On a tropical dual Nullstellensatz

Since a tropical Nullstellensatz fails even for tropical univariate polynomials we study a conjecture on a tropical dual Nullstellensatz for tropical polynomial systems in terms of solvability of a tropical linear system with the Cayley matrix associated to the tropical polynomial system. The conjecture on a tropical effective dual Nullstellensatz is proved for tropical univariate polynomials.     

Note on a conjecture of toft

A conjecture of Toft [17] asserts that any 4-critical graph (or equivalently, every 4-chromatic graph) contains a fully odd subdivision ofK ₄. We show that if a graphG has a degree three nodev such thatG-v is 3-colourable, then eitherG is 3-colourable or it contains a fully oddK ₄. This resolves Toft's conjecture in the special case where a 4-critical graph has a degree three node, which is in turn used to prove the conjecture for line-graphs. The proof is constructive and yields a polynomial algorithm which given a 3-degenerate graph either finds a 3-colouring or exhibits a subgraph that is a fully odd subdivision ofK ₄. (A graph is 3-degenerate if every subgraph has some node of degree at most three.)     

On variance of exponents for isolated surface singularities with modality ⩽ 2

Using the theory of the mixed Hodge structure one can define a notion of exponents of a singularity.In 2000,Hertling proposed a conjecture about the variance of the exponents of a singularity.Here,we prove that the Hertling conjecture is true for isolated surface singularities with modality ≤ 2.     

Melham's Conjecture on Odd Power Sums of Fibonacci Numbers

Ozeki and Prodinger showed that the odd power sum of the first several consecutive Fibonacci numbers of even order is equal to a polynomial evaluated at a certain Fibonacci number of odd order. We prove that this polynomial and its derivative both vanish at 1, and will be an integer polynomial after multiplying it by a product of the first consecutive Lucas numbers of odd order. This presents an a?rmative answer to a conjecture of Melham.     

Ehrhart polynomials of cyclic polytopes

The Ehrhart polynomial of an integral convex polytope counts the number of lattice points in dilates of the polytope. In (Coefficients and roots of Ehrhart polynomials, preprint), the authors conjectured that for any cyclic polytope with integral parameters, the Ehrhart polynomial of it is equal to its volume plus the Ehrhart polynomial of its lower envelope and proved the case when the dimension d=2. In our article, we prove the conjecture for any dimension.     

Spanning trees and a conjecture of Kontsevich

Kontsevich conjectured that the number of zeros over the fieldF _q of a certain polynomialQ _G associated with the spanning trees of a graphG is a polynomial function ofq. We show the connection between this conjecture, the Matrix-Tree Theorem, and orthogonal geometry. We verify the conjecture in certain cases, such as the complete graph, and discuss some modifications and extensions.Partially supported by NSF grant #DMS-9743966.     

Chip firing and the tutte polynomial

It is shown that the generating function of critical configurations of a version of a chip firing game on a graphG is an evaluation of the Tutte polynomial ofG, thus proving a conjecture of Biggs [3]. Supported by a grant from D.G.A.P.A.     

Measuring the closeness to singularities of a planar parallel manipulator using geometric algebra

A new index for measuring the closeness to the singularities of parallel manipulators using geometric algebra is proposed in this paper. Constraint wrenches acting on the moving platform of a parallel manipulator are derived using the outer product and dual operations. Removing the redundant constraint wrenches, a singularity polynomial is obtained when the coefficient of the outer product of all the non-redundant constraint wrenches equals zero. A singularity surface can be drawn using the singularity polynomial. Similarly, an approximate singularity polynomial and approximate singularity surface can be obtained by imposing a threshold to the singular polynomial. Then the singularity volume is calculated as the space between singularity surface and approximate singularity surface. The new index is derived by calculating the ratio of the non-singularity workspace volume (the workspace volume minus the singularity volume) to the workspace volume. The proposed index is coordinate-free and has a clear geometrical and physical interpretation. This index can be a basis for selecting structural parameters, path planning and mechanism design.     

Some results on the spectral reconstruction problem

This paper is concerned with the spectral version of the reconstruction conjecture: Whether a graph with n>2 vertices is determined (up to isomorphism) by the collection of its spectrum and the spectrum of its vertex-deleted graphs? Some positive results as well as a method for constructing counterexamples to the problem are provided.     

Generalizations and applications of the nowhere zero linear mappings in network coding

In [2], Alon and Tarsi proposed a conjecture about the nowhere-zero point in linear mappings. In this paper, we first study some generalizations of this problem, and obtain necessary and sufficient conditions for the existence of nowhere point in these generalized problems under the assumption |F|?n+2, where n is the number of rows of the matrix A. Then we apply the results in these generalizations to give a polynomial time algebraic construction of the acyclic network codings.     

Cyclic orders: Equivalence and duality

Cyclic orders of graphs and their equivalence have been promoted by Bessy and Thomassé’s recent proof of Gallai’s conjecture. We explore this notion further: we prove that two cyclic orders are equivalent if and only if the winding number of every circuit is the same in the two. The proof is short and provides a good characterization and a polynomial algorithm for deciding whether two orders are equivalent. We then derive short proofs of Gallai’s conjecture and a theorem “polar to” the main result of Bessy and Thomassé, using the duality theorem of linear programming, total unimodularity, and the new result on the equivalence of cyclic orders.     

