldots ,X_m] \rightarrow {\mathbb{F}}[t]$$ where $$\varphi (X_i ) =t^{a_i}.$$ As is well-known, for any monomial ordering \( \prec \) there exists a reduced Gröbner basis \( {\mathcal{G}} \) for \( I_\Gamma \) consisting of binomials $$ \vec{X}^{\vec{\alpha}}-\vec{X}^{\vec{\beta}}$$ This Gröbner basis then by Buchberger's algorithm is independent of the field under consideration. The number of polynomials (binomials) in \( {\mathcal{G}} \) is always at least \( m-1 \) (recall that \( m \) is the number of elements in the only minimal set of generators of the semigroup).

Motivated by applications in order domain theory and algebraic function field theory we are interested in those cases where \(|{\mathcal{G}}|\) is as close to \(m-1\) as possible. The genus of a numerical semigroup by definition is the number of gaps in it. I.e. the genus \( g \) of a numerical semigroup \( \Gamma \) equals \(g=| {\mathbb{N}} \backslash \Gamma| \). As a beginning for semigroups of small genus and not too many generators we list \( {\mathcal{G}} \) for all possible choices of lexicographic ordering.

Genus = 1

Semigroup Choice of lex-ordering Gröbner basis
\( \{2,3\} \) \( Y \prec X \) \( \{ X^3-Y^2 \} \)
\( X \prec Y \) \( \{ Y^2 - X^3 \}\)


Genus = 2

Semigroup Choice of lex-ordering Gröbner basis
\( \{2,5\} \) \( Y \prec X \) \( \{ X^2-Y^5 \} \)
\( X \prec Y \) \( \{ Y^5- X^2 \}\)


Semigroup Choice of lex-ordering Gröbner basis
\( \{3, 4, 5\} \) \( Z \prec Y \prec X \) \( \{ Y^5-Z^4, XZ-Y^2, XY^3-Z^3, X^3-YZ, X^2Y-Z^2 \} \)
\( Y \prec Z \prec X \) \( \{ Z^4-Y^5, XY^3-Z^3, XZ-Y^2,X^2Y-Z^2,X^3-ZY \}\)
\( Z \prec X \prec Y \) \( \{ X^5-Z^3, YZ-X^3,YX^2-Z^2,Y^2-XZ \}\)
\( X \prec Z \prec Y \) \( \{ Z^3-X^5, YZ-X^3, YX^2-Z^2, Y^2-XZ \}\)
\( Y \prec X \prec Z \) \( \{ X^4-Y^3,ZY-X^3, ZX-Y^2,Z^2-X^2Y \}\)
\( X \prec Y \prec Z \) \( \{ Y^3-X^4, ZX-Y^2,ZY-X^3,Z^2-YX^2 \}\)



Genus = 3

Semigroup Choice of lex-ordering Gröbner basis
\( \{2,7\} \) \( Y \prec X \) \( \{ X^2-Y^7 \} \)
\( X \prec Y \) \( \{ Y^7- X^2 \}\)


Semigroup Choice of lex-ordering Gröbner basis
\( \{3,4\} \) \( Y \prec X \) \( \{ X^4-Y^3 \} \)
\( X \prec Y \) \( \{ Y^3- X^4 \}\)


Semigroup Choice of lex-ordering Gröbner basis
\( \{3, 5, 7\} \) \( Z \prec Y \prec X \) \( \{ Y^7-X^5, XZ-Y^2, XY^5-Z^4, X^2Y^3-Z^3, X^3Y-Z^2,X^4-YZ \} \)
\( Y \prec Z \prec X \) \( \{Z^5-Y^7, XY^5-Z^4, XZ-Y^2, X^2Y^3-Z^3, X^3Y-Z^2, X^4-ZY \}\)
\( Z \prec X \prec Y \) \( \{ X^7-Z^3, YZ-X^4, YX^3-Z^2, Y^2-XZ \}\)
\( X \prec Z \prec Y \) \( \{ Z^3-X^7, YX^3-Z^2,YZ-X^4,Y^2-ZX \}\)
\( Y \prec X \prec Z \) \( \{ X^5-Y^3,ZY-X^4,ZX-Y^2,Z^2-X^3Y \}\)
\( X \prec Y \prec Z \) \( \{Y^3-X^5, ZX-Y^2,ZY-X^4, Z^2-YX^3 \}\)


