Integrabilidade algébrica para r-formas polinomiais

Abstract

The theory of integrability developed by Darboux provides a relationship between the integrability of polynomial vector fields and the number of invariant algebraic hypersurfaces that they admit. This theory, initially proposed by Henri Poincar´e, has been successfully applied to the study of many physical models. Jean Pierre Jouanolou significantly improved Darboux’s theory by characterizing the existence of rational first integrals for polynomial 1-forms in Kn and P n K, where K is an algebraically closed field of characteristic zero. This work is based on the study of article [9], in which M. Corrˆea Jr., L. G. Maza, and M. G. Soares prove an algebraic integrability theorem of Darboux–Jouanolou type for polynomial r-forms. Thus, the first main result of this work is given by Theorem 4.0.1 Let K be an algebraically closed field and let ω be an r-form of degree d in Kn . If ω admits

d − 1 + n n

.

n r + 1 + r + 1 irreducible invariant algebraic hypersurfaces, then ω admits a rational first integral. Assuming that the form does not admit a rational first integral, the above equation provides an upper bound for the number of invariant hypersurfaces of ω, unfortunately, this bound is far from optimal. Thus, the second main result of this work is based on the Lagutinskii–Pereira theory of integrability, in which the concept of the r-th extactic polynomial of a vector field is introduced, providing another criterion for the existence of rational first integrals. In particular, we obtain an optimal bound for the number of invariant hyperplanes. Theorem 5.0.2 Let X be a polynomial vector field on C n of degree d. Suppose that X does not admit a rational first integral. Then the number of irreducible invariant hypersurfaces of degree k of X is at most (d − 1) n+k k

2

+ 1 k

k + 1 2 n + k k

− 1

. In particular, if k = 1, we have

n + 1 2

(d − 1) + n, and this last bound is optimal. Finally, in the last part of this work, we present an application to the theory of onedimensional holomorphic foliations on the projective space P n . Theorem 6.0.3 Let F be a one-dimensional holomorphic foliation on the complex projective space P n C with degree deg(F) = d. Suppose that F does not admit a rational first integral then 1. The number of irreducible invariant hypersurfaces of F of degree k, counted with multiplicity, is at most

n + k k

+ 1 k

n+k k

2

(d − 1). 2. For k = 1, we have (n + 1) + n + 1 2

(d − 1) = nd +

n 2

(d − 1) + 1.