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. Then G k[y] is a Gröbner basis of the Theorem ∩ The Extension and elimination ideal I k[y] for the monomial order on k[y] Closure Theorems ∩ Prove induced by >. Extension and Closure Theorems The Extension Proof Theorem The Closure ∑ Theorem f I k[y] has standard representation f = g G Ag g. If An Example ∈ ∩ ∈ Ag = 0, then LT(g) LT(Ag g) LT(f ) k[y], so g G k[y]. Constructible Sets 6 ≤ ≤ ∈ ∈ ∩ References SR Criterion G k[y] is a Gröbner basis of I k[y]. ⇒ ∩ ∩ Partial Solutions

Gröbner Bases Tutorial Given I k[x,y]= k[x1,...,xs,ys+1,...,yn], the elimination David A. Cox ⊆ ideal I k[y] will be denoted Gröbner ∩ Basics Notation and I := I k[y] k[y]. Definitions s Gröbner Bases ∩ ⊆ The Consistency and Finiteness Theorems

Elimination Definition Theory The Elimination The variety of partial solutions is Theorem The Extension and Closure Theorems V(I ) n s. Prove s A − Extension and ⊆ Closure Theorems Question The Extension Theorem n s The Closure How do the partial solutions V(I ) relate to the Theorem s A − n ⊆ An Example original variety V := V(I) A ? Constructible Sets ⊆ References Partial Solutions

Gröbner Bases Tutorial

David A. Cox Given coordinates x1,...,xs,ys+1,...,yn, let

Gröbner n n s Basics πs : A A − Notation and Definitions −→ Gröbner Bases The Consistency and denote projection onto the last n s coordinates. Finiteness Theorems − Elimination Theory An ideal I k[x,y] gives: The Elimination ⊆ Theorem n π n s The Extension and V = V(I) A and s(V ) A − . Closure Theorems ⊆ ⊆ n s Prove Is = I k[y] k[y] and V(Is) A − . Extension and ∩ ⊆ ⊆ Closure Theorems Lemma The Extension Theorem The Closure π Theorem s(V ) V(Is). An Example ⊆ Constructible Sets

References Partial Solutions Don’t Always Extend

Gröbner Bases Tutorial In A3, consider David A. Cox x V = V(xy 1,y z). Gröbner − − Basics Using lex order Notation and Definitions with x > y > z, y Gröbner Bases The Consistency and I = xy 1,y z Finiteness Theorems h − − i has Gröbner basis ← the plane y = z Elimination ↑ Theory xy 1,y z. Thus The Elimination − − ↑ ↓ Theorem I y z , so the z The Extension and 1 = Closure Theorems h − i ↓ partial solutions are ← the solutions Prove ← Extension and the line y = z. The the partial Closure solutions Theorems The partial solution The Extension Theorem (0,0) does not extend. the arrows ↑,↓ The Closure Theorem indicate the An Example projection π Constructible Sets 1

References The Extension Theorem

Gröbner Bases Tutorial n Let I k[x,y ,...,yn]= k[x,y] with variety V = V(I) A , David A. Cox ⊆ 2 ⊆ and let I1 := I k[y] be the first elimination ideal. We Gröbner ∩ Basics assume that k is algebraically closed. Notation and Definitions Gröbner Bases Theorem The Consistency and Finiteness Theorems Let b = (a2,...,an) V(I1) be a partial solution. If the ideal I Elimination ∈ Theory contains a polynomial f such that The Elimination Theorem The Extension and N Closure Theorems f = c(y)x + terms of degree < N in x Prove Extension and Closure with c(b) = 0, then there is a k such that Theorems 6 ∈ The Extension (a,b) = (a,a2,...,an) is a solution, i.e., Theorem The Closure Theorem An Example (a,a2,...,an) V . Constructible Sets ∈ References Zariski Closure

