Toric ideals finitely generated up to symmetry

Transcription

1 Toric ideals finitely generated up to symmetry Anton Leykin Georgia Tech MOCCA, Levico Terme, September 2014 based on [arxiv: ] Noetherianity for infinite-dimensional toric varieties (with Jan Draisma, Rob H. Eggermont, Robert Krone) and [arxiv: ] Equivariant lattice generators and Markov bases (with Thomas Kahle, Robert Krone)

2 Equivariant Gröbner bases (EGBs) Ideals in some -dimensional rings can be represented finitely: K[x 1, x 2,...] with the action of S is Noetherian up to symmetry: every equivariant ideal is generated by orbits of finitely many elements.... still true if S is replaced with the monoid Inc(N) = {monotonous maps N N}. E.g., x 1, x 2,... = x 2014 S = x 1 Inc(N). There is an algorithm to compute an equivariant GB.... or perhaps are represented finitely: K[y ij i, j N] and K[y {i,j} i, j N] with the diagonal action of S are not equivariantly Noetherian. However, if EGB algorithm terminates, then the output is an EGB.

3 Equivariant Gröbner bases (EGBs) Ideals in some -dimensional rings can be represented finitely: K[x 1, x 2,...] with the action of S is Noetherian up to symmetry: every equivariant ideal is generated by orbits of finitely many elements.... still true if S is replaced with the monoid Inc(N) = {monotonous maps N N}. E.g., x 1, x 2,... = x 2014 S = x 1 Inc(N). There is an algorithm to compute an equivariant GB.... or perhaps are represented finitely: K[y ij i, j N] and K[y {i,j} i, j N] with the diagonal action of S are not equivariantly Noetherian. However, if EGB algorithm terminates, then the output is an EGB.

4 Equivariant maps General goal: study families with S n symmetry as n. Let K[Y ] and K[X] be polynomial rings with S -actions. A map φ : K[Y ] K[X] is a S -equivariant map if σφ(f) = φ(σf) for all σ S, f K[Y ]. An ideal I K[Y ] is a S -invariant ideal if σi I for all σ S. φ is S -equivariant ker φ is a S -invariant ideal.

5 Example (Two theorems) [de Loera-Sturmfels-Thomas] Let φ : y {i,j} x i x j for i j N, ker φ = y {1,2} y {3,4} y {1,4} y {3,2} S. [Aoki-Takemura] Let φ : y ij x i x j for i j N, ker φ = y 12 y 34 y 14 y 32, y 12 y 23 y 31 y 21 y 32 y 13 S. Idea of a proof: eliminate (using EGB) in the ring K[x i, y ij i, j N]. Definition: Equivariant Markov basis of φ is a set generating ker φ up to symmetry.

