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Gr¨

• bner Bases – a short introduction

Elena Dimitrova AIMS, 2019

Elena Dimitrova Gr¨

• bner 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 x1, . . . , xn 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 Zp.

Theorem

Every function f : Zn

p → Zp can be represented as a polynomial of degree

at most p − 1.

Elena Dimitrova Gr¨

• bner 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¨

• bner Bases – a short introduction

AIMS, 2019 3 / 11

Multivariate polynomial rings

◮ n ≥ 2 k[x1, . . . , xn] is not a PID.

Theorem (Hilbert Basis Theorem)

Every ideal I ⊆ k[x1, . . . , xn] is finitely generated, i.e. I =< f1, . . . , ft > for some fi ∈ I.

Elena Dimitrova Gr¨

• bner Bases – a short introduction

AIMS, 2019 4 / 11

Ideal Membership Problem

Example

k = Z3, k[x, y] f1 = y + 1, f2 = y2 + xy, I =< f1, f2 > Is f = y2 − x ∈ I? Divide f first by f1: y y + 1 |y2 − x y2 + y −x − y y2 | − x. Does this mean that −x − y is the remainder? I.e., f / ∈ I?? (BTW, LT(f2) = y2 or xy? LT(−x − y) = −x or −y?)

Elena Dimitrova Gr¨

• bner Bases – a short introduction

AIMS, 2019 5 / 11

Ideal Membership Problem

Example

k = Z3, k[x, y] f1 = y + 1, f2 = y2 + xy, I =< f1, f2 > Is f = y2 − x ∈ I? Divide f first by f2: 1 −x y2 + xy |y2 − x y + 1 |−xy − x y2 + xy −xy − x −xy − x f = −xf1 + f2, so f ∈ I!

Elena Dimitrova Gr¨

• bner Bases – a short introduction

AIMS, 2019 6 / 11

Generating sets

Problem: {f1, f2} is not a “nice” generating set for I. “Definition” A nice generating set for an ideal will not do what {f1, f2} 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¨

• bner bases (GBs).

Notes: I can have multiple GBs because there is no unique way to order the monomials of a multivariate polynomial (x2 ≺ xy or x2 ≻ xy?) Each I = 0 has finitely many GBs even for k infinite.

Elena Dimitrova Gr¨

• bner Bases – a short introduction

AIMS, 2019 7 / 11

Monomial orders

Definition

A monomial order on k[x1, . . . , xn] is any relation ≺ on Zn

≥0, or

equivalently, on the set of monomials xα, α ∈ Zn

≥0, satisfying:

• 1. ≺ is a total order on Zn

≥0.

• 2. If α ≺ β and γ ∈ Zn

≥0, then α + γ ≺ β + γ.

• 3. ≺ is a well-ordering on Zn

≥0 (every nonempty subset of Zn ≥0 has a

smallest element under ≺). Examples: The lexicographic monomial order (lex) is analogous to ordering of words in dictionaries, e.g. x2y = xxy ≻ xyz. The graded lexicographic order (grlex) uses total degree for comparison and uses lex to breaks ties, e.g. xy2z4 ≻grlex xyz5 since both have total degree 7 and xy2z4 ≻lex xyz5.

Elena Dimitrova Gr¨

• bner Bases – a short introduction

AIMS, 2019 8 / 11

Gr¨

• bner bases

Definition

Let I ⊆ k[x1, . . . , xn] be an ideal other than 0. LT(I) = {cxα : there exists f ∈ I with LT(f ) = cxα} 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 = {g1, . . . , gt} is a Gr¨

• bner basis (GB) of I if

< LT(g1), . . . , LT(gt) >=< LT(I) >.

Example

Recall the example over Z3: f1 = y + 1, f2 = xy + y2, f = −x + y2, I =< f1, f2 >. Let’s use lex. We saw that −xf1 + f2 = −x + y2 = f ∈ I, so LT(f ) = −x ∈< LT(I) >. However, neither y = LT(f1) nor xy = LT(f2) divide −x, so LT(f ) / ∈< LT(f1), LT(f2) >.

Elena Dimitrova Gr¨

• bner Bases – a short introduction

AIMS, 2019 9 / 11

Gr¨

• bner bases

GBs are computable. Algorithms: Buchberger, Buchberger-M¨

• ller, Faug`

ere’s F4 and F5. Software: Macaulay 2, GAP, CoCoA, Magma, Maple, Mathematica, SINGULAR, Sage, SymPy (Python), AXIOM, REDUCE. “What is a Gr¨

• bner basis?” by B. Sturmfels in Notices of the AMS (2005).

“Ideals, Varieties, and Algorithms” by Cox, Little, O’Shea. Springer.

Elena Dimitrova Gr¨

• bner Bases – a short introduction

AIMS, 2019 10 / 11

Gr¨

• bner 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

• rders.

Elena Dimitrova Gr¨

• bner Bases – a short introduction

AIMS, 2019 11 / 11