Gröbner Complexes and Tropical Bases

Transcription

1 Gröbner Complexes and Tropical Bases Jan Verschelde University of Illinois at Chicago Department of Mathematics, Statistics, and Computer Science jan Graduate Computational Algebraic Geometry Seminar Jan Verschelde (UIC) Gröbner Complexes and Tropical Bases 3 April / 25

2 Gröbner Complexes and Tropical Bases 1 Introduction Introduction to Tropical Geometry 2 Gröbner Bases over Fields with Valuations local and global term orders initial forms of initial forms 3 Gröbner Complexes a universal Gröbner basis defining polyhedra the Gröbner complex 4 Tropical Bases Laurent Polynomials Jan Verschelde (UIC) Gröbner Complexes and Tropical Bases 3 April / 25

3 Gröbner Complexes and Tropical Bases 1 Introduction Introduction to Tropical Geometry 2 Gröbner Bases over Fields with Valuations local and global term orders initial forms of initial forms 3 Gröbner Complexes a universal Gröbner basis defining polyhedra the Gröbner complex 4 Tropical Bases Laurent Polynomials Jan Verschelde (UIC) Gröbner Complexes and Tropical Bases 3 April / 25

4 Introduction to Tropical Geometry Introduction to Tropical Geometry is the title of a forthcoming book of Diane Maclagan and Bernd Sturmfels. The web page papers/tropicalbook.html offers the pdf file of a book, dated 31 March Today we look at some building blocks... This seminar is based on sections 2.4, 2.5, and 2.6. Jan Verschelde (UIC) Gröbner Complexes and Tropical Bases 3 April / 25

5 Gröbner Complexes and Tropical Bases 1 Introduction Introduction to Tropical Geometry 2 Gröbner Bases over Fields with Valuations local and global term orders initial forms of initial forms 3 Gröbner Complexes a universal Gröbner basis defining polyhedra the Gröbner complex 4 Tropical Bases Laurent Polynomials Jan Verschelde (UIC) Gröbner Complexes and Tropical Bases 3 April / 25

6 a Gröbner basis that does not generate the ideal The initial ideal of a homogeneous ideal I in K[x 0, x 1,..., x n ] is in w (I) = in w (f) : f I K[x 0, x 1,..., x n ], K is the residue field. A Gröbner basis for I with respect to w is a finite set G = { g 1, g 2,..., g s } I, with in w (g 1 ), in w (g 2 ),..., in w (g s ) = in w (I). Consider a nonhomogeneous ideal I = x K[x] and w = 1. Then G = {x x 2 } is a Gröbner basis for I, as in (1) (G) = {x} and in (1) (I) = x. However, G does not generate I because x x x 2. There is no polynomial f K[x]: x = (x x 2 )f. x = (x x 2 )f 1 = (1 x)f 1 x x 1 = f x 0 f x 1 1 = f and 0 = f. Jan Verschelde (UIC) Gröbner Complexes and Tropical Bases 3 April / 25

7 local orders and homogeneous ideals The point is that in w is a local, not a global monomial order. Consider (1 x)x = x x 2 x = x x 2 1 x = (x x 2 ) x i. Instead of working with power series, a weak normal form is computed by Mora s normal form algorithm, which leads to standard basis. For homogeneous ideals, Mora s normal form algorithm becomes equal to Buchberger s algorithm to compute a Gröbner basis. Working with homogeneous ideals I over fields with valuations, the definition in w (g 1 ), in w (g 2 ),..., in w (g s ) = in w (I) for a finite set G = { g 1, g 2,..., g s } I gives a basis for I. i=0 Jan Verschelde (UIC) Gröbner Complexes and Tropical Bases 3 April / 25

8 initial forms of initial forms of polynomials The initial form of an initial form is an initial form. Lemma (Lemma 2.4.5) Fix f K[x 0, x 1,..., x n ], w Γ n+1 val, and v Q n+1. There exists an ǫ > 0 such that for all δ Γ val with 0 < δ < ǫ, we have Proof. Let f = in v (in w (f)) = in w+δv (f). u N n+1 c u x u. W = min(val(c u ) + u, w, c u 0). Then: in w (f) = u N n+1 c u t w,u W x u. Let W = min( v, u : val(c u ) + u, w = W). Then: in v (in w (f)) = c u t w,u W x u. v,u =W Jan Verschelde (UIC) Gröbner Complexes and Tropical Bases 3 April / 25

9 proof continued Recall the notations: W W = min(val(c u ) + u, w, c u 0) = trop(f)(w), = min( v, u : val(c u ) + u, w = W). For sufficiently small ǫ > 0, we have: trop(f)(w + ǫv) = min(val(c u ) + w, u + ǫ v, u ) = W + ǫw and {u : val(c u ) + w + δv, u = W + ǫw } = {u : val(c u ) + w, u = W, v, u = W }. This implies in w+δv (f) = in v (in w (f)) for δ Γ val with 0 < δ < ǫ. QED Jan Verschelde (UIC) Gröbner Complexes and Tropical Bases 3 April / 25

