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Gale

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18 September 2007 ArXiV:0706.3745GALE DUALITY FOR COMPLETE INTERSECTIONSFR´ED´ERIC BIHAN AND FRANK SOTTILEAbstract. We show that every complete intersection defined by Laurent polynomials inan algebraic torus is isomorphic to a complete intersection defined by master functions inthe complement of a hyperplane arrangement, and vice versa. We call systems definingsuch isomorphic schemes Gale dual systems because the exponents of the monomials in thepolynomials annihilate the weights of the master functions. We use Gale duality to give aKouchnirenko theorem for the number of solutions to a system of master functions and tocompute some topological invariants of master function complete intersections.IntroductionA complete intersection with support W is a subscheme of the torus (C×)m+nhaving puredimension m that may be defined by a systemf1(x1, . . . , xm+n) = f2(x1, . . . , xm+n) = · · · = fn(x1, . . . , xm+n) = 0of Laurent polynomials with support W.Let p1(y), . . . , pl+m+n(y) be degree 1 polynomials defining an arrangement A of hyper-planes in Cl+mand let β = (b1, . . . , bl+m+n) ∈ Zl+m+nbe a vector of integers. A masterfunction of weight β is the rational functionp(y)β:= p1(y)b1· p2(y)b2· · · pl+m+n(y)bl+m+n,which is defined on the complement MA:= Cl+m\A of the arrangement. A master functioncomplete intersection is a pure subscheme of MAwhich may be defined by a systemp(y)β1= p(y)β2= · · · = p(y)βl= 1of master functions.We describe a correspondence between systems of polynomials defining complete inter-sections and systems of master functions defining complete intersections that we call Galeduality, as the exponent vectors of the monomials in the polynomials and the weights ofthe master functions annihilate each other. There is also a second linear algebraic dualitybetween the degree 1 polynomials piand linear forms defining the polynomials fi. Ourmain result is that the schemes defined by a pair of Gale dual systems are isomorphic. Thisfollows from the simple geometric observation that a complete intersection with support Wis a linear section of the torus in an appropriate projective embedding, and that in turn is a2000 Mathematics Subject Classification. 14M25, 14P25, 52C35.Key words and phrases. sparse polynomial system, hyperplane arrangement, master function, fewnomial,complete intersection.Sottile supported by the Institute for Mathematics and its Applications, NSF CAREER grant DMS-0538734, and Peter Gritzmann of the Technische Universit¨at M¨unchen.12 FR´ED´ERIC BIHAN AND FRANK SOTTILEtorus section of a linear embedding of a hyperplane complement. We explain this geometryin Section 1.In Section 2 we describe this duality concretely in terms of systems of polynomials andsystems of master functions, for this concrete version is how it has been used.The value of this duality is that it allows us to transfer results about solutions to poly-nomial systems to results about solutions to master functions and vice versa. The versionof this valid for positive real-number solutions was used to give a new upper bound on thenumber of positive solutions of a zero-dimensional complete intersection of fewnomials [8],to give a continuation algorithm for finding all real solutions to such a system without alsocomputing all complex solutions [2], and to give a new upper bound on the sum of the Bettinumbers of a fewnomial hypersurface [6]. The version valid for the real numbers leads toa surprising upper bound for the number of real solutions to a system of fewnomials withprimitive exponents [1]. In Section 3, we offer two results about master function completeintersections that follow from well-known results about polynomial systems. The first is ananalog of Kouchnirenko’s bound [3] for the number of points in a zero-dimensional masterfunction complete intersection and the other is a formula for the Euler characteristic of amaster function complete intersection.Another application is afforded by tropical geometry [11]. Each subvariety in the torus(C×)m+nhas an associated tropical variety, which is a fan in Rm+n. Gale duality allowsus to associate certain tropical varieties to master function complete intersections in thecomplement of a hyperplane arrangement. We believe it is an interesting problem to extendthis to arbitrary subvarieties of the hyperplane complement defined by master functions.1. The geometry of Gale dualityLet l, m, and n be nonnegative integers with l, n > 0. We recall the standard geometricformulation of a system of Laurent polynomial in terms of toric varieties, then the lessfamiliar geometry of systems of master functions, and then deduce the geometric version ofGale duality.1.1. Sparse polynomial systems. An integer vector w = (a1, . . . , am+n) ∈ Zm+nis theexponent vector of a monomialxw:= xa11xa22· · · xam+nm+n,which is a function on the torus (C×)m+n. Let W = {w0, w1, . . . , wl+m+n} ⊂ Zm+nbe a setof exponent vectors. A (Laurent) polynomial f with support W is a linear combination ofmonomials with exponents in W,(1.1) f(x) :=l+m+nXi=0cixwiwhere ci∈ C .Acomplete intersection with support W is a subscheme of (C×)m+nof pure dimension nwhich may be defined by a system(1.2) f1(x1, . . . , xm+n) = · · · = fn(x1, . . . , xm+n) = 0GALE DUALITY FOR COMPLETE INTERSECTIONS 3of polynomials with support W. Since multiplying a polynomial f by a monomial doesnot change its zero scheme in (C×)m+n, we will always assume that w0= 0 so that ourpolynomials have a constant term.Consider the homomorphism of algebraic groupsϕW: (C×)m+n−→ {1} × (C×)l+m+n⊂ Pl+m+nx 7−→ (1, xw1, . . . , xwl+m+n) .This map ϕWis dual to the homomorphism of free abelian groups Zl+m+nιW−−→ Zm+nwhichmaps the ith basis element of Zl+m+nto wi. Write ZW for the image, which is the freeabelian subgroup generated by W.The kernel of ϕWis the dual Hom(CW, C×) of the cokernel CW := Zm+n/ZW of themap ιW. The vector configuration W isprimitive when ZW = Zm+n, which is equivalentto the map ϕWbeing a closed immersion.If we let [z0, z1, . . . , zl+m+n] be coordinates for Pl+m+n, then the polynomial f (1.1) equalsϕ∗W(Λ), where Λ is the linear form on Pl+m+n,Λ(z) =l+m+nXi=0cizi.In this way, polynomials on (C×)m+nwith support W are pullbacks of linear forms onPl+m+n. A system (1.2) of such polynomials defines the subscheme ϕ∗W(L), where L ⊂Pl+m+nis the linear space cut out by the forms corresponding to the polynomials fi. Anintersection


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