Toric Ideals, an Introduction
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1 The 20th National School on Algebra: DISCRETE INVARIANTS IN COMMUTATIVE ALGEBRA AND IN ALGEBRAIC GEOMETRY Mangalia, Romania, September 2-8, 2012 Hara Charalambous Department of Mathematics Aristotle University of Thessalonike
2 Toric ideals Let A = {a 1,, a m } Z n We consider NA := {l 1 a l m a m l i N 0 } Let K be a field Definition The toric ideal I A associated to A is the kernel of the K-algebra homomorphism given by φ : K[x 1,, x m ] K[t 1,, t n, t 1 1 t 1 n ] where a i = (a i,1, a i,2,, a i,n ) φ(x i ) = t a i = t a i,1 1 t a i,2 2 t a i,n n,
3 Example Let A = {2, 1, 1} N φ : K[x 1, x 2, x 3 ] K[t] : x 1 t 2, x 2 t, x 3 t Some elements in the kernel of φ (ie I A ) are: x 1 x 2 2 I A, x 1 x 2 3 I A, x 2 x 3 I A, x 5 2 x 2 1 x 3 I A, etc It is not hard to prove that there are at least two minimal generating sets, (exercise) I A = (x 1 x 2 2, x 2 x 3 ) = (x 1 x 2 3, x 2 x 3 )
4 Example Let A = {(2, 1, 0), (1, 2, 0), (2, 0, 1), (1, 0, 2), (0, 2, 1), (0, 1, 2)} A φ : K[x 1, x 2, x 3, x 4, x 5, x 6 ] K[t 1, t 2, t 3 ] x 1 t 2 1 t 2, x 2 t 1 t 2 2, x 3 t 2 1 t 3, x 4 t 1 t 2 3, x 5 t 2 2 t 3, x 6 t 2 t 2 3 One can prove that I A = (x 1 x 6 x 2 x 4, x 1 x 6 x 3 x 5, x 2 2 x 3 x 2 1 x 5, x 2 x 2 3 x 2 1 x 4, x 1 x 2 5 x 2 2 x 6, x 1 x 2 4 x 2 3 x 6, x 2 4 x 5 x 3 x 2 6, x 1x 4 x 5 x 2 x 3 x 6, x 4 x 2 5 x 2x 2 6 ) Note that the sum of the first and sixth column equals the sum of the second and fourth column, (dependence relation among the columns)
5 Example Definitions A I A = (x 1 x 3 x 2 x 4 ) The columns of the matrix of A correspond to the edges of the following graph: I A is called the toric ideal of the graph
6 Example The corresponding graph is A The toric ideal is 0: I A = 0
7 Properties of toric ideals toric ideals are prime ideals (easy proof) toric ideals are generated by binomials (needs more work initial terms of polynomials) Does the converse hold? How do we compute toric ideals?
8 Grading and Toric ideals A = {a 1,, a m } Z n We grade the polynomial ring K[x 1,, x m ] by the semigroup NA: deg A (x i ) = a i, i = 1,, m For u = (u 1,, u m ) N m and x u := x u 1 1 x u m m deg A (x u ) := u 1 a u m a m NA we let Theorem The toric ideal I A is generated by all the binomials x u x v such that deg A (x u ) = deg A (x v ) If x u x v I A we define deg A (x u x v ) := deg A (x u )
9 Example Example Let A = {2, 1, 1} In φ : K[x 1, x 2, x 3 ] we set deg A (x 1 ) = 2, deg A (x 2 ) = deg A (x 1 ) = 1 I A = (x 1 x 2 2, x 2 x 3 ) deg A (x 1 x 2 2 ) = 2, deg A (x 2 x 3 ) = 1
10 Example Example A I A = (x 1 x 6 x 2 x 4, x 1 x 6 x 3 x 5, x 2 2 x 3 x 2 1 x 5, x 2 x 2 3 x 2 1 x 4, x 1 x 2 5 x 2 2 x 6, x 1 x 2 4 x 2 3 x 6, x 2 4 x 5 x 3 x 2 6, x 1x 4 x 5 x 2 x 3 x 6, x 4 x 2 5 x 2x 2 6 ) deg A (x 1 x 6 x 2 x 4 ) = (2, 2, 2)
11 Generators of Toric Ideals Remark Ịf u Z m then u = u + u where u +, u N m Recall that A = {a 1,, a m } We let π : Z m ZA where (u 1,, u m ) u 1 a u m a m If w, v N m and deg A (w) = deg A (w) then u = w v ker π Theorem I A = x u+ x u : u ker π