Gröbner Bases Tutorial

David A. Cox Definition Gröbner n Basics Given a subset S A , the Zariski closure of S is the Notation and ⊆ n Definitions smallest variety S A containing S. Gröbner Bases ⊆ The Consistency and Finiteness Theorems

Elimination Theory Lemma The Elimination Theorem n The Extension and The Zariski closure of S A is S = V(I(S)). Closure Theorems ⊆ Prove Extension and Closure Theorems The Extension Example Theorem The Closure n n n n Theorem Over C, the set Z C has Zariski closure Z = C . An Example ⊆ Constructible Sets

References The Closure Theorem

Gröbner n Bases Tutorial Let V = V(I) A and let k be algebraically closed. ⊆ David A. Cox Theorem Gröbner Basics π Notation and V(Is)= s(V ). Definitions Gröbner Bases n s π The Consistency and Thus V(Is) is the smallest variety in A − containing s(V ). Finiteness Theorems

Elimination Furthermore, there is an affine variety Theory The Elimination n s Theorem W V(Is) A − The Extension and Closure Theorems ⊆ ⊆ Prove with the following properties: Extension and Closure Theorems V(Is) W = V(Is). The Extension \ Theorem V(Is) W πs(V ). The Closure Theorem \ ⊆ An Example Constructible Sets Thus “most” partial solutions in V(Is) come from actual References solutions, i.e, the projection of V fills up “most” of V(Is). Prove Extension and Closure Theorems

Gröbner Bases Tutorial

David A. Cox Traditional proofs of the Extension and Closure Theorems use resultants or more abstract methods from algebraic Gröbner Basics geometry. Notation and Definitions Gröbner Bases The Consistency and Recently, Peter Schauenberg wrote Finiteness Theorems Elimination “A Gröbner-based treatment of elimination theory for affine Theory The Elimination varieties” Theorem The Extension and Closure Theorems (Journal of Symbolic Computation, to appear). This paper Prove Extension and uses Gröbner bases to give new proofs of the Extension Closure Theorems and Closure Theorems. The Extension Theorem The Closure Theorem Part I of the tutorial will conclude with these proofs. We An Example Constructible Sets begin with the Extension Theorem. References Prove the Extension Theorem

Gröbner Bases Tutorial n 1 Fix a partial solution b = (a ,...,an) V(I ) A . David A. Cox 2 ∈ 1 ⊂ − Gröbner Notation for the Proof Basics Notation and Definitions Let f k[x,y2,...,yn]= k[x,y]. Gröbner Bases ∈ The Consistency and We write Finiteness Theorems Elimination M Theory f = c(y) x + terms of degree < M in x The Elimination Theorem The Extension and LC|{zx (f}) Closure Theorems

Prove Extension and We set Closure ¯ Theorems f := f (x,b) k[x]. The Extension ∈ Theorem The Closure Theorem An Example Constructible Sets By hypothesis, there is f I with LCx (f ) = 0. References ∈ 6 Lemma

Gröbner Let G be a Gröbner basis of I using that eliminates x. Bases Tutorial > David A. Cox Lemma

Gröbner Basics There is g G with LCx (g) = 0. Notation and ∈ 6 Definitions Gröbner Bases The Consistency and Proof Finiteness Theorems ∑ Elimination Let f = g G Ag g be a standard representation of f I with Theory ∈ ∈ The Elimination LCx (f ) = 0. Thus Theorem 6 The Extension and Closure Theorems LT(f )= max LT(Ag g) Ag = 0 . Prove { | 6 } Extension and Closure Theorems Since > eliminates x, it follows that The Extension Theorem The Closure Theorem degx (f )= max degx (Ag g) Ag = 0 An Example { | 6 } Constructible Sets LCx (f )= ∑ LCx (Ag) LCx (g) References degx (Ag g)=degx (f ) Main Claim

