Polyhedral Fan Research Articles

Dimension results for extremal-generic polynomial systems over complete toric varieties

We study polynomial systems with prescribed monomial supports in the Cox ring of a toric variety built from a complete polyhedral fan. We present combinatorial formulas for the dimension of their associated subvarieties under genericity assumptions on the coefficients of the polynomials. Using these formulas, we identify at which degrees generic systems in polytopal algebras form regular sequences. Our motivation comes from sparse elimination theory, where knowing the expected dimension of these subvarieties leads to specialized algorithms and to large speed-ups for solving sparse polynomial systems. As a special case, we classify the degrees at which regular sequences defined by weighted homogeneous polynomials can be found, answering an open question in the Gröbner bases literature. We also show that deciding whether a sparse system is generically a regular sequence in a polytopal algebra is hard from the point of view of theoretical computational complexity.

Tropical Poincaré duality spaces

Abstract The tropical fundamental class of a rational balanced polyhedral fan induces cap products between tropical cohomology and tropical Borel–Moore homology. When all these cap products are isomorphisms, the fan is said to be a tropical Poincaré duality space. If all the stars of faces also are such spaces, such as for fans of matroids, the fan is called a local tropical Poincaré duality space. In this article, we first give some necessary conditions for fans to be tropical Poincaré duality spaces and a classification in dimension one. Next, we prove that tropical Poincaré duality for the stars of all faces of dimension greater than zero and a vanishing condition implies tropical Poincaré duality of the fan. This leads to necessary and sufficient conditions for a fan to be a local tropical Poincaré duality space. Finally, we use such fans to show that certain abstract balanced polyhedral spaces satisfy tropical Poincaré duality.

Real phase structures on matroid fans and matroid orientations

We introduce the notion of real phase structure on rational polyhedral fans in Euclidean space. Such a structure consists of an assignment of affine spaces over $\mathbb{Z}/2\mathbb{Z}$ to each top dimensional face of the fan subject to two conditions. Given an oriented matroid we can construct a real phase structure on the fan of the underlying matroid. Conversely, we show that from a real phase structure on a matroid fan we can produce an orientation of the underlying matroid. Thus real phase structures are cryptomorphic to matroid orientations. The topes of the orientated matroid are recovered immediately from the real phase structure. We also provide a direct way to recover the signed circuits of the oriented matroid from the real phase structure.

Extension of period maps by polyhedral fans

Kato and Usui developed a theory of partial compactifications for quotients of period domains D by arithmetic groups Γ, in an attempt to generalize the toroidal compactifications of Ash-Mumford-Rapoport-Tai to non-classical cases. Their partial compactifications, which aim to fully compactify the images of period maps, rely on the choice of a fan which is strongly compatible with Γ. In particular, they conjectured the existence of a complete fan, which would serve to simultaneously compactify all period maps of a given type.In this article, we briefly review the theory, and construct a fan which compactifies the image of a period map arising from a particular two-parameter family of Calabi-Yau threefolds studied by Hosono and Takagi, with Hodge numbers (1,2,2,1). On the other hand, we disprove the existence of complete fans in some general cases, including the (1,2,2,1) case.

Sublinear circuits and the constrained signomial nonnegativity problem

Conditional Sums-of-AM/GM-Exponentials (conditional SAGE) is a decomposition method to prove nonnegativity of a signomial or polynomial over some subset X of real space. In this article, we undertake the first structural analysis of conditional SAGE signomials for convex sets X. We introduce the X-circuits of a finite subset {mathcal {A}}subset {mathbb {R}}^n, which generalize the simplicial circuits of the affine-linear matroid induced by {mathcal {A}} to a constrained setting. The X-circuits serve as the main tool in our analysis and exhibit particularly rich combinatorial properties for polyhedral X, in which case the set of X-circuits is comprised of one-dimensional cones of suitable polyhedral fans. The framework of X-circuits transparently reveals when an X-nonnegative conditional AM/GM-exponential can in fact be further decomposed as a sum of simpler X-nonnegative signomials. We develop a duality theory for X-circuits with connections to geometry of sets that are convex according to the geometric mean. This theory provides an optimal power cone reconstruction of conditional SAGE signomials when X is polyhedral. In conjunction with a notion of reduced X-circuits, the duality theory facilitates a characterization of the extreme rays of conditional SAGE cones. Since signomials under logarithmic variable substitutions give polynomials, our results also have implications for nonnegative polynomials and polynomial optimization.

