Gr obner Bases a short introduction Elena Dimitrova AIMS, 2019 - PowerPoint PPT Presentation

Gr¨ obner Bases – a short introduction Elena Dimitrova AIMS, 2019 Elena Dimitrova Gr¨ obner Bases – a short introduction AIMS, 2019 1 / 11

Polynomial rings Let k be a field. Let R be the set of all polynomials in variables x 1 , . . . , x n and coefficients in k . R is called a polynomial ring . Definition An ideal I of R is a nonempty subset of R such that for any x , y ∈ I and r ∈ R , x + y and rx are in I . We are mostly interested in finite fields of the form Z p . Theorem Every function f : Z n p → Z p can be represented as a polynomial of degree at most p − 1 . Elena Dimitrova Gr¨ obner Bases – a short introduction AIMS, 2019 2 / 11

Univariate polynomial rings ◮ n = 1 k [ x ] is a PID. I = < g > for some g ∈ k [ x ], unique up to a constant multiple. Question: Is f ∈ I ? Divide f by g : • Remainder is 0 = > yes. • Remainder is not 0 = > no. Elena Dimitrova Gr¨ obner Bases – a short introduction AIMS, 2019 3 / 11

Multivariate polynomial rings ◮ n ≥ 2 k [ x 1 , . . . , x n ] is not a PID. Theorem (Hilbert Basis Theorem) Every ideal I ⊆ k [ x 1 , . . . , x n ] is finitely generated, i.e. I = < f 1 , . . . , f t > for some f i ∈ I. Elena Dimitrova Gr¨ obner Bases – a short introduction AIMS, 2019 4 / 11

Ideal Membership Problem Example k = Z 3 , k [ x , y ] f 1 = y + 1 , f 2 = y 2 + xy , I = < f 1 , f 2 > Is f = y 2 − x ∈ I ? Divide f first by f 1 : y | y 2 − x y + 1 y 2 + y − x − y y 2 � | − x . Does this mean that − x − y is the remainder? I.e., f / ∈ I ?? (BTW, LT ( f 2 ) = y 2 or xy ? LT ( − x − y ) = − x or − y ?) Elena Dimitrova Gr¨ obner Bases – a short introduction AIMS, 2019 5 / 11

Ideal Membership Problem Example k = Z 3 , k [ x , y ] f 1 = y + 1 , f 2 = y 2 + xy , I = < f 1 , f 2 > Is f = y 2 − x ∈ I ? Divide f first by f 2 : 1 − x y 2 + xy | y 2 − x y + 1 |− xy − x y 2 + xy − xy − x − xy − x 0 f = − xf 1 + f 2 , so f ∈ I ! Elena Dimitrova Gr¨ obner Bases – a short introduction AIMS, 2019 6 / 11

Generating sets Problem: { f 1 , f 2 } is not a “nice” generating set for I . “Definition” A nice generating set for an ideal will not do what { f 1 , f 2 } did for I , i.e. ◮ The remainder of division will be the same regardless of the order of division. ◮ Remainder is 0 iff f ∈ I . Theorem (Buchberger, 1965) Nice generating sets exist for any polynomial ideal I � = 0 and are called Gr¨ obner bases (GBs). Notes: I can have multiple GBs because there is no unique way to order the monomials of a multivariate polynomial ( x 2 ≺ xy or x 2 ≻ xy ?) Each I � = 0 has finitely many GBs even for k infinite. Elena Dimitrova Gr¨ obner Bases – a short introduction AIMS, 2019 7 / 11

Monomial orders Definition A monomial order on k [ x 1 , . . . , x n ] is any relation ≺ on Z n ≥ 0 , or equivalently, on the set of monomials x α , α ∈ Z n ≥ 0 , satisfying: 1. ≺ is a total order on Z n ≥ 0 . 2. If α ≺ β and γ ∈ Z n ≥ 0 , then α + γ ≺ β + γ . 3. ≺ is a well-ordering on Z n ≥ 0 (every nonempty subset of Z n ≥ 0 has a smallest element under ≺ ). Examples: The lexicographic monomial order (lex) is analogous to ordering of words in dictionaries, e.g. x 2 y = xxy ≻ xyz . The graded lexicographic order (grlex) uses total degree for comparison and uses lex to breaks ties, e.g. xy 2 z 4 ≻ grlex xyz 5 since both have total degree 7 and xy 2 z 4 ≻ lex xyz 5 . Elena Dimitrova Gr¨ obner Bases – a short introduction AIMS, 2019 8 / 11

Gr¨ obner bases Definition Let I ⊆ k [ x 1 , . . . , x n ] be an ideal other than 0. LT ( I ) = { cx α : there exists f ∈ I with LT ( f ) = cx α } and < LT ( I ) > , called the initial ideal of I , is the ideal generated by the elements of LT ( I ). Definition Fix a monomial order. G = { g 1 , . . . , g t } is a Gr¨ obner basis (GB) of I if < LT ( g 1 ) , . . . , LT ( g t ) > = < LT ( I ) > . Example Recall the example over Z 3 : f 1 = y + 1 , f 2 = xy + y 2 , f = − x + y 2 , I = < f 1 , f 2 > . Let’s use lex. We saw that − xf 1 + f 2 = − x + y 2 = f ∈ I , so LT ( f ) = − x ∈ < LT ( I ) > . However, neither y = LT ( f 1 ) nor xy = LT ( f 2 ) divide − x , so LT ( f ) / ∈ < LT ( f 1 ) , LT ( f 2 ) > . Elena Dimitrova Gr¨ obner Bases – a short introduction AIMS, 2019 9 / 11

Gr¨ obner bases GBs are computable. Algorithms: Buchberger, Buchberger-M¨ oller, Faug` ere’s F4 and F5. Software: Macaulay 2, GAP, CoCoA, Magma, Maple, Mathematica, SINGULAR, Sage, SymPy (Python), AXIOM, REDUCE. “What is a Gr¨ obner basis?” by B. Sturmfels in Notices of the AMS (2005). “Ideals, Varieties, and Algorithms” by Cox, Little, O’Shea. Springer. Elena Dimitrova Gr¨ obner Bases – a short introduction AIMS, 2019 10 / 11

Gr¨ obner bases No uniqueness is guaranteed by the GB existence theorem, not even for a fixed monomial order. To remedy this for a fixed monomial order... Definition Fix a monomial order. A reduced GB for a polynomial ideal I is a GB G for I such that: 1. For all g ∈ G , g is monic. 2. LT ( g ) does not divide any term of any h ∈ G − { g } . Theorem Let I � = 0 . For a given monomial order, I has a unique reduced GB. Note: An ideal can still have multiple reduced GBs for different monomial orders. Elena Dimitrova Gr¨ obner Bases – a short introduction AIMS, 2019 11 / 11