Gröbner Bases Tutorial David A. Cox By the lemma, we can pick g G with LCx (g) = 0 and ∈ 6 M := deg (g) minimal. Note that M = deg (g¯) > 0. Gröbner x x Basics Notation and Definitions Main Claim Gröbner Bases The Consistency and ¯ Finiteness Theorems f f I = g¯ k[x]. { | ∈ } h i⊆ Elimination Theory The Elimination Consequence: If g¯(a)= 0 for some a k, then Theorem ∈ The Extension and Closure Theorems f (a,b)= ¯f (a)= 0 Prove Extension and Closure Theorems for all f I, so (a,b) = (a,a2,...,an) V = V(I). The Extension ∈ ∈ Theorem This proves the Extension Theorem! The Closure Theorem An Example Strategy to prove Main Claim: Show h¯ g¯ for all h G. Constructible Sets ∈ h i ∈ References Claim

Gröbner Bases Tutorial

David A. Cox Consider h G with degx (h) < M = degx (g). M minimal ∈ ¯ Gröbner implies LCx (h)= 0, so degx (h) < degx (h). Basics Notation and Definitions Claim Gröbner Bases The Consistency and ¯ Finiteness Theorems h = 0. Elimination Theory Proof. Let m := deg (h) < M and set The Elimination x Theorem The Extension and M m Closure Theorems S := LCx (g)x − h LCx (h)g I, Prove − ∈ Extension and Closure with standard representation S = ∑` G A` `. Observe: Theorems ∈ The Extension M m ¯ ∑ ¯ Theorem LCx (g)x − h = S = ` G A` ` The Closure ∈ Theorem max deg (A )+ deg (`) A = 0 = deg (S) < M An Example x ` x ` x Constructible Sets { | 6 }

References Prove the Claim

Gröbner Bases Tutorial M m ¯ ¯ LCx (g)x − h = S = ∑` G A` ` David A. Cox ∈ max degx (A`)+ degx (`) = degx (S) < M Gröbner { } Basics First bullet implies: Since LCx (g) = 0 and m = deg (h), Notation and x Definitions 6 Gröbner Bases ¯ ¯ The Consistency and M degx (h)+ degx (h) max degx (A`)+ degx (`) Finiteness Theorems − ≤ { } Elimination Theory so that The Elimination Theorem The Extension and Closure Theorems ¯ ¯ degx (h) degx (h) min M (degx (A`)+ degx (`)) . Prove − ≥ { − } Extension and Closure Second bullet implies: All ` G in S have degx (`) < M, so Theorems ∈ The Extension deg (`¯) < deg (`). Hence Theorem x x The Closure Theorem ¯ An Example degx (A`)+ degx (`) < degx (A`)+ degx (`) < M. Constructible Sets References The two strict inequalities give deg (h) deg (h¯) 2. x − x ≥ Finish the Claim

Gröbner Bases Tutorial The inequality deg (h) deg (h¯) 2 applies to all h G x − x ≥ ∈ David A. Cox with deg (h) < M and hence to all ` G in S = ∑ A `. x ∈ ` ` Gröbner Arguing as before gives Basics Notation and Definitions Gröbner Bases degx (A`)+ degx (`) < degx (A`)+ degx (`) < M, The Consistency and Finiteness Theorems Elimination ↑ Theory drops by at least 2 The Elimination Theorem The Extension and Closure Theorems and Prove Extension and Closure deg (h) deg (h¯) min M (deg (A`)+ deg (`¯)) 3. Theorems x − x ≥ { − x x }≥ The Extension Theorem | {z3 } The Closure Theorem ≥ An Example Constructible Sets Continuing this way, we see that h¯ = 0 for all h G with References ∈ degx (h) < M, as claimed. Prove the Main Claim

Gröbner Proof. For h G, we show h¯ g¯ by induction on deg (h). Bases Tutorial ∈ ∈ h i x David A. Cox Base Case: deg (h) < M implies h¯ = 0 g¯ by Claim. x ∈ h i Gröbner Basics Inductive Step: Assume h¯ g¯ for all h G with Notation and ∈ h i ∈ Definitions degx (h) < m M. Take h G with degx (h)= m. Then Gröbner Bases ≥ ∈ The Consistency and Finiteness Theorems m M S := LC (g)h LC (h)x g I Elimination x x − Theory − ∈ The Elimination Theorem has standard representation S = ∑` G A` `, so The Extension and Closure Theorems ∈

