Derivations Without Darboux Polynomials

This thesis is a survey of the research on derivations without Darboux polynomials.First, we introduce the notion of Jouanolou derivation and brie?y sketch the purely algebraicproof, due to Ollagnier et al, of the fact that Jouanolou derivation has no Darboux poly-nomials. Next, we present several examples of homogeneous derivations without Darbouxpolynomials, due to Ollagnier et al. Finally, with the conclusion that simple derivations haveno Darboux polynomials, some inhomogeneous derivations are given.Consider s?homogeneous derivations of k[X]. If n 2 or s = 1, every homogeneousderivation has a Darboux polynomial; if n = 3 or s 2, Jouanolou derivation has no Darbouxpolynomials,Theorem 1.1 The derivation , where s 2, has no Darboux polyno-mials.Chapter 2 is concerned with the proof of Theorem 1.1 due to Ollagnier et al by algebraicmethod.Chapter 3 is concerned with homogeneous derivations without Darboux polynomials.Section 1 is concerned with a result of Shamsuddin and its modification due to Ollagnier etal:Theorem 3.3 Let R be a commutative domain containing Q and let d be a derivationwithout Darboux elements. Let derivation D be the expansion of d in R[t] such that D(t) =at + b, for some a, b∈R, then the following two conditions are equivalent:(1) The derivation D has no Darboux elements.(2) There exist no elements r of R such that d(r) = ar + b.In Section 2, we introduce a series of examples of homogeneous derivations withoutDarboux polynomials, due to Ollagnier et al, which are produced with Theorem 3.3 andJouanolou derivation. Example 3.1 If n 4, s 2, then the derivation of k[x1, ..., xn]has no Darboux polynomials.Example 3.2 If n 4, then the derivation of k[x1, ..., xn]has no Darboux polynomials.Example 3.3 If n 4, s 3, then the derivation of k[x1, ..., xn]has no Darboux polynomials.Example 3.4 The derivation of k[x, y, z, t]has no Darboux polynomials.Example 3.5 If n 4, then the derivation of k[x1, ..., xn]has no Darboux polynomials.In Section 3, we pay attention to homogeneous monomial derivations of k[x,y,z] due toOllagnier and Nowicki, which have no Darboux polynomials.Proposition 3.2 Let d : k[x, y, z]→k[x, y, z] be an irreducible derivation such thatd(x), d(y), d(z) are monic monomials of the same degree 2. Then d has no Darboux polyno-mials if and only if d can be found in the following list. The derivations in each part are thesame, up to permutations of variables.Proposition 3.4 Let d : k[x, y, z]→k[x, y, z] be an irreducible derivation such thatd(x), d(y), d(z) are monic monomials of the same degree 3. Then d has no Darboux polyno-mials if and only if d can be found in the following list. The derivations in each part are thesame, up to permutations of variables. Proposition 3.6 Let d : k[x, y, z]→k[x, y, z] be an irreducible derivation such thatd(x), d(y), d(z) are monic monomials of the same degree 4. Then d has no Darboux polyno-mials if and only if d can be found in the following list. The derivations in each part are thesame, up to permutations of variables.In Chapter 4, we introduce many inhomogeneous simple derivations given by Nowicki,Jordan and Maciejewski et al, which have no Darboux polynomials. Nowicki got four simplederivations of k[x, y]. Then Maciejewski et al found two classes of simple derivations of k[x, y].Example 4.9Example 4.10If degb is odd and deg b > 2 deg a.For n 2, three derivations were found by Maciejewski et al via Theorem 3.1.Example 4.11Example 4.12Example 4.13...