Gröbner Complexes and Tropical Bases

Gröbner Complexes and Tropical Bases Jan Verschelde University of Illinois at Chicago Department of Mathematics, Statistics, and Computer Science http://www.math.uic.edu/ jan jan@math.uic.edu Graduate Computational Algebraic Geometry Seminar Jan Verschelde (UIC) Gröbner Complexes and Tropical Bases 3 April 2014 1 / 25

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 2014 2 / 25

Introduction to Tropical Geometry Introduction to Tropical Geometry is the title of a forthcoming book of Diane Maclagan and Bernd Sturmfels. The web page http://homepages.warwick.ac.uk/staff/d.maclagan/ papers/tropicalbook.html offers the pdf file of a book, dated 31 March 2014. 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 2014 4 / 25

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 0 + 0 x 1 = f x 0 f x 1 1 = f and 0 = f. Jan Verschelde (UIC) Gröbner Complexes and Tropical Bases 3 April 2014 6 / 25

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 2014 7 / 25

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 2014 8 / 25

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 2014 9 / 25

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 2014 10 / 25

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 2014 12 / 25

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 2014 13 / 25

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 2014 14 / 25

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 2014 15 / 25

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 2014 16 / 25

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 2014 17 / 25

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 2014 18 / 25

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 2014 19 / 25

an example f = t x 2 1 + 2x 1x 2 + 3t x 2 2 + 4x 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 2014 20 / 25

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 2014 22 / 25

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 2014 23 / 25

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 2014 24 / 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 2014 25 / 25