Prove degx (S) < m degx (`) < m ` in S. Extension and ⇒ ∀ Closure Theorems ¯ The Extension By inductive hypothesis, ` g¯ . Hence Theorem ∈ h i The Closure Theorem ¯ m M ∑ ¯ An Example LCx (g)h LCx (h)x − g¯ = S = ` G A` `. Constructible Sets − ∈ References Then LCx (g) = 0 implies h¯ g¯ . 6 ∈ h i The Closure Theorem

Gröbner Bases Tutorial We next give Schauenberg’s proof of the Closure Theorem.

David A. Cox n s Fix a partial solution b = (as+1,...,an) V(Is) A − and a Gröbner ∈ ⊂ Basics Gröbner basis G of I for > that eliminates x = (x1,...,xs). Notation and Definitions Gröbner Bases The Consistency and Notation for the Proof Finiteness Theorems

Elimination Let f k[x1,...,xs,ys+1,...,yn]= k[x,y]. Theory ∈ The Elimination We write Theorem The Extension and Closure Theorems α α f = c(y) x (f ) + terms < x (f ). Prove Extension and Closure |LC{zs(f )} Theorems The Extension Theorem We set The Closure Theorem ¯ An Example f := f (x,b) k[x]. Constructible Sets ∈

References A Special Case

Gröbner Bases Tutorial

David A. Cox Special Case Gröbner Basics If b V(Is) satisfies Notation and ∈ Definitions Gröbner Bases The Consistency and LCs(g) = 0 for all g G k[y], Finiteness Theorems 6 ∈ \ Elimination Theory then b πs(V ). The Elimination ∈ Theorem The Extension and Closure Theorems Proof. If we can find a = (a1,...,as) such that g¯(a)= 0 for Prove all g G, then g(a,b)= 0 for all g G. This implies Extension and ∈ ∈ Closure Theorems The Extension (a,b) V , Theorem ∈ The Closure Theorem π An Example and b s(V ) follows. Constructible Sets ∈ References Prove the Special Case

Gröbner Bases Tutorial Let G = `¯ ` G k[y] . Take g,h G k[y] and set David A. Cox { | ∈ \ } ∈ \ γ γ Gröbner x x Basics S := LCs(g) h LCs(h) g Notation and α(h) α(g) Definitions x − x Gröbner Bases The Consistency and γ α(g) α(h) Finiteness Theorems where x = lcm(x ,x ). A standard representation Elimination ∑ Theory S = ` G A` ` gives The Elimination ∈ Theorem γ γ The Extension and x x Closure Theorems LC (g) h¯ LC (h) g¯ = S = ∑ A `.¯ s α(h) s α(g) `¯ G ` Prove x − x ∈ Extension and Closure Theorems Then LCs(g) = 0 and LCs(h) = 0 imply: The Extension 6 6 Theorem S = ∑ A `¯is a lcm representation. The Closure `¯ G ` Theorem ∈ An Example S is the S-polynomial of g¯ h¯ up to a constant. Constructible Sets ,

References Finish the Special Case

Gröbner Bases Tutorial David A. Cox ∑ ¯ S = `¯ G A` ` is a lcm representation. ∈ Gröbner ¯ Basics S is the S-polynomial of g¯,h up to a nonzero constant. Notation and Definitions Gröbner Bases These tell us that for every The Consistency and Finiteness Theorems ¯ ¯ Elimination g¯,h G = ` ` G k[y] , Theory ∈ { | ∈ \ } The Elimination Theorem ¯ ¯ The Extension and the S-polynomial of g,h has a lcm representation with Closure Theorems respect to G. LCM Criterion G is a Gröbner basis of G . Prove ⇒ h i Extension and Closure Theorems Since LTx (g) = 0 for g G k[y], g¯ is nonconstant for every The Extension 6 ∈ \ Theorem g¯ G. Consistency Theorem V(G) = /0. The Closure ∈ ⇒ 6 Theorem An Example Constructible Sets Hence there exits a such that g¯(a)= 0 for all g G. ∈ References Saturation