The Gröbner fan of the Hilbert scheme

We give a notion of "combinatorial proximity" among strongly stable ideals in a given polynomial ring with a fixed Hilbert polynomial. We show that this notion guarantees "geometric proximity" of the corresponding points in the Hilbert scheme. We define a graph whose vertices correspond to strongly stable ideals and whose edges correspond to pairs of adjacent ideals. Every term order induces an orientation of the edges of the graph. This directed graph describes the behavior of the points of the Hilbert scheme under Gr\"obner degenerations with respect to the given term order. Then, we introduce a polyhedral fan that we call Gr\"obner fan of the Hilbert scheme. Each cone of maximal dimension corresponds to a different directed graph induced by a term order. This fan encodes several properties of the Hilbert scheme. We use these tools to present a new proof of the connectedness of the Hilbert scheme. Finally, we improve the technique introduced in the paper "Double-generic initial ideal and Hilbert scheme" by Bertone, Cioffi and Roggero to give a lower bound on the number of irreducible components of the Hilbert scheme.

Secondary Fans and Secondary Polyhedra of Punctured Riemann Surfaces

A famous construction of Gel'fand, Kapranov and Zelevinsky associates to each finite point configuration a polyhedral fan, which stratifies the space of weight vectors by the combinatorial types of regular subdivisions of A. That fan arises as the normal fan of a convex polytope. In a completely analogous way, we associate to each hyperbolic Riemann surface ℛ with punctures a polyhedral fan. Its cones correspond to the ideal cell decompositions of ℛ that occur as the horocyclic Delaunay decompositions which arise via the convex hull construction of Epstein and Penner. Similar to the classical case, this secondary fan of ℛ turns out to be the normal fan of a convex polyhedron, the secondary polyhedron of ℛ.

Gröbner fans of x-homogeneous ideals in R〚t〛[x

We generalise the notion of a Gröbner fan to ideals in R〚t〛[x1,…,xn] for certain classes of coefficient rings R and give a constructive proof that the Gröbner fan is a rational polyhedral fan. For this we introduce the notion of initially reduced standard bases and show how these can be computed in finite time. We deduce algorithms for computing the Gröbner fan, implemented in the computer algebra system Singular. The problem is motivated by the wish to compute tropical varieties over the p-adic numbers.

Lossy Gossip and Composition of Metrics

We study the monoid generated by $$n \times n$$n×n distance matrices under tropical (or min-plus) multiplication. Using the tropical geometry of the orthogonal group, we prove that this monoid is a finite polyhedral fan of dimension $$\left( {\begin{array}{c}n 2\end{array}}\right) $$n2, and we compute the structure of this fan for $$n$$n up to $$5$$5. The monoid captures gossip among $$n$$n gossipers over lossy phone lines, and contains the gossip monoid over ordinary phone lines as a submonoid. We prove several new results about this submonoid as well. In particular, we establish a sharp bound on chains of calls in each of which someone learns something new.

Classification of normal toric varieties over a valuation ring of rank one

Normal toric varieties over a field or a discrete valuation ring are classified by rational polyhedral fans. We generalize this classification to normal toric varieties over an arbitrary valuation ring of rank one. The proof is based on a generalization of Sumihiro's theorem to this non-noetherian setting. These toric varieties play an important role for tropicalizations.

Distance-Based Phylogenetic Methods Around a Polytomy.

Distance-based phylogenetic algorithms attempt to solve the NP-hard least-squares phylogeny problem by mapping an arbitrary dissimilarity map representing biological data to a tree metric. The set of all dissimilarity maps is a Euclidean space properly containing the space of all tree metrics as a polyhedral fan. Outputs of distance-based tree reconstruction algorithms such as UPGMA and neighbor-joining are points in the maximal cones in the fan. Tree metrics with polytomies lie at the intersections of maximal cones. A phylogenetic algorithm divides the space of all dissimilarity maps into regions based upon which combinatorial tree is reconstructed by the algorithm. Comparison of phylogenetic methods can be done by comparing the geometry of these regions. We use polyhedral geometry to compare the local nature of the subdivisions induced by least-squares phylogeny, UPGMA, and neighbor-joining when the true tree has a single polytomy with exactly four neighbors. Our results suggest that in some circumstances, UPGMA and neighbor-joining poorly match least-squares phylogeny.

Computing tropical linear spaces

We define and study the cyclic Bergman fan of a matroid M, which is a simplicial polyhedral fan supported on the tropical linear space T(M) of M and is amenable to computational purposes. It slightly refines the nested set structure on T(M), and its rays are in bijection with flats of M which are either cyclic flats or singletons. We give a fast algorithm for calculating it, making some computational applications of tropical geometry now viable. Our C++ implementation, called TropLi, and a tool for computing vertices of Newton polytopes of A-discriminants, are both available online.

