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Gröbner Bases Tutorial Gröbner Bases Tutorial David A. Cox Gröbner Basics Gröbner Bases Tutorial Notation and Definitions Gröbner Bases Part I: Gröbner Bases and the Geometry of Elimination The Consistency and Finiteness Theorems Elimination Theory The Elimination Theorem David A. Cox The Extension and Closure Theorems Department of Mathematics and Computer Science Prove Extension and Amherst College Closure ¡ ¢ £ ¢ ¤ ¥ ¡ ¦ § ¨ © ¤ ¥ ¨ Theorems The Extension Theorem ISSAC 2007 Tutorial The Closure Theorem An Example Constructible Sets References Outline Gröbner Bases Tutorial 1 Gröbner Basics David A. Cox Notation and Definitions Gröbner Gröbner Bases Basics Notation and The Consistency and Finiteness Theorems Definitions Gröbner Bases The Consistency and 2 Finiteness Theorems Elimination Theory Elimination The Elimination Theorem Theory The Elimination The Extension and Closure Theorems Theorem The Extension and Closure Theorems 3 Prove Extension and Closure Theorems Prove The Extension Theorem Extension and Closure The Closure Theorem Theorems The Extension Theorem An Example The Closure Theorem Constructible Sets An Example Constructible Sets 4 References References Begin Gröbner Basics Gröbner Bases Tutorial David A. Cox k – field (often algebraically closed) Gröbner α α α Basics x = x 1 x n – monomial in x ,...,x Notation and 1 n 1 n Definitions α ··· Gröbner Bases c x , c k – term in x1,...,xn The Consistency and Finiteness Theorems ∈ k[x]= k[x1,...,xn] – polynomial ring in n variables Elimination Theory An = An(k) – n-dimensional affine space over k The Elimination Theorem n The Extension and V(I)= V(f1,...,fs) A – variety of I = f1,...,fs Closure Theorems ⊆ nh i Prove I(V ) k[x] – ideal of the variety V A Extension and ⊆ ⊆ Closure √I = f k[x] m f m I – the radical of I Theorems { ∈ |∃ ∈ } The Extension Theorem The Closure Theorem Recall that I is a radical ideal if I = √I. An Example Constructible Sets References Monomial Orders Gröbner Bases Tutorial David A. Cox Definition Gröbner Basics A monomial order is a total order > on the set of monomials Notation and α Definitions x satisfying: Gröbner Bases α β α γ β γ γ The Consistency and Finiteness Theorems x > x implies x x > x x for all x α α Elimination x > 1 for all x = 1 Theory 6 The Elimination Theorem The Extension and We often think of a monomial order as a total order on the Closure Theorems set of exponent vectors α n. Prove N Extension and ∈ Closure Theorems Lemma The Extension Theorem A monomial order is a well-ordering on the set of all The Closure Theorem monomials. An Example Constructible Sets References Examples of Monomial Orders Gröbner Bases Tutorial David A. Cox Examples α β Gröbner Lex order with x1 > > xn: x >lex x iff Basics ··· Notation and Definitions α β or α β and α β or Gröbner Bases 1 > 1, 1 = 1 2 > 2, ... The Consistency and Finiteness Theorems n Elimination Weighted order using w R+ and lex to break ties: α β ∈ Theory x x iff The Elimination >w,lex Theorem The Extension and Closure Theorems α β w α > w β, or w α = w β and x >lex x Prove · · · · Extension and Closure Theorems Graded lex order with x1 > > xn: This is >w,lex for The Extension ··· Theorem w = (1,...,1) The Closure Theorem An Example Constructible Sets Weighted orders will appear in Part II when we discuss the References Gröbner walk. Leading Terms Gröbner Bases Tutorial David A. Cox Definition Gröbner Fix a monomial order > and let f k[x] be nonzero. Write Basics ∈ Notation and Definitions α Gröbner Bases f = c x + terms with exponent vectors β = α The Consistency and 6 Finiteness Theorems α β β Elimination such that c = 0 and x > x wherever β = α and x Theory 6 6 The Elimination appears in a nonzero term of f . Then: Theorem The Extension and α Closure Theorems LT(f )= c x is the leading term of f Prove α Extension and LM(f )= x is the leading monomial of f Closure Theorems LC(f )= c is the leading coefficient of f The Extension Theorem The Closure Theorem The leading term LT(f ) is sometimes called the initial term, An Example Constructible Sets denoted in(f ). References The Division Algorithm Gröbner Bases Tutorial Division Algorithm David A. Cox Given nonzero polynomials f ,f1,...,fs k[x] and a Gröbner ∈ Basics monomial order >, there exist r,q1,...,qs k[x] with the Notation and ∈ Definitions following properties: Gröbner Bases The Consistency and f = q1 f1 + + qs fs + r. Finiteness Theorems ··· Elimination No term of r is divisible by any of LT(f1),..., LT(fs). Theory The Elimination Theorem LT(f )= max> LT(qi ) LT(fi ) qi = 0 . The Extension and { | 6 } Closure Theorems Prove Extension and Definition Closure Theorems Any representation The Extension Theorem The Closure Theorem f = q1 f1 + + qs fs An Example ··· Constructible