Gröbner Fix ideals I,J k[x]. Bases Tutorial ⊆ David A. Cox Definition Gröbner Basics The saturation of I with respect to J is Notation and Definitions Gröbner Bases ∞ M The Consistency and I : J := f k[x] J f I for M 0 . Finiteness Theorems { ∈ | ⊆  } Elimination Theory To see what this means geometrically, let The Elimination Theorem The Extension and Closure Theorems V := V(I) Prove W := V(J). Extension and Closure Theorems The Extension Then: Theorem The Closure Theorem Lemma An Example Constructible Sets ∞ V W = V(I : J ). References \ Proof of the Closure Theorem

Gröbner Proof. The goal is to find a variety W V(Is) such that Bases Tutorial ⊆ David A. Cox V(Is) W πs(V ) and V(Is) W = V(Is). Gröbner \ ⊆ \ Basics Let G be a reduced Gröbner basis of I that eliminates x. Set Notation and Definitions Gröbner Bases ∏ LC The Consistency and J := Is + g G k[y] s(g) . Finiteness Theorems ∈ \ Elimination Theory Then The Elimination Theorem V(J)= [ V(Is) V(LCs(g)). The Extension and ∩ Closure Theorems g G k[y] ∈ \ Prove Extension and Notice that Closure Theorems The Extension b V(Is) V(J) LCs(g) = 0 g G k[y]. Theorem ∈ \ ⇒ 6 ∀ ∈ \ The Closure Theorem By Special Case, b π (V ). Hence An Example s Constructible Sets ∈ References V(Is) V(J) πs(V ). \ ⊆ Two Cases

Gröbner Bases Tutorial Case 1. V(Is) V(LCs(g)) = V(Is) for all g G k[y]. \ ∈ \ David A. Cox Intersecting finitely many open dense sets is dense, so Gröbner Basics Notation and Definitions V(Is) V(J)= Tg G k[y] V(Is) V(LCs(g)) = V(Is). Gröbner Bases \ ∈ \ \ The Consistency and Finiteness Theorems Thus the theorem holds with W = V(J). Elimination Theory The Elimination Theorem Case 2. V(Is) V(LCs(g)) ( V(Is) for some g G k[y]. The Extension and Closure Theorems \ ∈ \ Prove The strategy will be to enlarge I. First suppose that Extension and Closure V(Is) V(LCs(g)). Then: Theorems ⊆ The Extension Theorem LC (g) vanishes on V(I ) and hence on V . Hence The Closure s s Theorem √ An Example Nullstellensatz LCs(g) I. Constructible Sets ⇒ ∈ G reduced LCs(g) / I (since LT(LCs(g)) divides LT(g)). References ⇒ ∈ Finish the Proof

Gröbner Together, these bullets imply that Bases Tutorial

David A. Cox I ( I + LCs(g) √I, Gröbner h i⊂ Basics Notation and so in particular, V(I)= V(I + LCs(g) ). So it suffices to prove Definitions h i Gröbner Bases the theorem for I + LCs(g) when V(Is) V(LCs(g)). The Consistency and h i ⊆ Finiteness Theorems Elimination On the other hand, when V(Is) * V(LCs(g)), we have a union Theory The Elimination of proper subsets Theorem The Extension and Closure Theorems V(Is)= V(Is) V(LCs(g)) (V(Is) V( LCs(g) )) Prove \ ∞ ∪ ∩ h i Extension and V I LC g V I LC g Closure = ( s : s( ) ) ( s + s( ) ) Theorems ∪ h i The Extension Theorem Hence it suffices to prove the theorem for The Closure Theorem An Example ∞ I ( I + Is : LCs(g) and I ( I + LCs(g) . Constructible Sets h i h i References ACC these enlargements occur only finitely often. ⇒ Example

