Results 1 
6 of
6
Ideal Membership in Polynomial Rings over the Integers
 J. Amer. Math. Soc
"... Abstract. We present a new approach to the ideal membership problem for polynomial rings over the integers: given polynomials f0, f1,..., fn ∈ Z[X], where X = (X1,..., XN) is an Ntuple of indeterminates, are there g1,..., gn ∈ Z[X] such that f0 = g1f1 + · · · + gnfn? We show that the degree of th ..."
Abstract

Cited by 16 (2 self)
 Add to MetaCart
Abstract. We present a new approach to the ideal membership problem for polynomial rings over the integers: given polynomials f0, f1,..., fn ∈ Z[X], where X = (X1,..., XN) is an Ntuple of indeterminates, are there g1,..., gn ∈ Z[X] such that f0 = g1f1 + · · · + gnfn? We show that the degree of the polynomials g1,..., gn can be bounded by (2d) 2O(N2) (h + 1) where d is the maximum total degree and h the maximum height of the coefficients of f0,..., fn. Some related questions, primarily concerning linear equations in R[X], where R is the ring of integers of a number field, are also treated.
Meandering of trajectories of polynomial vector fields in the affine nspace
 Publ. Math
, 1997
"... We give an explicit upper bound for the number of isolated intersections between an integral curve of a polynomial vector field in Rn and an affine hyperplane. The problem turns out to be closely related to finding an explicit upper bound for the length of ascending chains of polynomial ideals spann ..."
Abstract

Cited by 5 (5 self)
 Add to MetaCart
We give an explicit upper bound for the number of isolated intersections between an integral curve of a polynomial vector field in Rn and an affine hyperplane. The problem turns out to be closely related to finding an explicit upper bound for the length of ascending chains of polynomial ideals spanned by consecutive derivatives. This exposition constitutes an extended abstract of a forthcoming paper: only the basic steps are outlined here, with all technical details being either completely omitted or at best indicated. Contents 1. Geometry:Trajectories of polynomial vector fields and their meandering 224
Trajectories Of Polynomial Vector Fields And Ascending Chains Of Polynomial Ideals
, 1999
"... We give an explicit upper bound for the number of isolated intersections between an integral curve of a polynomial vector field in R^n and an algebraic hypersurface. The answer is polynomial in the height (the magnitude of coefficients) of the equation and the size of the curve in the spacetime, ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
We give an explicit upper bound for the number of isolated intersections between an integral curve of a polynomial vector field in R^n and an algebraic hypersurface. The answer is polynomial in the height (the magnitude of coefficients) of the equation and the size of the curve in the spacetime, with the exponent depending only on the degree and the dimension. The problem
ORDERINGS OF MONOMIAL IDEALS
, 2003
"... We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal invariants for the complexity of this ordered set. In particular ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal invariants for the complexity of this ordered set. In particular, we give an interpretation of the height function in terms of the HilbertSamuel polynomial, and we compute upper and lower bounds on the maximal order type.
Quantitative theory of ordinary differential equations and tangential Hilbert 16th problem, ArXiv Preprint math.DS/0104140
 Department of Mathematics, Weizmann Institute of Science, P.O.B. 26, Rehovot 76100, Israel Email address: yakov@wisdom.weizmann.ac.il WWW
, 2001
"... Abstract. These highly informal lecture notes aim at introducing and explaining several closely related problems on zeros of analytic functions defined by ordinary differential equations and systems of such equations. The main incentive for this study was its potential application to the tangential ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
Abstract. These highly informal lecture notes aim at introducing and explaining several closely related problems on zeros of analytic functions defined by ordinary differential equations and systems of such equations. The main incentive for this study was its potential application to the tangential Hilbert 16th problem on zeros of complete Abelian integrals. The exposition consists mostly of examples illustrating various phenomena related to this problem. Sometimes these examples give an insight concerning the proofs, though the complete exposition of the latter is mostly relegated to separate expositions.
On Functions And Curves Defined By Ordinary Differential Equations
, 1999
"... . These notes constitute a substantially extended version of a talk given in the Fields Institute (Toronto) during the semester "Singularities and Geometry", that culminated by Arnoldfest in celebration of V. I. Arnold's 60th anniversary. We give a survey of different results showing how an upper bo ..."
Abstract
 Add to MetaCart
. These notes constitute a substantially extended version of a talk given in the Fields Institute (Toronto) during the semester "Singularities and Geometry", that culminated by Arnoldfest in celebration of V. I. Arnold's 60th anniversary. We give a survey of different results showing how an upper bound for the number of isolated zeros for functions satisfying ordinary differential equations, may be obtained without solving these equations. The main source of applications is the problem on zeros of complete Abelian integrals, one of the favorite subjects discussed on Arnold's seminar in Moscow for over quarter a century. Data aequatione quotcunque fluentes quantitae involvente fluxiones invenire et vice versa. Isaac Newton It is useful to solve differential equations. Translation by Vladimir Arnold x1. Introduction 1.1. Equations and solutions. One of the illusions that are pleasant to nourish is the claim that simple equations cannot have complicated solutions. Though completely re...