We study equilibrium computation for exchange markets. We show that the market equilibrium of either of the following two markets: 1. The Fisher market with several classes of concave non-homogeneous utility functions; 2. A mixed Fisher and Arrow-Debreu market with homogeneous and log-concave utility functions can be computed as convex programming and by interior-point algorithms in polynomial ...

This paper introduces the umodules, a generalisation of modules for the homogeneous relations. We first present some properties of the umodule family, then show that, if the homogeneous relation fulfills some natural axioms, the umodule family has a unique decomposition tree. We show that this tree can be computed in polynomial time, under a certain size assumption. We apply this theory to a ne...

The parametric or geometric continuity of a rational polynomial curve has often been obtained by requiring the homogeneous polynomial curve associated with the rational curve to possess parametric or geometric continuity, respectively. Recently this approach has been shown overly restrictive. We make use of the necessary and su cient conditions of rational parametric continuity for de ning basi...

Let R be a polynomial ring over a field of dimension n. Let m be the maximal homogeneous ideal of R. Let I be a complete intersection homogeneous ideal of R minimally generated by polynomials of degrees d1, d2, . . . , dn. F. S. Macaulay showed that for all integers i = 0, 1, . . . , δ + 1, I : m = I + m where δ = d1 + d2 + . . .+ dn − n. We provide a modern proof of a generalization of this re...

In this paper we introduce homogeneously orderable graphs which are a common generalization of distance-hereditary graphs, dually chordal graphs and homogeneous graphs. We present a characterization of the new class in terms of a tree structure of the closed neighborhoods of homogeneous sets in 2-graphs which is closely related to the defining elimination ordering. Moreover, we characterize the...

The geometric mean and the function (det(·))1/m (on the m-by-m positive definite matrices) are examples of “hyperbolic means”: functions of the form p1/m , where p is a hyperbolic polynomial of degree m. (A homogeneous polynomial p is “hyperbolic” with respect to a vector d if the polynomial t → p(x+ td) has only real roots for every vector x.) Any hyperbolic mean is positively homogeneous and ...

We consider natural Hamiltonian systems of n > 1 degrees of freedom with polynomial homogeneous potentials of degree k. We show that under a genericity assumption, for a fixed k, at most only a finite number of such systems is integrable. We also explain how to find explicit forms of these integrable potentials for small k.

We investigate the use of non-homogeneous spherical poly-nomials for the approximation of functions deened on the sphere S 2. A spherical polynomial is the restriction to S 2 of a polynomial in the three coordinates x; y; z of R 3. Let P d be the space of spherical polynomials with degree d. We show that P d is the direct sum of H d and H d?1 , where H d denotes the space of homogeneous degree-...

Let k be a field and S = k[x1, . . . , xn] be a polynomial ring over k. We consider finite sequences of homogeneous polynomials of positive degrees and their operation on non-trivial finitely generated graded S-modules (with grading in Z). Here and in the following, by a polynomial, we always mean an element of S. Moreover, by an S-module we mean in the following a graded S-module with grading ...

In this paper we first characterize all quasi–homogeneous but non–homogeneous planar polynomial differential systems of degree five, and then among which we classify all the ones having a center at the origin. Finally we present the topological phase portrait of the systems having a center at the origin.