12 Example Example Let A = {2, 1, 1} We have seen that I A = (x 1 x 2 2, x 2 x 3 ) Let π : Z 3 ZA where u = (u 1, u 2, u 3 ) 2u 1 + u 2 + u 3 u = (1, 2, 0) ker π and u = (1, 0, 0) (0, 2, 0) Note that another way to get u is as the difference u = (3, 1, 2) (2, 3, 2) The corresponding binomials are x 1 x 2 2 and x 3 1 x 2 x 2 3 x 2 1 x 3 2 x 2 3 I A
13 Important binomials of I A primitive binomials circuit binomials binomials that belong to a reduced Grobner basis of I A binomials that belong to a minimal generating set of I A (Markov binomials) indispensable binomials that belong to all generating sets of I A
14 Graver basis Definition An irreducible binomial x u+ x u I A is called primitive if there exists no other binomial x v+ x v I A such that x v+ divides x u+ and x v divides x u The set of all primitive binomials of a toric ideal I A is called the Graver basis of I A Example We have seen that I = (x 1 x2 2, x 2 x 3 ) is the toric ideal corresponding to A = {2, 1, 1} All elements of the minimal generating set of I are primitive x 2 5 x 1 2x 3 I is not primitive since x2 2 x 1 I
15 Circuits and supp(x u ) := {i x i divides x u } supp(x u x v ) := supp(x u ) supp(x v ) Definition An irreducible binomial B I A is called a circuit if there is no binomial B I A such that supp(b ) supp(b) No two circuits can have the same support, (why?)
16 Example Example A = {1, 2, 4} Then I A = (x1 2 x 2, x1 4 x 3, x2 2 x 3) All 3 elements of this generating set of I A are circuits: supp(x 2 1 x 2) = {1, 2}, supp(x 4 1 x 3) = {1, 3}, supp(x 2 2 x 3) = {2, 3} All 3 elements of this generating set of I A are primitive The element x 3 x1 2x 2 is primitive but is not a circuit
17 Circuits and Primitive binomials What is the relation between Circuits and Primitive binomials? Theorem (B Sturmfels) Circuits are primitive Theorem (B Sturmfels) The set of circuits of I A generate up to radical I A
18 Review on Monomial Orders By T n we denote the set of monomials x a in k[x 1,, x n ], where x a = x a 1 1 x a 2 2 x a n n and a = (a 1, a 2,, a n ) Definition By a monomial order on T n we mean a total order on T n such that 1 < x a for all x a T n with x a 1 If x a < x b then x a x c < x b x c for all x c T n If n 2 then there are infinitely many monomial orders on T n
19 Review on leading monomials: initial terms Let < be a monomial order on k[x 1,, x n ] Let f be a nonzero polynomial in k[x 1,, x n ] We may write f = a 1 x u 1 + a 2 x u 2 + a r x ur, where a i 0 and x u 1 > x u 2 > > x u r If a 1 = 1 we call f monic Definition For f 0 in k[x 1,, x n ], we define the leading monomial of f to be in < (f ) = x u 1
20 Proof that I A is generated by binomials A = {a 1,, a m }, φ : K[x 1,, x m ] K[t 1,, t n, t1 1 tn 1 ], f (x 1,, x m ) f (t a 1,, t a m ) We will show that I A = x u x v : deg A (x u ) = deg A (x v ) Sketch of Proof One direction (RHS LHS) is clear For the other containment fix a monomial order < If LHS RHS, find f I A, f monic, f RHS Choose f so that in < (f ) = x u is minimal among all elements with this property Since f is in I A = ker φ, f must have at least one other term x v, whose image will cancel the image of x u ( so that deg A (x u ) = deg A (x v )) But then f (x u x v ) I A and this gives a contradiction, (why?)
