On the second part of Hilbert's 16th problem

At the 1900 International Congress of Mathematicians, held that year in Paris, the German mathematician David Hilbert put forth a list of 23 problems that he saw as being the greatest challenges for twentieth-century mathematics. Hilbert's 10th problem, to find a method (what we now call an algorithm) for deciding whether a Diophantine equation has an integral solution, was solved by Yuri Matiyasevich in 1970. Proving the undecidability of Hilbert's 10th problem is clearly one of the great mathematical results of the century.This book presents the full, self-contained negative solution of Hilbert's 10th problem. In addition it contains a number of diverse, often striking applications of the technique developed for that solution (scattered previously in journals), describes the many improvements and modifications of the original proof - since the problem was unsolved 20 years ago, and adds several new, previously unpublished proofs.Included are numerous exercises that range in difficulty from the elementary to small research problems, open questions,and problems. Each chapter concludes with a commentary providing a historical view of its contents. And an extensive bibliography contains references to all of the main publications directed to the negative solution of Hilbert's 10th problem as well as the majority of the publications dealing with applications of the solution.Intended for young mathematicians, Hilbert's 10th Problem requires only a modest mathematical background. A few less well known number-theoretical results are presented in the appendixes. No knowledge of recursion theory is presupposed. All necessary notions are introduced and defined in the book, making it suitable for the first acquaintance with this fascinating subject.Yuri Matiyasevich is Head of the Laboratory of Mathematical Logic, Steklov Institute of Mathematics, Russian Academy of Sciences, Saint Petersburg.

Small limit cycles bifurcating from fine focus points in quartic order Z 3-equivariant vector fields

In this chapter we deal with abelian integrals. They play a key role in the infinitesimal version of the 16th Hilbert problem. Recall that 16th Hilbert problem and its ramifications is one of the principal research subject of Christiane Rousseau and of the first author. We recall briefly the definition and explain the role of abelian integrals in 16th Hilbert problem. We also give a simple well-known proof of a property of abelian integrals. The reason for presenting it here is that it serves as a model for more complicated and more original treatment of abelian integrals in the study of Hamiltonian monodromy of fully integrable systems, which is the main subject of this chapter. We treat in particular the simplest example presenting non-trivial Hamiltonian monodromy: the spherical pendulum.

Hilbert's 16th problem, about the maximum number of limit cycles of planar polynomial vector fields of a given degree , has been one of the most important driving forces for new developments in the qualitative theory of vector fields. Increasing the dimension, one cannot expect the existence of a finite upper bound for the number of limit cycles of, for instance, 3D polynomial vector fields of a given degree . Here, as an extension of such a problem in the 3D space, we investigate the number of isolated invariant tori in 3D polynomial vector fields. In this context, given a natural number , we denote by the upper bound for the number of isolated invariant tori of 3D polynomial vector fields of degree . Based on a recently developed averaging method for detecting invariant tori, our first main result provides a mechanism for constructing 3D differential vector fields with a number of normally hyperbolic invariant tori from a given planar differential vector field with hyperbolic limit cycles. The strength of our mechanism in studying the number lies in the fact that the constructed 3D differential vector field is polynomial provided that the given planar differential vector field is polynomial. Accordingly, our second main result establishes a lower bound for in terms of lower bounds for the number of hyperbolic limit cycles of planar polynomial vector fields of degree . Based on this last result, we apply a methodology due to Christopher and Lloyd to show that grows as fast as . Finally, the above‐mentioned problem is also formulated for higher dimensional polynomial vector fields.

PP-graphics with a nilpotent elliptic singularity in quadratic systems and Hilbert's 16th problem

Hermitian analogues of Hilbert's 17-th problem

In 1957, Kolmogorov and Arnold gave a solution to the 13th problem which had been formulated by Hilbert in 1900. Actually, it is known that there exist many open problems which can be derived from the original problem. From the function-theoretic point of view, Hilbert's 13th problem can be exactly characterized as the superposition representability problem for continuous functions of several variables. In this paper, the solution to the superposition representability problem for infinitely differentiable functions of several variables is given.

In this work we study the Positive Extension property and Hilbert's 17th problem for real analytic germs and sets. A real analytic germ X0 of has the property if every positive semidefinite analytic function germ on X0 has a positive semidefinite analytic extension to ; analogously one states the property for a global real analytic set X in an open set Ω of ℝn. These properties are natural variations of Hilbert's 17th problem. Here, we prove that: (1) A real analytic germ has the property if and only if every positive semidefinite analytic function germ on X0 is a sum of squares of analytic function germs on X0; and (2) a global real analytic set X of dimension ≦ 2 and local embedding dimension ≦ 3 has the property if and only if it is coherent and all its germs have the property. If that is the case, every positive semidefinite analytic function on X is a sum of squares of analytic functions on X. Moreover, we classify the singularities with the property.