Semigroup Choice of lex-ordering Gröbner basis
\( \{4, 5, 6, 7\} \) tbd tbd



Genus = 4

Semigroup Choice of lex-ordering Gröbner basis
\( \{2,9\} \) \( Y \prec X \) \( \{ X^2-Y^9 \} \)
\( X \prec Y \) \( \{ Y^9- X^2 \}\)


Semigroup Choice of lex-ordering Gröbner basis
\( \{3,5\} \) \( Y \prec X \) \( \{ X^5-Y^3 \} \)
\( X \prec Y \) \( \{ Y^3- X^5 \}\)


Semigroup Choice of lex-ordering Gröbner basis
\( \{3, 7, 8\} \) \( Z \prec Y \prec X \) \( \{ Y^8 Z^7,XZ^4-Y^5,XY^3-Z^3,X^2Z-Y^2,X^3Y-Z^2,X^5-YZ\} \)
\( Y \prec Z \prec X \) \( \{ Z^7-Y^8,XY^3-Z^3,XZ^4-Y^5,X^2Z_Y^2,X^3Y-Z^2,X^5-ZY \}\)
\( Z \prec X \prec Y \) \( \{ X^8-Z^3,YZ-X^5,YX^3-Z^2,Y^2-X^2Z \}\)
\( X \prec Z \prec Y \) \( \{ Z^3-X^7,YX^3-Z^2,YZ-X^4,Y^2-ZX \}\)
\( Y \prec X \prec Z \) \( \{ X^5-Y^3,ZY-X^4,ZX-Y^2,Z^2-X^3Y \}\)
\( X \prec Y \prec Z \) \( \{ Y^3-X^5,ZX-Y^2,ZY-X^4,Z^2-YX^3 \}\)




Semigroup Choice of lex-ordering Gröbner basis
\( \{4, 5, 6\} \) \( Z \prec Y \prec X \) \( \{Y^6-Z^5,XZ-Y^2,XY^4-Z^4,X^2Y^2-Z^3,X^3-Z^2 \} \)
\( Y \prec Z \prec X \) \( \{ Z^5-Y^6,XY^4-Z^4,XZ-Y^2,X^2Y^2-Z^3,X^3-Z^2 \}\)
\( Z \prec X \prec Y \) \( \{ X^3-Z^2, Y^2-XZ \}\)
\( X \prec Z \prec Y \) \( \{ Z^2-X^3,Y^2-ZX \}\)
\( Y \prec X \prec Z \) \( \{ Y^4-X^5,ZX-Y^2,ZY^2-X^4,Z^2-X^3 \}\)
\( X \prec Y \prec Z \) \( \{ X^5-Y^4,ZY^2-X^4,ZX-Y^2,Z^2-X^3 \}\)




Semigroup Choice of lex-ordering Gröbner basis
\( \{4, 5, 7\} \) \( Z \prec Y \prec X \) \( \{ Y^7-Z^5,XZ^3-Y^5,XY^2-Z^2, X^2Z-Y^3,X^3-YZ \} \)
\( Y \prec Z \prec X \) \( \{X^7-Z^5,XZ^3-Y^5,XY^2-Z^2,X^2 Z-Y^3,X^3-YZ \}\)
\( Z \prec X \prec Y \) \( \{ X^7-Z^4,YZ-X^3,YX^4-Z^3,Y^2X-Z^2,Y^3-X^2Z \}\)
\( X \prec Z \prec Y \) \( \{ Z^4-X^7,YX^4-Z^3,YZ-X^3,Y^2X-Z^2,Y^3-ZX^2 \}\)
\( Y \prec X \prec Z \) \( \{ X^5-Y^4,ZY-X^3,ZX^2-Y^3,Z^2-XY^2 \}\)
\( X \prec Y \prec Z \) \( \{ Y^4-X^5, ZX^2-Y^3, ZY-X^3,Z^2-Y^2X \}\)


Semigroup Choice of lex-ordering Gröbner basis
\( \{5, 6, 7, 8, 9\} \) tbd tbd


Genus = 5

Semigroup Choice of lex-ordering Gröbner basis
\( \{2,11\} \) \( Y \prec X \) \( \{ X^2-Y^{11} \} \)
\( X \prec Y \) \( \{ Y^{11}- X^2 \}\)