21 Review: Grobner bases Definition A set of non-zero polynomials G = {g 1,, g t } contained in an ideal I, is called Grobner basis for I if and only if for all nonzero f I there exists i {1,, t} such that in < (g i ) divides in < (f ) For any nonzero ideal I and for any term order there exists a Grobner basis for I There are infinitely many Grobner bases for I
22 Reduced Grobner bases Definition A Grobner basis G = {g 1,, g t } is called a reduced Grobner basis for I if The polynomials g i are monic for i {1,, t} and no monomial of g i is divisible by any in < (g j ) for any j i Theorem (Buchberger) Let < be a monomial order on k[x 1,, x n ] and I a nonzero ideal Then I has a unique reduced Grobner basis with respect to the monomial order <
23 Reduced Grobner bases of toric ideals Theorem Let I be a toric ideal and < a monomial order on the underlying ring The reduced Grobner basis of I consists of binomials (Buchberger s algorithm applied to a binomial generating set of I )
24 How to compute a binomial generating set of I A Usually one starts with the generators of an ideal and then finds a Grobner basis of it For the task in question we work in the opposite direction! Suppose that A = {a 1,, a m } N n Consider the ideal J in K[t 1,, t n, x 1,, x m ]: J = x 1 t a 1,, x m t a m Lemma Let φ : K[x 1,, x m ] K[t 1,, t n ] given by x i t a i = t a i,1 1 t a i,2 2 t a i,n n for i = 1, m Then Sketch of proof I A = ker φ = J K[x 1,, x m ]
25 How to compute a binomial generating set of I A ker φ = x 1 t a 1,, x m t am K[x 1,, x m ] (one direction: inclusion RHS to LHS) Let f be an element on the RHS of the equality Then f (x 1,, x m ) = m (x i t a i )h i (t 1,, t n, x 1,, x m ) i=1 We see that f (t a 1,, t a m ) = 0 and f ker φ = I A
26 How to compute a binomial generating set of I A ker φ = x 1 t a 1,, x m t am K[x 1,, x m ] (Other direction: LHS inside the RHS) Let g(x 1,, x m ) = u c ux 1 u1 x m u m ker φ Then g(t a 1,, t am ) = u c ut u 1a1 t umam = 0 Thus in K[t 1,, t n, x 1,, x n ] g(x 1,, x m ) = g(x 1,, x m ) g(t a 1,, t a m ) = u1 u c u (x 1 x m m t u 1a1 t umam ) J u
27 How to compute a binomial generating set of I A Theorem Consider an elimination monomial order in K[t 1,, t n, x 1,, x m ], with t 1,, t n all greater than x 1,, x m Let G be the reduced Grobner basis of J in K[t 1,, t n, x 1,, x m ] Then G K[x 1,, x m ] is a Grobner basis of J K[x 1,, x m ] = I A The above gives the algorithm for finding a reduced Grobner basis of I A when A N n This is a generating set of I A Exercise Find an algorithm to compute I A when A Z n
28 Initial ideals Definition The Universal Grobner basis of an ideal I is the union of all reduced Grobner bases Is this set finite? Let < be a monomial order on k[x 1,, x n ] and I a nonzero ideal The initial ideal of ideal of I is the monomial ideal generated by the leading monomials of any polynomial of I Theorem Let I be a nonzero ideal of k[x 1,, x n ] Then I has finitely many distinct initial ideals
29 Universal Grobner basis of a toric ideal Theorem ( Weispfenning, N Schwartz) The universal Grobner basis of any ideal I is a finite set Let I A be a toric ideal Theorem Ṭhe universal Grobner basis of I A is a finite set of binomials The universal Grobner basis is a finite subset of I A and is a Grobner basis for I A with respect to all term orders simultaneously
30 Circuits, Universal Grobner bases and Primitive polynomials Let UGB A denote the Universal Grobner basis of A Theorem (Sturmfels) For any toric ideal I A the following containments hold: Circuits A UGB A Graver A
31 B Sturmfels, Grobner bases and Convex Polytopes, University Lecture Series, Vol 8, AMS (1995) B Sturmfels, Equations defining toric varieties, Algebraic Geometry - Santa Cruz 1995, Proc Sympos Pure Math, 62, Part 2, Amer Math Soc, Providence, RI, 1997, pp
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