6 Kernel of φ : y ij x 2 i x j Using EGB for elimination get a Markov basis: y 1,3 y 0,2 y 1,2 y 0,3 y 2,0 y 1,0 y 1,2 y 0,2 y 2,1 y 0,1 y 1,2 y 0,2 y 2,3 y 0,1 y 2,1 y 0,3 y 2,3 y 1,0 y 2,0 y 1,3 y 3,1 y 2,0 y 3,0 y 2,1 y 3,2 y 0,1 y 3,1 y 0,2 y 3,2 y 1,0 y 3,0 y 1,2 y 1,2 y 2 0,1 y2 1,0 y 0,2 y 2,0 y 0,1 2 y 1,0 y2 0,2 y 2,1 y 0,2 2 y2 2,0 y 0,1 y 2,1 y 1,0 y 0,2 y 2,0 y 1,2 y 0,1 y 2,1 y 1,0 2 y2 1,2 y 0,1 y 2 2,1 y 0,2 y2 2,0 y 1,2 y 2 2,1 y 1,0 y 2,0 y2 1,2 y 2,1 y 1,0 y 0,3 y 2,0 y 1,3 y 0,1 y 2 2,1 y 0,3 y2 2,0 y 1,3 y 2,3 y 1,2 y 0,2 y 2 2,0 y 1,3 y 3,0 y 1,2 y 0,2 y 2,0 y 1,3 y 0,3 y 3,0 y 1,2 2 y 2,0 y2 1,3 y 3,0 y 2 2,1 y 2,3 y 2,0 y 1,3 y 3,1 y 2 0,2 y 2,1 y2 0,3 y 3,1 y 1,0 y 0,2 y 3,0 y 1,2 y 0,1 y 3,1 y 1,2 y 0,2 y 2,1 y 1,3 y 0,3 y 3,1 y 2,3 y 0,3 y 2 3,0 y 2,1 y 2 3,1 y 0,2 y2 3,0 y 1,2 y 3,2 y 1,3 y 0,3 y 2 3,0 y 1,2 y 3,2 y 2,0 y 1,3 y 3,0 y 2,3 y 1,2 y 3,2 y 2,0 y 1,4 y 3,0 y 2,4 y 1,2 y 3,2 y 2,1 y 0,3 y 3,1 y 2,3 y 0,2 y 3,2 y 2,1 y 0,4 y 3,1 y 2,4 y 0,2 y 4,0 y 2,3 y 1,3 y 3,0 y 2,4 y 1,4 y 4,1 y 2,3 y 0,3 y 3,1 y 2,4 y 0,4 y 4,2 y 1,3 y 0,3 y 3,2 y 1,4 y 0,4 y 4,2 y 2,0 y 1,3 y 4,0 y 2,3 y 1,2 y 4,2 y 2,1 y 0,3 y 4,1 y 2,3 y 0,2 y 2,1 y 1,2 y 0,3 y 0,2 y 2,0 2 y 1,3 y 0,1 y 3,1 y 2,3 y 1,3 y 0,4 y 3,0 2 y 2,1 y 1,4 y 3,1 y 2,3 2 y 0,4 y2 3,0 y 2,4 y 2,1 y 3,2 y 2,3 y 1,3 y 0,4 y 3,0 2 y 2,4 y 1,2 y 4,1 y 2,3 y 1,4 y 0,4 y 4,0 2 y 2,1 y 1,3 y 4,1 y 3,2 y 1,4 y 0,4 y 4,0 2 y 3,1 y 1,2 y 4,1 y 3,4 y 2,4 y 0,5 y 4,0 2 y 3,1 y 2,5 y 2,1 y 1,2 2 y2 0,3 y2 2,0 y2 1,3 y 0,1 y 2,1 y 1,2 2 y 0,4 y 0,3 y2 2,0 y 1,4 y 1,3 y 0,1 y 3,2 y 2,3 2 y 1,4 y 0,4 y2 3,0 y2 2,4 y 1,2 y 3,2 y 2,3 2 y 1,4 y 0,5 y2 3,0 y 2,5 y 2,4 y 1,2 y 4,1 y 2,3 y 1,4 2 y 0,5 y2 4,0 y 2,1 y 1,5 y 1,3 y 4,1 y 3,2 y 1,4 2 y 0,5 y2 4,0 y 3,1 y 1,5 y 1,2 y 4,3 y 4,0 2 y 3,2 y 3,1 y 4,2 y 4,1 y2 3,4 y 0,3 y 5,1 y 4,2 y 3,5 2 y 0,3 y2 5,0 y 4,3 y 3,2 y 3,1 51 generators, width 6, degree 5. (Computation in 4ti2 by Draisma, EquivariantGB in M2 by Krone)

7 Questions Does a finite equivariant Markov basis (or lattice generating set) exist? Can we compute one? Can we find small bases? Size: number of generators. Degree: maximum degree of the generators. Width: largest index value used by the generators. Can we answer these for the one-monomial map φ : K[y ij i j N] K[x 1, x 2,...] where a > b are coprime? y ij x a i x b j

8 Questions Does a finite equivariant Markov basis (or lattice generating set) exist? Can we compute one? Can we find small bases? Size: number of generators. Degree: maximum degree of the generators. Width: largest index value used by the generators. Can we answer these for the one-monomial map φ : K[y ij i j N] K[x 1, x 2,...] where a > b are coprime? y ij x a i x b j

9 Factor φ as φ : K[Y ] π K [ ] z11, z 12,... ψ K[x z 21, z 22,... 1, x 2,...] π : y ij z 1i z 2j ψ : z 1i x a i ψ : z 2i x b i ker φ = ker π + π 1 (im π ker ψ). ker π = y 12 y 34 y 14 y 32, y 12 y 23 y 31 y 21 y 32 y 13 S. (ker π)k[y ± ] = y 12 y 34 y 14 y 32 S. What does im π ker ψ look like?

10 Lattice picture A ψ : M 2 N M 1 N [ ] c11 c 12 [ ac c 21 c bc 21 ac 12 + bc 22 ]. A ψ multiplies matrices by [a b] on the left. Any v ker A ψ must have all columns in the span of [ ] b. a Lemma: im π ker ψ is generated by S -orbits of z C1 z C2 such that [ ] b b 0 C 1 C 2 =. a a 0 z b 11z a 22 z b 12z a 21 generates the lattice.

