Simple Darboux points of polynomial planar vector fields

AbstractWe are interested in some aspects of the integrability of complex polynomial planar vector fields in finite form. Especially, in the case of simple Darboux points, we deduce the famous Baum–Bott formula from a kind of global residue theorem; our elementary proof essentially relies on Hilbert's Nullstellensatz.As a corollary of our result, we propose formulas relating the various integers involved in the Lagutinskii–Levelt procedure for a Darboux polynomial at the various Darboux points. In particular, from the whole set of our formulas, it is possible to deduce an upper bound on the degree of irreducible Darboux polynomials in classical cases; with respect to such applications, this corollary seems to provide an alternate tool to usual genus formulas.As many people do in these subjects, we illustrate our corollary by giving a new simple proof of the fact that the polynomial Jouanolou derivation ys∂x+zs∂y+xs∂z, with s⩾2, has no Darboux polynomial

Proof of Dejean's conjecture for alphabets with 5, 6, 7, 8, 9, 10 and 11 letters

AbstractAxel Thue proved that overlapping factors could be avoided in arbitrarily long words on a two-letter alphabet while, on the same alphabet, square factors always occur in words longer than 3. Françoise Dejean stated an analogous result for three-letter alphabets: every long enough word has a factor, which is a fractional power with an exponent at least 7/4 and there exist arbitrary long words in which no factor is a fractional power with an exponent strictly greater than 7/4. The number 7/4 is called the repetition threshold of the three-letter alphabets.Thereafter, she proposed the following conjecture: the repetition threshold of the k-letter alphabets is equal to k/(k−1) except in the particular cases k=3, where this threshold is 7/4, and k=4, where it is 7/5.For k=4, this conjecture was proved by J.J. Pansiot (1984).In this paper, we give a computer-aided proof of Dejean's conjecture for several other values: 5, 6, 7, 8, 9, 10 and 11

Generic polynomial vector fields are not integrable

AbstractWe study some generic aspects of polynomial vector fields or polynomial derivations with respect to their integration. In particular, using a well-suited presentation of Darboux polynomials at some Darboux point as power series in local Darboux coordinates, it is possible to show, by algebraic means only, that the Jouanolou derivation in four variables has no polynomial first integral for any integer value s ≥ 2 of the parameter.Using direct sums of derivations together with our previous results we show that, for all n ≥ 3 and s ≥ 2, the absence of polynomial first integrals, or even of Darboux polynomials, is generic for homogeneous polynomial vector fields of degree s in n variables