Tropical hyperelliptic curves

We study the locus of tropical hyperelliptic curves inside the moduli space of tropical curves of genus g. We define a harmonic morphism of metric graphs and prove that a metric graph is hyperelliptic if and only if it admits a harmonic morphism of degree 2 to a metric tree. This generalizes the work of Baker and Norine on combinatorial graphs to the metric case. We then prove that the locus of 2-edge-connected genus g tropical hyperelliptic curves is a (2g−1)-dimensional stacky polyhedral fan whose maximal cells are in bijection with trees on g−1 vertices with maximum valence 3. Finally, we show that the Berkovich skeleton of a classical hyperelliptic plane curve satisfying a certain tropical smoothness condition is a standard ladder of genus g.

Equivariant Chow cohomology of nonsimplicial toric varieties

For a toric variety X_P determined by a rational polyhedral fan P in a lattice N, Payne shows that the equivariant Chow cohomology of X_P is the Sym(N)--algebra C^0(P) of integral piecewise polynomial functions on P. We use the Cartan-Eilenberg spectral sequence to analyze the associated reflexive sheaf on Proj(N), showing that the Chern classes depend on subtle geometry of P and giving criteria for the splitting of the sheaf as a sum of line bundles. For certain fans associated to the reflection arrangement A_n, we describe a connection between C^0(P) and logarithmic vector fields tangent to A_n.

On the spectral side of Arthur's trace formula --- combinatorial setup

In Arthur’s trace formula, a ubiquitous role is played by certain limiting expressions arising from piecewise smooth functions with respect to projections of the Coxeter fan ((G;M)-families). These include terms resulting from intertwining operators on the spectral side and volumes of polytopes on the geometric side. We introduce the combinatorial concept of a compatible family with respect to an arbitrary polyhedral fan and obtain new formulas for the corresponding limiting expressions in this general framework. Our formulas can be regarded as algebraic generalizations of certain volume formulas for convex polytopes. In a companion paper, the results are used to study the spectral side of the trace formula.

The homology of tropical varieties

Given a closed subvarietyX of an algebraic torusT, the associated tropical variety is a polyhedral fan in the space of 1-parameter subgroups of the torus which describes the behaviour of the subvariety at infinity. We show that the link of the origin has only top rational homology if a genericity condition is satisfied.

Three Counter-Examples on Semi-Graphoids

Semi-graphoids are combinatorial structures that arise in statistical learning theory. They are equivalent to convex rank tests and to polyhedral fans that coarsen the reflection arrangement of the symmetric group Sn. In this paper we resolve two problems on semi-graphoids posed in Studený's book (2005), and we answer a related question of Postnikov, Reiner and Williams on generalized permutohedra. We also study the semigroup and the toric ideal associated with semi-graphoids.

Computing Tropical Varieties

The tropical variety of a d-dimensional prime ideal in a polynomial ring with complex coefficients is a pure d-dimensional polyhedral fan. This fan is shown to be connected in codimension one. We present algorithmic tools for computing the tropical variety, and we discuss our implementation of these tools in the Grobner fan software Gfan. Every ideal is shown to have a finite tropical basis, and a sharp lower bound is given for the size of a tropical basis for an ideal of linear forms.

Local Gröbner fans

This Note summarizes results concerning local Gröbner fans. Any Gröbner fan of an ideal in a power series ring, or in an analytic or formal differential operator ring, is found to be a polyhedral fan. We also compare the notions of global and local Gröbner fans, and discuss applications of our results. To cite this article: R. Bahloul, N. Takayama, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

The circuit ideal of a vector configuration

Given a configuration A = { a 1 , … , a n } ⊂ Z d , a basis ideal of A is an ideal J B = 〈 x u + − x u − : u ∈ B 〉 ⊂ k [ x 1 , … , x n ] where B spans the lattice L A = { u ∈ Z n : ∑ a i u i = 0 } . Our main interest is to understand when the toric ideal, I A , of A equals a basis ideal J B with radical I A . The circuit ideal, J C A , of A is an example of such a basis ideal. We study such a J B in relation to I A from various algebraic and combinatorial perspectives with a special focus on J C A . We prove that the obstruction to equality of the ideals is the existence of certain polytopes. This result is based on a complete characterization of the standard pairs/associated primes of a monomial initial ideal of J B and their differences from those for the corresponding toric initial ideal. Eisenbud and Sturmfels proved that the embedded primes of J B are indexed by certain faces of the cone spanned by A . We provide a necessary condition for a particular face to index an embedded prime and a partial converse. Finally, we compare various polyhedral fans associated to I A and J C A . The Gröbner fan of J C A is shown to refine that of I A when the codimension of the ideals is at most two.