11 im π ker ψ is generated by S -orbits of z C1 z C2 with the following C 1, C 2 (i) For each 0 n a b, [ ] [ ] b + n n c13 c C 1 = 14 n b + n c13 c, C 0 a c 23 c 24 2 = 14 a 0 c 23 c 24 where j 3 c 1j = a b n and j 3 c 2j = n. (ii) For each 1 n b, [ ] b 0 a b + n 0 C 1 =, C 0 a n 0 2 = (Experiments using 4ti2 and Macaulay2 helped.) [ ] 0 b a b + n 0. a 0 n 0

12 Markov basis formula [Kahle-Krone-L.] A S -equivariant Markov basis for φ is ( for (a, b) = (2, 1) ) (i) y 12 y 34 y 14 y 32 ; (ii) y 12 y 23 y 31 y 21 y 32 y 13 ; (...) (iii) for each 0 n a b, y b+n 12 j 3 y c1j j2 yc2j yb+n 21 j 3 y c1j j1 yc2j 1j where j 3 c 1j = a b n and j 3 c 2j = n; (iv) for each 1 n b, y b n 12 yn 13y a b+n 32 y b n 21 yn 23y a b+n 31. (...) ( y12 y 32 y 21 y 31, n = 0; y12y 2 23 y21y 2 13, n = 1; ( y13 y 2 32 y 23 y ) Size = O((a b) (a b) ), (5) degree = max(a + b, 2a b), (3) width = max(4, a b + 2). (4) )

13 S -equivariant lattice generating set For width = 2, φ : y ij x a i xb j, [Kahle, Krone, L.] Up to symmetry, the lattice generators are y 12y 34 y 14y 32; y21y b a b 31 y12y b a b 32. Note: Size = 2, degree = a, width = 4. These are small! In general, φ : y (i1,...,i k ) x a1 i 1 x a k i k. [Hillar-delCampo] The lattice (equivariant binomial ideal) can be generated in width 2d 1, where d = i a i. [Kahle-Krone-L.] There is a formula for lattice generators with width k + 2 and (k 2 + k 2)/2 elements or width 2k and k 1 elements, Markov bases seem involved for k 3.

14 S -equivariant lattice generating set For width = 2, φ : y ij x a i xb j, [Kahle, Krone, L.] Up to symmetry, the lattice generators are y 12y 34 y 14y 32; y21y b a b 31 y12y b a b 32. Note: Size = 2, degree = a, width = 4. These are small! In general, φ : y (i1,...,i k ) x a1 i 1 x a k i k. [Hillar-delCampo] The lattice (equivariant binomial ideal) can be generated in width 2d 1, where d = i a i. [Kahle-Krone-L.] There is a formula for lattice generators with width k + 2 and (k 2 + k 2)/2 elements or width 2k and k 1 elements, Markov bases seem involved for k 3.

15 Theorem (Draisma, Eggermont, Krone, L.) Any S -equivariant toric map x 11, x 12,... φ : K[Y ] K. x k1, x k2,... where Y has a finite number of S orbits, has a finite S -equivariant Markov basis. There exists an algorithm to construct a Markov basis above... (not implemented!) Implies [de Loera-Sturmfels-Thomas], [Aoki-Takemura], and... [Hillar-Sullivant] Let E be a hypergraph on [k] and let φ E : y (i1,...,i k ) x e,(ij1,...,i js ). e=(j 1,...,j s) E If e has at most one infinite index, a finite equivariant Markov basis.

16 Factoring the map Main idea: factor the map φ : K[Y ] K[X] as φ = ψ π. Let Y = {y p,j p, J}, where p [N], N is the number of S -orbits; J is a k p-tuple of distinct natural numbers. π : K[Y ] K[Z] is given by ψ... captures the rest. y p,j l [k p] z p,l,jl. Want: show that ker π is finitely generated and im π is Noetherian up to symmetry.

17 Matching monoid Generate a monoid by π(y p,j ) = z A, where A p [N] N[kp] N is an [N]-tuple of finite-by- matrices A p [ ] x p φ(y p,(0,1) y p,(2,4) ) φ(y p,(0,4) y p,(2,1) )

18 Proposition: For an N-tuple A p [N] N[kp] N, z A im π iff p [N] the matrix A p N [kp] N has all row sums equal to a number d p N and all column sums d p. We call such A good. Proposition: The Inc(N) divisibility order on the matching monoid (of good A) is a wpo. This settles the Noetherianness of im π.

19 Putting things together Proposition: ker π is generated by binomials in y p,j of degree at most 2 max p k p 1. Recall: ker φ = ker π + π 1 (im π ker ψ). Therefore, ker φ is finitely generated. It is possible to generalize Buchberger s algorithm to K[M] for a monoid M with a wpo... if all ingredients are made effective. One can find an equivariant GB for ker π of the same degree as in Proposition above (w.r.t. a certain monomial order). There exists an algorithm to construct a Markov basis for ker φ.

20 Summary Computing equivariant Markov bases is hard for machines. It is possible to find relatively small bases for some families. [DEKL] main theorem implies Noetherianness up to symmetry for the kernel of a monomial map in a large class of maps with the image in a S -Noetherian ring. What parts of commutative algebra transplant to the -dimensional S -equivariant setting?