Class Of Polynomial Systems Research Articles

Sylvester-type matrices for sparse resultants

The resultant matrix of a polynomial system depends on the geometry of its input Newton polytopes. Therefore for sparse inputs, the matrix is lower in dimension. The aim of the study is to infer conditions on the class of polynomial systems that can give a resultant matrix whose size is minimized, that is an optimal or Sylvester-type sparse resultant matrix. From the work of Emiris, the ‘incremental algorithm’ has been claimed to produce optimal matrices for the class of multi-homogeneous (or multigraded) systems of special structure. Cyclic polynomial systems for n-root problems also fall under this classification. We have applied the Maple multires package to obtain Sylvester-type matrices for some examples. The ultimate aim of the study is to verify whether the multigraded systems constitute to the only class of polynomial systems that can give sparse resultant optimal matrix; hence giving a necessary and sufficient condition for producing exact sparse resultants.

Poincaré Bifurcations of Two Classes of Polynomial Systems

Using bifurcation methods and the Abelian integral, we investigate the number of the limit cycles that bifurcate from the period annulus of the singular point when we perturb the planar ordinary differential equations of the form <svg style="vertical-align:-3.56265pt;width:98.675003px;" id="M1" height="16.625" version="1.1" viewBox="0 0 98.675003 16.625" width="98.675003" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,9.262,11.95)"><path id="x307" d="M-161 571q0 -21 -15 -36t-36 -15q-20 0 -35 15.5t-15 35.5q0 21 15 36.5t35 15.5q22 0 36.5 -15t14.5 -37z" /></g><g transform="matrix(.017,-0,0,-.017,.062,12.138)"><path id="x1D465" d="M536 404q0 -17 -13.5 -31.5t-26.5 -14.5q-8 0 -15 10q-11 14 -25 14q-22 0 -67 -50q-47 -52 -68 -82l37 -102q31 -88 55 -88t78 59l16 -23q-32 -48 -68.5 -78t-65.5 -30q-19 0 -37.5 20t-29.5 53l-41 116q-72 -106 -114.5 -147.5t-79.5 -41.5q-21 0 -34.5 14t-13.5 37&#xA;q0 16 13.5 31.5t28.5 15.5q12 0 17 -11q5 -10 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Inversion of Polynomial Systems and Separation of Nonlinear Mixtures of Finite-Alphabet Sources

In this contribution, multiple-input multiple-output (MIMO) mixing systems are considered, which are instantaneous and nonlinear but polynomial. We first address the problem of invertibility, searching the inverse in the class of polynomial systems. It is shown that Grobner bases techniques offer an attractive solution for testing the existence of an exact inverse and computing it. By noticing that any nonlinear mapping can be interpolated by a polynomial on a finite set, we tackle the general nonlinear case. Relying on a finite alphabet assumption of the input source signals, theoretical results on polynomials allow us to represent nonlinear systems as linear combinations of a finite set of monomials. We then generalize the first results to give a condition for the existence of an exact nonlinear inverse. The proposed method allows to compute this inverse in polynomial form. In the light of the previous results, we go further to the blind source separation problem. It is shown that for sources in a finite alphabet, the nonlinear problem is tightly connected with both problems of under determination and of dependent sources. We concentrate on the case of two binary sources, for which an easy solution can be found. By simulation, this solution is compared to techniques borrowed from classification methods.

An analogue rotated vector field of polynomial system

A class of polynomial system was structured, which depends on a parameter δ. When δ monotonous changes, more than one neighbouring limit cycles located in the vector fiel of this polynomial system can expand (or reduce) together with the δ. But the expansion (or reduction) of these limit cycles is not surely monotonous. This vector field is like the rotated vector field. So these limit cycles of the polynomial system are called to constitute an “analogue rotated vector field” with δ. They may become an effective tool to study the bifurcation of multiple limit cycle or fine separatrix cycle.

Controllability of structured polynomial systems

Two algorithms, based on the Grobner basis method, which facilitate the controllability analysis for a class of polynomial systems are presented. The authors combine these algorithms with some results on output dead-beat controllability in order to obtain sufficient, as well as necessary, conditions for complete and state dead-beat controllability for a surprisingly large class of polynomial systems. Our results are generically applicable to the class of polynomial systems in strict feedback form.

ON THE BASIS PROPERTY OF CERTAIN POLYNOMIAL SYSTEMS IN SPACES OF ENTIRE FUNCTIONS OF EXPONENTIAL TYPE

A class of polynomial systems (, ) is considered such that each polynomial , starting with , has together with its derivatives of order up to and including at least one zero in the closed unit disc. It is shown that each polynomial system of the class forms a quasipower basis in the space of entire functions of exponential type less than (), provided does not exceed a certain absolute constant .