PDF: Tutorial Handout
Gröbner bases
Toric ideals
Integer programming
Triangulations
Contingency tables
Gröbner bases of toric ideals and their application Hidefumi Ohsugi Kwansei Gakuin University
ISSAC 2014 tutorial, Kobe, July 22, 2014
1 / 75
Gröbner bases
Toric ideals
Integer programming
Triangulations
Contingency tables
Contents Part I. Introduction to Gröbner bases 1
Gröbner bases
Part II. Gröbner bases of toric ideals 2 3 4 5 6
Toric ideals Application to integer programming Application to triangulations of convex polytopes Application to contingency tables (statistics) Quadratic Gröbner bases (if possible)
2 / 75
Gröbner bases
Toric ideals
Integer programming
Triangulations
Contingency tables
Gröbner bases and toric ideals Gröbner bases A “very good” set of polynomials keyword: division of a polynomial (by several polynomials in n variables.) invented by B. Buchberger in 1965. (“standard bases” H. Hironaka in 1964.) Elimination Theorem for systems of polynomial equations implemented in a lot of mathematical software Mathematica, Maple, Macauley2, Singular, CoCoA, Risa/Asir, .... 3 / 75
Gröbner bases
Toric ideals
Integer programming
Triangulations
Contingency tables
Gröbner bases and toric ideals
Toric ideals Prime ideals generated by binomials Gröbner bases of toric ideals have a lot of application commutative algebra, algebraic geometry triangulations of convex polytopes integer programming contingency tables (statistics) ···
4 / 75
Gröbner bases
Toric ideals
Integer programming
Triangulations
Contingency tables
System of linear equations Example f1 = x1 + x3 + 3x4 = 0 f2 = x2 − x3 − 2x4 = 0 f3 = 2x1 + 3x2 − x3 = 0
1 0 1 3 1 0 1 3 1 0 1 3 0 1 −1 2 → 0 1 −1 2 → 0 1 −1 2 2 3 −1 0 0 3 −3 −6 0 0 0 0
f3 = 2f1 + 3f2 . 5 / 75
Gröbner bases
Toric ideals
Integer programming
Triangulations
Contingency tables
Division?
x −1
x +1 ) x2 x 2 −x x x −1 1
For example, which monomial in f = x12 + 2x1 x2 x3 − 3x1 + x35 + 5 shoud be the largest?
6 / 75
Gröbner bases
Toric ideals
Integer programming
Triangulations
Contingency tables
Monomial order Definition Mn : set of all monomials in the variables x1 , . . . , xn A total order < on Mn is called a monomial order if < satisfies the following: 1 u ∈ Mn , u 6= 1 =⇒ 1 < u. 2 u, v , w ∈ Mn , u < v =⇒ uw < vw.
x −1
x +1 ) x2 x 2 −x x x −1 1
−1 + x
−x 2 −x 3 ) x2 x 2 −x 3 x3 x 3 −x 4 x4 .. . 7 / 75
Gröbner bases
Toric ideals
Integer programming
Triangulations
Contingency tables
Lexicographic order Example (Lexicographic order (x1 > · · · > xn )) def
x1a1 x2a2 · · · xnan >lex x1b1 x2b2 · · · xnbn ⇐⇒ a1 > b1 or a1 = b1 and a2 > b2 or a1 = b1 , a2 = b2 , and a3 > b3 or .. . For example, x1 >lex x2100 x3 x12 x22 x5 >lex x12 x2 x3 x4 8 / 75
Gröbner bases
Toric ideals
Integer programming
Triangulations
Contingency tables
(Degree) Reverse lexicographic order Example (Reverse lexicographic order (x1 > · · · > xn )) def
x1a1 x2a2 · · · xnan >revlex x1b1 x2b2 · · · xnbn ⇐⇒ Pn Pn a > i i=1 i=1 bi or Pn Pn and an < bn i=1 ai = i=1 bi or Pn Pn a = b , a and an−1 < bn−1 n = bn i=1 i i=1 i .. . For example, x1 x3 > x4 > x5 ) Gröbner bases of IA with respect to