The algebraic form of Hilbert's 13th problem asks for the resolvent degree \operatorname{RD}(n) of the general polynomial f(x) = x^n + a_1 x^{n-1} + \cdots + a_n of degree n , where a_1, \ldots, a_n are independent variables. The resolvent degree is the minimal integer d such that every root of f(x) can be obtained in a finite number of steps, starting with \mathbb{C}(a_1, \ldots, a_n) and adjoining algebraic functions in \leqslant\nobreak d variables at each step. Recently Farb and Wolfson defined the resolvent degree \operatorname{RD}_k(G) for every finite group G and any base field k of characteristic 0 . In this setting \operatorname{RD}(n) = \operatorname{RD}_{\mathbb{C}}(\operatorname{S}_n) , where \operatorname{S}_n denotes the symmetric group. In this paper we extend their definition of \operatorname{RD}_k(G) to an arbitrary algebraic {group} G over an arbitrary field k . We investigate the dependency of this quantity on k and show that \operatorname{RD}_k(G) \leqslant 5 for any field k and any connected group G . The question whether \operatorname{RD}_k(G) can be bigger than 1 for any field k and any algebraic group G over k (not necessarily connected) remains open.

The Stokes Phenomenon and Hilbert's 16th Problem

Global bifurcation of limit cycles in a perturbed integrable non-Hamiltonian system is investigated using bifurcation method of limit cycles. The study reveals that, for the integrable non-Hamiltonian system under polynomial perturbations [equation (8) in the introduction], the upper bound for the number of limit cycles is [(n+m-1/2)] + 1 when n ≥ m + 2; it is m + 1 when n = m, m + 1; and it is m when 1 ≤ n ≤ m - 1. The results presented here are helpful for further investigating the Hilbert's 16th problem.

Flow invariant subspaces for lattice dynamical systems by F. Antoneli, A. P. S. Dias, M. Golubitsky, and Y. Wang Waves in extended systems: From low- to high-dimensional behavior by O. Batiste and E. Knobloch Mixed mode oscillations due to the generalized Canard phenomenon by M. Brons, M. Krupa, and M. Wechselberger Bioremediation of waste in a porous medium by C. Chen and J. Chadam Bifurcations of gyroscopic systems near a 0: 1 resonance by R. J. McDonald, N. S. Namachchivaya, and W. Nagata High dimensional data clustering from a dynamical systems point of view by J. Wu Computation of limit cycles-The second part of Hilbert's 16th problem by P. Yu.

In this article we determine the global geometry of the planar quadratic differential systems with a weak focus of third order. This class plays a significant role in the context of Hilbert's 16-th problem. Indeed, all examples of quadratic differential systems with at least four limit cycles, were obtained by perturbing a system in this family. We use the algebro-geometric concepts of divisor and zero-cycle to encode global properties of the systems and to give structure to this class. We give a theorem of topological classification of such systems in terms of integer-valued affine invariants. According to the possible values taken by them in this family we obtain a total of 18 topologically distinct phase portraits. We show that inside the class of all quadratic systems with the topology of the coefficients, there exists a neighborhood of the family of quadratic systems with a weak focus of third order and which may have graphics but no polycycle in the sense of [15] and no limit cycle, such that any quadratic system in this neighborhood has at most four limit cycles.

Planar quadratic differential systems occur in many areas of applied mathematics. Although more than one thousand papers have been written on these systems, a complete understanding of this family is still missing. Classical problems, and in particular, Hilbert's 16th problem [Hilbert, 1900, 1902], are still open for this family. Our goal is to make a global study of the family QsnSN of all real quadratic polynomial differential systems which have a finite semi-elemental saddle-node and an infinite saddle-node formed by the collision of two infinite singular points. This family can be divided into three different subfamilies, all of them with the finite saddle-node in the origin of the plane with the eigenvectors on the axes and with the eigenvector associated with the zero eigenvalue on the horizontal axis and (A) with the infinite saddle-node in the horizontal axis, (B) with the infinite saddle-node in the vertical axis and (C) with the infinite saddle-node in the bisector of the first and third quadrants. These three subfamilies modulo the action of the affine group and time homotheties are three-dimensional and we give the bifurcation diagram of their closure with respect to specific normal forms, in the three-dimensional real projective space. The subfamilies (A) and (B) have already been studied [Artés et al., 2013b] and in this paper we provide the complete study of the geometry of the last family (C). The bifurcation diagram for the subfamily (C) yields 371 topologically distinct phase portraits with and without limit cycles for systems in the closure [Formula: see text] within the representatives of QsnSN(C) given by a chosen normal form. Algebraic invariants are used to construct the bifurcation set. The phase portraits are represented on the Poincaré disk. The bifurcation set of [Formula: see text] is not only algebraic due to the presence of some surfaces found numerically. All points in these surfaces correspond to either connections of separatrices, or the presence of a double limit cycle.

This article is a survey of the results on Hilbert's 16th problem from 1876 to the present. 1. HARNACK'S THEOREM. The number of branches of a non-singular curve of order m in RP2 does not exceed Harnack's method for constructing curves with the greatest number of branches (M-curves). The methods of Hilbert, Brusotti, and Wiman for constructing M-curves. M-curves on a quadric. 2. The concepts of roughness and degrees of non-roughness. Brusotti's theorem on the independence of the simplifications (small corrections) of simple double points and cusps. 3. The theorems of Petrovskii and Oleinik on algebraic curves and surfaces. The generalization due to Kharlamov. 4. Sextic curves in RP2. Cubic and quartic surfaces in RP3. 5. An oval of a non-singular curve is said to be odd if it lies inside an odd number of other ovals, and even otherwise. Let A be an M-curve of even order m, and suppose that the number of its even (odd) ovals is P(L). Then Broad generalizations were given by Rokhlin and others. 6. Some conjectures.