10 initial forms of initial forms of ideals Lemma (Lemma 2.4.6) Let I be a homogeneous ideal in K[x 0, x 1,...,x n ] and fix w Γ val. There exists a v Q n+1 and ǫ > 0 such that 1 in v (in w (I)) and in w+ǫv (I) are monomial ideals; and 2 in v (in w (I)) in w+ǫv (I). Corollary (Corollary 2.4.9) Let I be a homogeneous ideal in K[x 0, x 1,...,x n ]. For any w Γ n+1 val and v Q n+1 there exists ǫ > 0 such that in v (in w (I)) = in w+δv (I) for all 0 < δ < ǫ with δv Γ n+1 val. Jan Verschelde (UIC) Gröbner Complexes and Tropical Bases 3 April / 25

11 Gröbner Complexes and Tropical Bases 1 Introduction Introduction to Tropical Geometry 2 Gröbner Bases over Fields with Valuations local and global term orders initial forms of initial forms 3 Gröbner Complexes a universal Gröbner basis defining polyhedra the Gröbner complex 4 Tropical Bases Laurent Polynomials Jan Verschelde (UIC) Gröbner Complexes and Tropical Bases 3 April / 25

12 motivation For homogeneous ideals in K[x 0, x 1,..., x n ] over a field with a valuation we can define Gröbner bases. There is no natural intrinsic notion for Gröbner bases for ideals in the Laurent polynomial ring K[x ±1 ] = K[x ±1 1, x ±1 ±1 2,...,x n ]. The tropical basis is the natural analogue to the notion of a universal Gröbner basis. The goal of section 2.5 is to construct a polyhedral complex from a given homogeneous ideal I K[x 0, x 1,..., x n ]. Jan Verschelde (UIC) Gröbner Complexes and Tropical Bases 3 April / 25

13 a universal Gröbner basis Consider a homogeneous ideal I K[x 0, x 1,...,x n ]. A universal Gröbner basis for I is a finite subset U of I, such that: for all w Γval n+1, in w (U) = { in w (f) : f U } generates in w (I) in K[x 0, x 1,..., x n ]. Example: { x1 (x U = 2 + x 3 x 1 ), x 2 (x 1 + x 3 x 2 ), x 3 (x 1 + x 2 x 3 ), x 1 x 2 (x 1 x 2 ), x 1 x 3 (x 1 x 3 ), x 2 x 3 (x 2 x 3 ) }. is a universal Gröbner basis for I = U = x 1 x 2, x 3 x 1 x 3, x 2 x 2 x 3, x 1. However, I contains x 1 x 2 x 3, a unit in K[x ±1 ], so U is not tropical basis. Jan Verschelde (UIC) Gröbner Complexes and Tropical Bases 3 April / 25

14 defining polyhedra For a homogeneous ideal I K[x 0, x 1,...,x n ] and for w Γ n+1 val we set C I [w] = { v Γ n+1 val : in v (I) = in w (I) }. Let C I [w] be the closure of C I [w] in R n+1 in the Euclidean topology. Consider a Gröbner basis {g 1, g 2,..., g s } of I with respect to w, and let in w (g i ) = x u i, for g i = v N n+1 c i,v x v. If C I [w] has the inequality description { z R n+1 : u i, z val(c i,v ) + v, z, for 1 i s, v N n+1 }, then C I [w] is a Γ val -rational polyhedron. A polyhedron P = { x R n : Ax b } is Γ-rational if A Q d n and b Γ d. Jan Verschelde (UIC) Gröbner Complexes and Tropical Bases 3 April / 25

15 the inequality description The inequality description of C I [w] { z R n+1 : u i, z val(c i,v ) + v, z, for 1 i s, v N n+1 }, is proven in the proof of the following: Proposition (Proposition 2.5.2) The set C I [w] is a Γ-rational polyhedron which contains the line R(1, 1,...,1) as its largest affine subspace. If in w (I) is not a monomial ideal, then there exists v Γval n+1 such that in v (I) is a monomial ideal and C I [w] is a proper face of C I [v]. Jan Verschelde (UIC) Gröbner Complexes and Tropical Bases 3 April / 25

16 properties of C I [w] The lineality space V of a polyhedron P is the largest affine subspace contained in P. We have that x P, v V implies x + v P. Denote by 1 = (1, 1,..., 1) R n+1. Recall that I is homogeneous. The line R1 is the lineality space of C I [w]. Theorem (Theorem 2.5.3) The polyhedra C I [w] as w varies over Γ n+1 val form a Γ val -polyhedral complex inside the n-dimensional space R n+1 /R1. Lemma (Lemma 2.5.4) Let I be a homogeneous ideal in K[x 0, x 1,...,x n ]. There are only finitely many distinct monomial initial ideals in w (I) as w runs over Γ n+1 val. Jan Verschelde (UIC) Gröbner Complexes and Tropical Bases 3 April / 25