Toric ideals
Integer programming
Triangulations
Contingency tables
Gröbner bases of toric ideals and their application Hidefumi Ohsugi Kwansei Gakuin University
ISSAC 2014 tutorial, Kobe, July 22, 2014
1 / 75
Gröbner bases
Toric ideals
Integer programming
Triangulations
Contingency tables
Contents Part I. Introduction to Gröbner bases 1
Gröbner bases
Part II. Gröbner bases of toric ideals 2 3 4 5 6
Toric ideals Application to integer programming Application to triangulations of convex polytopes Application to contingency tables (statistics) Quadratic Gröbner bases (if possible)
2 / 75
Gröbner bases
Toric ideals
Integer programming
Triangulations
Contingency tables
Gröbner bases and toric ideals Gröbner bases A “very good” set of polynomials keyword: division of a polynomial (by several polynomials in n variables.) invented by B. Buchberger in 1965. (“standard bases” H. Hironaka in 1964.) Elimination Theorem for systems of polynomial equations implemented in a lot of mathematical software Mathematica, Maple, Macauley2, Singular, CoCoA, Risa/Asir, .... 3 / 75
Gröbner bases
Toric ideals
Integer programming
Triangulations
Contingency tables
Gröbner bases and toric ideals
Toric ideals Prime ideals generated by binomials Gröbner bases of toric ideals have a lot of application commutative algebra, algebraic geometry triangulations of convex polytopes integer programming contingency tables (statistics) ···
4 / 75
Gröbner bases
Toric ideals
Integer programming
Triangulations
Contingency tables
System of linear equations Example f1 = x1 + x3 + 3x4 = 0 f2 = x2 − x3 − 2x4 = 0 f3 = 2x1 + 3x2 − x3 = 0
1 0 1 3 1 0 1 3 1 0 1 3 0 1 −1 2 → 0 1 −1 2 → 0 1 −1 2 2 3 −1 0 0 3 −3 −6 0 0 0 0
f3 = 2f1 + 3f2 . 5 / 75
Gröbner bases
Toric ideals
Integer programming
Triangulations
Contingency tables
Division?
x −1
x +1 ) x2 x 2 −x x x −1 1
For example, which monomial in f = x12 + 2x1 x2 x3 − 3x1 + x35 + 5 shoud be the largest?
6 / 75
Gröbner bases
Toric ideals
Integer programming
Triangulations
Contingency tables
Monomial order Definition Mn : set of all monomials in the variables x1 , . . . , xn A total order < on Mn is called a monomial order if < satisfies the following: 1 u ∈ Mn , u 6= 1 =⇒ 1 < u. 2 u, v , w ∈ Mn , u < v =⇒ uw < vw.
x −1
x +1 ) x2 x 2 −x x x −1 1
−1 + x
−x 2 −x 3 ) x2 x 2 −x 3 x3 x 3 −x 4 x4 .. . 7 / 75
Gröbner bases
Toric ideals
Integer programming
Triangulations
Contingency tables
Lexicographic order Example (Lexicographic order (x1 > · · · > xn )) def
x1a1 x2a2 · · · xnan >lex x1b1 x2b2 · · · xnbn ⇐⇒ a1 > b1 or a1 = b1 and a2 > b2 or a1 = b1 , a2 = b2 , and a3 > b3 or .. . For example, x1 >lex x2100 x3 x12 x22 x5 >lex x12 x2 x3 x4 8 / 75
Gröbner bases
Toric ideals
Integer programming
Triangulations
Contingency tables
(Degree) Reverse lexicographic order Example (Reverse lexicographic order (x1 > · · · > xn )) def
x1a1 x2a2 · · · xnan >revlex x1b1 x2b2 · · · xnbn ⇐⇒ Pn Pn a > i i=1 i=1 bi or Pn Pn and an < bn i=1 ai = i=1 bi or Pn Pn a = b , a and an−1 < bn−1 n = bn i=1 i i=1 i .. . For example, x1 x3 > x4 > x5 ) Gröbner bases of IA with respect to
