Throughout this paper, k will denote a commutative ring containing the rational numbers Q, and k = k[x{, . . . , xn] will be the polynomial ring over k . If ƒ : k —• k is a polynomial map (i.e., a fc-algebra homomorphism), then ƒ is a polynomial automorphism provided there is an inverse ƒ " which is also a polynomial map. Very little is known about the group of polynomial automorphisms, and ind...

In this paper we propose an efficient method for solving the spherically constrained homogeneous polynomial optimization problem. The new approach has the following three main ingredients. First, we establish a block coordinate descent type search method for nonlinear optimization, with the novelty being that we only accept a block update that achieves the maximum improvement, hence the name of...

Hyperbolic polynomials have their origins in partial diierential equations. We show in this paper that they have applications in interior point methods for convex programming. Each homogeneous hyperbolic polynomial p has an associated open and convex cone called its hyperbolicity cone. We give an explicit representation of this cone in terms of polynomial inequalities. The function F (x) = ? lo...

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...

In this paper we shall primarily study polynomial integrability of the differential system ẋ = −y + Pn(x, y), ẏ = x + Qn(x, y), n = 2, 3, where Pn and Qn are homogeneous polynomials of degree n. By taking various yet very elementary ways, we not only straightforwardly find the necessary and sufficient integrability conditions but also explicitly present the corresponding polynomial first integr...

Let P : E → K be an N−homogeneous polynomial, where E is a Banach space over K = R or C . We study properties of the set CP = {x ∈ E : P is weakly sequentially continuous at x}. Introduction Our interest in set of points of weak sequential continuity of a polynomial arises from the following simple observations. If P is any 2−homogeneous scalar valued polynomial on E which is weakly sequentiall...

Let p be a homogeneous polynomial of degree n in n variables, p(z1, . . . , zn) = p(Z), Z ∈ C. We call such a polynomial p H-Stable if p(z1, . . . , zn) 6= 0 provided the real parts Re(zi) > 0, 1 ≤ i ≤ n. This notion from Control Theory is closely related to the notion of Hyperbolicity used intensively in the PDE theory. The main theorem in this paper states that if p(x1, . . . , xn) is a homog...

Three primal-dual interior-point algorithms for homogeneous cone programming are presented. They are a short-step algorithm, a large-update algorithm, and a predictor-corrector algorithm. These algorithms are described and analyzed based on a characterization of homogeneous cone via T -algebra. The analysis show that the algorithms have polynomial iteration complexity.

This paper provides an estimate of the sum of a homogeneous polynomial P over the lattice points inside a sphere of radius R. The polynomial P is assumed to be of degree ν and have zero mean over the sphere. It is proved that ∑ x∈Z3 |x|≤R P (x) = O ,P (R ν+83/64+ ) ∗MIT, supported by Summer Program of Undergraduate Research

The multigraded Hilbert scheme parametrizes all homogeneous ideals in a polynomial ring graded by an abelian group with a fixed Hilbert function. We prove that any multigraded Hilbert scheme is smooth and irreducible when the polynomial ring is Z[x, y], which establishes a conjecture of Haiman and Sturmfels.