17 the polyhedral complex Given a tropical polynomial function F : R n+1 R, we write Σ F for the coarsest polyhedral complex such that F is linear on each cell in Σ F. The maximal cells of the polyhedral complex Σ F have the form σ = { w R n+1 : F(w) = a + w, u } where a x u runs over monomials of F. We have Σ F = R n+1. If the coefficients a in a x u lie in a subgroup Γ R, then the complex Σ F is Γ-rational. A polyhedral complex Σ is Γ-rational if every P Σ is Γ-rational. Jan Verschelde (UIC) Gröbner Complexes and Tropical Bases 3 April / 25

18 fix an arbitrary homogeneous ideal By Lemma 2.5.4, there exists D N such that any initial monomial ideal in w (I) has generators of degree at most D. Let M d be the set of monomials of degree d in K[x 0, x 1,..., x n ]. The coefficients of a basis {f 1, f 2,...,f s } are stored in the matrix A d. The rows of A d are indexed by M d. For J = s, A J d is the s-by-s minor of A d with column indexed by J. We define the polynomial D g := g d, where g d := d=1 J M d J = s det(a J d ) u J x u Theorem (Theorem 2.5.6) If w Γ n+1 lies in the interior of a maximal cell σ Σ trop(g), then σ = C I [w]. Jan Verschelde (UIC) Gröbner Complexes and Tropical Bases 3 April / 25

19 the Gröbner complex Definition (Definition 2.5.7) For a homogeneous ideal I in K[x 0, x 1,..., x n ], the Gröbner complex Σ(I) consists of the polyhedra C I [w] as w ranges over Γ n+1 val. The line R1 is the lineality space of Σ(I). We identify Σ(I) with the quotient complex in R n+1 /R1. R n+1 /R1 is called the tropical projective torus. Points in R n+1 /R1 can be uniquely represented by vectors of the form (0, v 1, v 2,..., v n ). Jan Verschelde (UIC) Gröbner Complexes and Tropical Bases 3 April / 25

20 an example f = t x x 1x 2 + 3t x x 0x 1 + 5x 0 x 2 + 6t x 2 0 C{{t}}[x 0, x 1, x 2 ], I = f. x 2 1 4x 0 x 1 6x 2 0 5x 0 x 2 2x 1 x 2 3x 2 2 The initial ideal in w (I) contains a monomial if and only if the corresponding cell is full dimensional. Jan Verschelde (UIC) Gröbner Complexes and Tropical Bases 3 April / 25

21 Gröbner Complexes and Tropical Bases 1 Introduction Introduction to Tropical Geometry 2 Gröbner Bases over Fields with Valuations local and global term orders initial forms of initial forms 3 Gröbner Complexes a universal Gröbner basis defining polyhedra the Gröbner complex 4 Tropical Bases Laurent Polynomials Jan Verschelde (UIC) Gröbner Complexes and Tropical Bases 3 April / 25

22 Laurent polynomials Consider f K[x ±1 ] = K[x ±1 1, x ±1 ±1 2,...,x n ]. Initial forms and initial ideals are defined as before, however... For generic w: the initial form in w (f) is just one monomial in K[x ±1 ], any monomial in K[x ±1 ] is a unit, and therefore in w (I) = 1 = K[x ±1 ]. Tropical geometry is concerned with the study of those w for which in w (I) is proper in K[x ±1 ]. Jan Verschelde (UIC) Gröbner Complexes and Tropical Bases 3 April / 25

23 computing with Laurent ideals Consider a Laurent ideal I in K[x ±1 ]. The homogenization of I K[x ±1 ] is the ideal I proj K[x 0, x 1,...,x n ] of all polynomials ( x0 m f x1, x 2,..., x ) n, f I, x 0 x 0 x 0 where m is the smallest integer that clears the denominator. To compute in w (I), the weight vectors for the homogeneous ideal I proj live in R n+1 /R1, and we identify R n+1 /R1 with R n, via w (0, w). Jan Verschelde (UIC) Gröbner Complexes and Tropical Bases 3 April / 25

24 initial ideals of Laurent ideals Proposition (Proposition 2.6.2) Let I be an ideal in K[x ±1 ] and fix w Γ n val. Then in w (I) is the image of in (0,w) (I Proj ) obtained by x 0 = 1. Every element of in w (I) has the form x u g where x u is a Laurent monomial and g = f(1, x 1, x 2,..., x n ) for some f in (0,w) (I proj ). Lemma (Lemma 2.6.3) Let I be an ideal in K[x ±1 ] and fix w Γ n val. 1 If g in w (I), then g = in w (h) for some h I. 2 If in v (in w (I)) = in w (I) for some v Z n, then in w (I) is homogeneous with respect to the grading given by deg(x i ) = v i. 3 If f, g K[x ±1 ], then in w (fg) = in w (f)in w (g). Jan Verschelde (UIC) Gröbner Complexes and Tropical Bases 3 April / 25

25 tropical basis Let I be an ideal in the Laurent polynomial ring K[x ±1 ] over a field K with a valuation. A finite generating set T of I is a tropical basis if for all w Γ n val, in w (I) contains a unit in w (T ) = { in w (f) : f T } contains a unit. Theorem (Theorem 2.6.5) Every ideal I in K[x ±1 ] has a finite tropical basis. Jan Verschelde (UIC) Gröbner Complexes and Tropical Bases 3 April / 25