Semigroup Choice of lex-ordering Gröbner basis
\( \{3, 7, 11\} \) \( Z \prec Y \prec X \) \( \{ Y^{11}-Z^7,XZ-Y^2,XY^9-Z^6,X^2Y^7-Z^5,X^3Y^5-Z^4,X^4Y^3-Z^3,X^5Y-Z^2,X^6-YZ\} \)
\( Y \prec Z \prec X \) \( \{ X^{11}-Z^3,YZ-X^6,YX^5-Z^2,Y^2-XZ \}\)
\( Z \prec X \prec Y \) \( \{ XY^9-Z^6, XZ-Y^2,X^2Y^7-Z^5,X^3Y^5-Z^4,X^4Y^3-Z^3,X^5Y-Z^2,X^6-ZY \}\)
\( X \prec Z \prec Y \) \( \{ Z^3-X^{11},YX^5-Z^2,YZ-X^6,Y^2-ZX \}\)
\( Y \prec X \prec Z \) \( \{ X^7-Y^3,ZY-X^6,ZX-Y^2,Z^2-X^5Y \}\)
\( X \prec Y \prec Z \) \( \{ Y^3-X^7,ZX-Y^2,ZY-X^6,Z^2-YX^5 \}\)




Semigroup Choice of lex-ordering Gröbner basis
\( \{3, 8, 10\} \) \( Z \prec Y \prec X \) \( \{ Y^5-Z^4, X^2Z-Y^2,X^2Y^3-Z^3, X^4Y-Z^2, X^6-YZ \} \)
\( Y \prec Z \prec X \) \( \{ Z^4-Y^5, X^2Y^3-Z^3, X^2Z-Y^2, X^4Y-Z^2, X^6-ZY \}\)
\( Z \prec X \prec Y \) \( \{ X^{10}-Z^3, YZ-X^6,YX^4-Z^2, Y^2-X^2Z \}\)
\( X \prec Z \prec Y \) \( \{ Z^3-X^{10},YX^4-Z^2, YZ-X^6, Y^2-ZX^2 \}\)
\( Y \prec X \prec Z \) \( \{X^8-Y^3,ZY-X^6,ZX^2-Y^2,Z^2-X^4Y \}\)
\( X \prec Y \prec Z \) \( \{ Y^3-X^8, ZX^2-Y^2, ZY-X^6, Z^2-YX^4 \}\)




Semigroup Choice of lex-ordering Gröbner basis
\( \{4, 6, 7\} \) \( Z \prec Y \prec X \) \( \{ Y^7-Z^6,XZ^2-Y^3,XY^4-Z^4,X^2Y-Z^2,X^3-Y^2 \} \)
\( Y \prec Z \prec X \) \( \{Z^6-Y^7,XY^4-Z^4,XZ^2-Y^3,X^2Y-Z^2,X^3-Y^2 \}\)
\( Z \prec X \prec Y \) \( \{ X^7-Z^4, YZ^2-X^5, YX^2-Z^2,Y^2-X^3 \}\)
\( X \prec Z \prec Y \) \( \{ Z^4-X^7,YX^2-Z^2,YZ^2-X^5,Y^2-X^3 \}\)
\( Y \prec X \prec Z \) \( \{ X^3-Y^2, Z^2-X^2Y \}\)
\( X \prec Y \prec Z \) \( \{ Y^2-X^3,Z^2-YX^2 \}\)


Semigroup Choice of lex-ordering Gröbner basis
\( \{5, 6, 7, 8\} \) tbd tbd


Semigroup Choice of lex-ordering Gröbner basis
\( \{5, 6, 7, 9\} \) tbd tbd


Semigroup Choice of lex-ordering Gröbner basis
\( \{5, 6, 8, 9\} \) tbd tbd


Semigroup Choice of lex-ordering Gröbner basis
\( \{5, 7, 8, 9, 11\} \) tbd tbd


Semigroup Choice of lex-ordering Gröbner basis
\( \{6, 7, 8, 9, 10, 11\} \) tbd tbd




  1. I. Bermejo and I. García-Marco. Gomplete intersections in simplicial toric varieties. Journal of Symbolic Computation, Vol. 68, 2015, pp. 265--286.
  2. J. C. Rosales and P. A. García-Sänchez. "Numerical Semigroups". Developments in Mathematics, Vol. 20. Springer, New Your, 2009.
  3. B. Sturmfels. Gröbner Bases and Convex Polytopes. Univ. Lect. Ser., Vol. 8. American Mathematical Society, Providence, RI, 1996.


The present homepage is a result of a project which grew out of a one week internship where Ali visited Department of Mathematical Sciences, Aalborg University, as part of his teaching obligations in primary school. Ali at the moment (ultimo 2017) is a student at HTX (high school).