Sets References satisfying the third bullet is a standard representation of f . The Ideal of Leading Terms Gröbner Bases Tutorial David A. Cox Definition Gröbner Basics Given an ideal I k[x] and a monomial order >, the ideal of Notation and ⊆ Definitions leading terms is the monomial ideal Gröbner Bases The Consistency and Finiteness Theorems LT(I) := LT(f ) f I . Elimination Theory h i h | ∈ i The Elimination Theorem If I = f1,...,fs , then The Extension and h i Closure Theorems Prove LT(f1),..., LT(fs) LT(I) , Extension and h i ⊆ h i Closure Theorems though equality need not occur. The Extension Theorem The Closure Theorem An Example This is where Gröbner bases enter the picture! Constructible Sets References Gröbner Bases Gröbner Fix a monomial order on k x . Bases Tutorial > [ ] David A. Cox Definition Gröbner Basics Given an ideal I k[x] a finite set G I 0 is a Gröbner Notation and ⊆ ⊆ \ { } Definitions basis for I under > if Gröbner Bases The Consistency and Finiteness Theorems LT(g) g G = LT(I) . Elimination h | ∈ i h i Theory The Elimination Theorem Definition The Extension and Closure Theorems A Gröbner basis G is reduced if for every g G, Prove ∈ Extension and LT Closure (g) divides no term of any element of G g . Theorems \ { } The Extension LC(g)= 1. Theorem The Closure Theorem An Example Theorem Constructible Sets References Every ideal has a unique reduced Gröbner basis under >. Criteria to be a Gröbner Basis Gröbner Bases Tutorial Given > and g,h k[x] 0 , we get the S-polynomial David A. Cox ∈ \ { } γ γ x x γ Gröbner S(g,h) := g h, x = lcm(LM(g), LM(h)). Basics LT(g) − LT(h) Notation and Definitions Gröbner Bases The Consistency and Finiteness Theorems Three Criteria Elimination Theory (SR) G I is a Gröbner basis of I every f I has The Elimination ⊆ ⇐⇒ ∈ Theorem a standard representation using G. The Extension and Closure Theorems (Buchberger) G is a Gröbner basis of G for Prove h i ⇐⇒ Extension and every g,h G, S(g,h) has a standard representation Closure ∈ Theorems using G. The Extension Theorem The Closure (LCM) G is a Gröbner basis of G for every Theorem ∑ h i ⇐⇒ An Example g,h G, S(g,h)= ` G A` `, where A` = 0 implies Constructible Sets ∈ ∈ 6 LT(A` `) < lcm(LM(g), LM(h)) (a lcm representation). References The Consistency Theorem Gröbner Fix an ideal I k[x], where k is algebraically closed. Bases Tutorial ⊆ David A. Cox Nullstellensatz Gröbner Basics (Strong) I(V(I)) = √I. Notation and Definitions (Weak) V(I)= /0 1 I I = k[x]. Gröbner Bases The Consistency and ⇐⇒ ∈ ⇐⇒ Finiteness Theorems Elimination The Consistency Theorem Theory The Elimination Theorem The following are equivalent: The Extension and Closure Theorems I = k[x]. Prove 6 Extension and 1 / I. Closure ∈ Theorems V(I) = /0. The Extension Theorem 6 The Closure I has a Gröbner basis consisting of nonconstant Theorem An Example polynomials. Constructible Sets References I has a reduced Gröbner basis = 1 . 6 { } The Finiteness Theorem Gröbner Fix an ideal I k[x], where k is algebraically closed. Also Bases Tutorial ⊆ David A. Cox fix a monomial order >. Gröbner The Finiteness Theorem Basics Notation and The following are equivalent: Definitions Gröbner Bases n The Consistency and V(I) A is finite. Finiteness Theorems ⊆ Elimination k[x]/I is a finite-dimensional vector space over k. Theory The Elimination I has a Gröbner basis G where i, G has a element Theorem The Extension and ∀ Closure Theorems whose leading monomial is a power of xi . Prove Only finitely many monomials are not in LT(I) . Extension and Closure h i Theorems The Extension When these conditions are satisfied: Theorem The Closure Theorem # solutions dimk k[x]/I. An Example ≤ Constructible Sets Equality holds I is radical. References ⇐⇒ dimk k[x]/I = # solutions counted with multiplicity. Begin Elimination Theory Gröbner Bases Tutorial David A. Cox Given k[x,y]= k[x1,...,xs,ys+1,...,yn], Gröbner Basics α β Notation and we write monomials as x y . Definitions Gröbner Bases The Consistency and Definition Finiteness Theorems Elimination A monomial order > on k[x,y] eliminates x whenever Theory The Elimination Theorem α β α γ β δ The Extension and x > x x y > x y Closure Theorems ⇒ Prove γ δ Extension and for all y ,y . Closure Theorems The Extension Theorem Example The Closure Theorem An Example Lex with x1 > > xn eliminates x = x1,...,xs s. Constructible Sets ··· { }∀ References The Elimination Theorem Gröbner Fix an ideal I k[x,y]. Bases Tutorial ⊆ David A. Cox Definition Gröbner Basics I k[y] is the elimination ideal of I that eliminates x. Notation and ∩ Definitions Gröbner Bases The Consistency and Theorem Finiteness Theorems Elimination Let G be a Gröbner basis of I for a monomial order > that Theory The Elimination eliminates x.
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