Gröbner Consider A4 with variables w,x,y,z. Let π : A4 A3 be Bases Tutorial 1 → David A. Cox projection onto the last three coordinates. Set 5 4 Gröbner f = x + x + 2 yz Basics − Notation and Definitions and I = (y z)(fw 1),(yw 1)(fw 1) . This defines Gröbner Bases The Consistency and h − − − − i Finiteness Theorems V := V(I)= V(y z,yw 1) V((x 5 +x 4 +2 yz)w 1) A4. Elimination − − ∪ − − ⊆ Theory The Elimination Then: Theorem The Extension and Closure Theorems g ,g = (y z)(fw 1),(yw 1)(fw 1) is a { 1 2} { − − − − } Prove Gröbner basis of I for lex with w > x > y > z. Extension and Closure I1 = 0. Theorems The Extension Furthermore, Theorem The Closure Theorem 5 4 An Example g1 = (y z)(x + x + 2 yz)
w + Constructible Sets − − ··· g = y(x 5 + x 4 + 2 yz) w 2 + References 2 −
··· Continue Example

Gröbner Bases Tutorial As in the proof of the Closure Theorem, let David A. Cox ∏ Gröbner J = I1 + g G k[x,y,z] LC1(g) ∈ \ Basics 5 4 5 4 Notation and = (y z)(x + x + 2 yz) y(x + x + 2 yz) Definitions h − − · − i Gröbner Bases 5 4 2 The Consistency and = y(y z)(x + x + 2 yz) . Finiteness Theorems h − − i Elimination Theory This satisfies Case 1 of the proof, so that The Elimination Theorem The Extension and 3 Closure Theorems V(I1) V(J)= A V(J) π1(V ). Prove \ \ ⊆ Extension and Closure However, Theorems The Extension Theorem π V 3 V x 5 x 4 2 yz The Closure 1( )= A ( + + )
Theorem \ − ∪ An Example 5 4 5 4 Constructible Sets V(y z,x + x + 2 yz) V(x + x + 2,y,z) . − − \
References Constructible Sets

Gröbner Bases Tutorial Definition David A. Cox A set S An is constructible if there are affine varieties Gröbner ⊆ n Basics Wi Vi A , i = 1,...,N, such that Notation and ⊆ ⊆ Definitions Gröbner Bases N The Consistency and Finiteness Theorems S = [ Vi Wi ). Elimination \ Theory i=1 The Elimination Theorem The Extension and Closure Theorems Theorem Prove π n n s Extension and Let k be algebraically closed and s : A A − be Closure → n Theorems projection onto the last n s coordinates. If V A is an The Extension π − n s ⊆ Theorem affine variety, then s(V ) A − is constructible. The Closure ⊆ Theorem An Example Constructible Sets The proof of the Closure Theorem can be adapted to give π References an algorithm for writing s(V ) as a constructible set. References

Gröbner Bases Tutorial

David A. Cox

Gröbner Basics T. Becker, V. Weispfenning, Gröbner Bases, Graduate Notation and Definitions Texts in Mathematics 141, Springer, New York, 1993. Gröbner Bases The Consistency and Finiteness Theorems D. Cox, J. Little, D. O’Shea, Ideals, Varieties and Elimination Theory Algorithms, Third Edition, Undergraduate Texts in The Elimination Theorem Mathematics, Springer, New York, 2007. The Extension and Closure Theorems P. Schauenberg, A Gröbner-based treatment of Prove Extension and elimination theory for affine varieties, Journal of Closure Theorems Symbolic Computation, to appear. The Extension Theorem The Closure Theorem An Example Constructible Sets

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