Polynomial Flows in the Plane: A Classification Based on Spectra of Derivations

We study the classification of polynomial vector fields in two complex variables under the hypotheses that the singularities are isolated and the flow is complete. Normal forms are obtained for the case the generic orbit is di¤eomorphic to C. For the case the generic orbit is di¤eomorphic to Cnf0g and there is an a‰ne singularity we classify the linear part of the vector field and prove the existence of entire linearization or first integral.2000 Mathematics Subject Classification: 37L75; 32M25. IntroductionWe consider a vector field X ¼ aðx; yÞ q qx þ bðx; yÞ q qy on C 2 , where aðx; yÞ and bðx; yÞ are polynomials in ðx; yÞ. In what follows we assume that X has isolated singularities, i.e., the singular set SingðX Þ :¼ fa ¼ b ¼ 0g is finite. We denote by j t the local flow of X and we will say that X is complete if for every point m A C 2 , the holomorphic map t 7 ! j t ðmÞ is defined on C. The map ðm; tÞ 7 ! j t ðmÞ is therefore holomorphic from C 2 to C 3 and we have qj t ðmÞ qt ¼ X ðj t ðmÞÞ. The vector field X induces a foliation F X on C 2 with singularities at the points of SingðX Þ; if m A C 2 we denote by L m the leaf of F X passing through m. If X is complete, and this is what we will suppose in what follows, the leaf L m can be of three types: a. L m ¼ fmg, if m is a singular point b. of type C, this is the case if and only if t 7 ! j t ðmÞ is injective c. of type C Ã , this is the case if m is non singular and t 7 ! j t ðmÞ is not injective (and therefore periodic). In this last case there exists t A C À f0g such that j tþt ðmÞ ¼ j t ðmÞ.These are the only possible cases, for t 7 ! j t ðmÞ cannot be doubly-periodic because if so then its image would be a (compact) elliptic curve in the a‰ne space C 2 , what is not possible. A vector field X can exhibit all three types of leaves, for instance a linear vector field X ¼ x q qx þ ly q qy , l A C À Q:Brought to you by | Purdue University Libraries Authenticated Download Date | 5/24/15 8:49 PM

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“…C 2 is polynomial. The description of polynomial flows is well known ( [2], [24] for instance); up to polynomial automorphism j t is of type…”

Section: Complete Vector Fields and Algebraic Invariant Curves-polynomentioning

confidence: 99%

Complete polynomial vector fields in two complex variables

Cerveau

Scárdua

2005

Forum Mathematicum

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“…Coomes and Zurkowski [8] show that 0 is a polynomial flow if and only if T(D) = C[n]. (See [3,5,6,7,10,11,12,13,14,15,17] for other results about polynomial flows.) Since the torsion part T(D) can play an important role in determining whether a vector field has a polynomial flow, we wish to gain a deeper understanding of the properties of T(D).…”

Section: Introductionmentioning

confidence: 99%