Chern classes of reflexive sheaves on normal surfaces

We define Chern classes of reflexive sheaves using Wahl's relative local Chern classes of vector bundles. The main result of the paper bounds contributions of singularities of a sheaf to the Riemann–Roch formula. Using it we are able to prove inequality in Wahl's conjecture on relative asymptotic RR formula for rank 2 vector bundles. Moreover, we prove that if Wahl's conjecture is true for a singularity then it is true for any its quotient. This implies Wahl's conjecture for quotient singularities and for quotients of cones over elliptic curves. Received March 2, 1998; in final form March 24, 1999 / Published online September 14, 2000     

The 16th Hilbert problem on algebraic limit cycles

For real planar polynomial differential systems there appeared a simple version of the 16th Hilbert problem on algebraic limit cycles: Is there an upper bound on the number of algebraic limit cycles of all polynomial vector fields of degree m? In [J. Llibre, R. Ramírez, N. Sadovskaia, On the 16th Hilbert problem for algebraic limit cycles, J. Differential Equations 248 (2010) 1401-1409] Llibre, Ramírez and Sadovskaia solved the problem, providing an exact upper bound, in the case of invariant algebraic curves generic for the vector fields, and they posed the following conjecture: Is1+(m−1)(m−2)/2the maximal number of algebraic limit cycles that a polynomial vector field of degree m can have?In this paper we will prove this conjecture for planar polynomial vector fields having only nodal invariant algebraic curves. This result includes the Llibre et al.?s as a special one. For the polynomial vector fields having only non-dicritical invariant algebraic curves we answer the simple version of the 16th Hilbert problem.     

Ehrhart polynomials of polytopes and spectrum at infinity of Laurent polynomials

Journal of Algebraic Combinatorics - Gathering different results from singularity theory, geometry and combinatorics, we show that the spectrum at infinity of a tame Laurent polynomial counts...     

Some phases of oscillatory integrals

The first example of a phase is presented for which Arhold’s conjecture on the validity of uniform estimates for oscillatory integrals with maximal singularity index is true, while his conjecture on the semicontinuity of the singularity index is false. A rough upper bound for the Milnor number such that the latter conjecture fails is obtained. The corresponding counterexample is simpler than Varchenko’s well-known counterexample to Arnold’s conjecture on the semicontinuity of the singularity index. This gives hope to decrease codimension and the Milnor number for which the conjecture on the semicontinuity of the singularity index fails.     

The negative answer to Kameko's conjecture on the hit problem

The goal of this paper is to give a negative answer to Kameko's conjecture on the hit problem stating that the cardinal of a minimal set of generators for the polynomial algebra Pk, considered as a module over the Steenrod algebra A, is dominated by an explicit quantity depending on the number of the polynomial algebra's variables k. The conjecture was shown by Kameko himself for k?3 in his PhD thesis in the Johns Hopkins University in 1990, and recently proved by us and Kameko for k=4. However, we claim that it turns out to be wrong for any k>4.In order to deny Kameko's conjecture we study a minimal set of generators for A-module Pk in some so-call generic degrees. What we mean by generic degrees is a bit different from that of other authors in the fields such as Crabb-Hubbuck, Nam, Repka-Selick, Wood …We prove an inductive formula for the cardinal of the minimal set of generators in these generic degrees when the number of the variables, k, increases. As an immediate consequence of this inductive formula, we recognize that Kameko's conjecture is no longer true for any k>4.     

Construction of determinantal representation of trigonometric polynomials

For a pair of n×n Hermitian matrices H and K, a real ternary homogeneous polynomial defined by F(t,x,y)=det(tI_n+xH+yK) is hyperbolic with respect to (1,0,0). The Fiedler conjecture (or Lax conjecture) is recently affirmed, namely, for any real ternary hyperbolic polynomial F(t,x,y), there exist real symmetric matrices S₁ and S₂ such that F(t,x,y)=det(tI_n+xS₁+yS₂). In this paper, we give a constructive proof of the existence of symmetric matrices for the ternary forms associated with trigonometric polynomials.     

Algebra retracts and Stanley–Reisner rings

In a paper from 2002, Bruns and Gubeladze conjectured that graded algebra retracts of polytopal algebras over a field k are again polytopal algebras. Motivated by this conjecture, we prove that graded algebra retracts of Stanley–Reisner rings over a field k are again Stanley–Reisner rings. Extending this result further, we give partial evidence for a conjecture saying that monomial quotients of standard graded polynomial rings over k descend along graded algebra retracts.     

A polynomial approach to the spectral corrections for Sturm–Liouville problems

A technique based on the evaluation of the zeros of a polynomial is proposed to estimate the spectral errors and set up a correcting procedure in Sturm–Liouville problems. The method suggested shows its effectiveness both in the regular and nonregular case and can be successfully applied also to those problems containing the eigenvalue parameter rationally. Some numerical experiments clearly confirm the theoretical results. In the case of an eigenvalue embedded in the essential spectrum the correcting procedure seems to be particularly helpful because the inner singularity gives rise to possible decay of the performance of some classical methods for the numerical integration.     

Counting points on varieties over finite fields related to a conjecture of Kontsevich

We describe a characteristic-free algorithm for reducing an algebraic variety defined by the vanishing of a set of integer polynomials. In very special cases, the algorithm can be used to decide whether the number of points on a variety, as the ground field varies over finite fields, is a polynomial function of the size of the field. The algorithm is then used to investigate a conjecture of Kontsevich regarding the number of points on a variety associated with the set of spanning trees of any graph. We also prove several theorems describing properties of a (hypothetical) minimal counterexample to the conjecture, and produce counterexamples to some related conjectures.Partially supported by NSF Grant DMS-9700787 and RIMS, Kyoto University.