Simple Derivations In Two Variables

Let R be a ring. A mapping d : R→R is said to be a derivation of R, if d(f + g) = d(f) + d(g) and d(fg) = fd(g) + d(f)g, for all f,g∈R. Two derivations d andδare said to be equivalent if there exists an algebra automorphismσsuch thatδ=σdσ^-1. An ideal I is called to be a differential ideal if d(I) (?) I, and d is simple if there are no other differential ideals than 0 and R.The derivation theory. is a promotion of differential operator. It is not only a powerful tool for mathematical research, but has become an important study topic in mathematics. As a special kind of derivations, simple derivations have played an important part in many branches of mathematics. Simple derivations play an unparalleled role in the construction of examples in noncommutative algebra. For example, simple derivations are useful for constructions of simple noncommutative rings and simple Lie algebras. Simple derivations can be used to determine whether a ring is a regular ring in commutative algebra. In addition, simple derivations also have been connected to other interesting topics such as the produce of new examples of nonholonomic irreducible modules over the Weyl algebra in the theory of holonomic modules and the question of the existence of the first integrals and Darboux polynomials of certain algebraic differential equations. Examples, applications and various properties of simple derivations can be found in many literatures.It is important but difficult to find new examples of simple derivations and examine their structure. Even over polynomial rings, only few families of simple derivations are known.Assume that R = k[x₁,…, x_n] is the polynomial ring over k in n variables. Consider the derivation d of R given by d(x₁) = f₁,…,d(x_n) = f_n. We say that a polynomial F∈R is a Darboux polynomial of d if F (?) k and d(F) = (?)F, for some (?)∈R, and (?) is said to be a polynomial eigenvalue of F. Since simple derivations have various applications in many branches of mathematics, it is of considerable interest to find sufficient and necessary conditions on f₁ ,…,f_n for d to be simple.The answer is obvious only for n = 1.If n = 2, only few examples of simple derivations of R = k[x,y] are known. The problem seems to be very difficult even under the additional condition d(x) = 1. In this case, Nowicki proves that if d : k[x, y]→k[x, y] is a derivation such that d(x) = 1, then d is simple if and only if d has no Darboux polynomials.In 1994, Nowicki gave a description of all simple derivations of k[x,y] such that d(x) = 1 and d(y) = a(x)y + b(x), where a(x), b(x)∈k[x].Proposition 1.1 Let R be a commutative domain containing Q and let d : R→R be a dimple derivation. Let R[t] be the polynomial ring in one variable over R, and (?) : R[t]→R[t] be an extension of the derivation d such that (?)(t) = at+b, for some a,b∈R. Then (?) is simple if and only if there is no element r of R such that d(r) = ar + b.If a = 0 or b - 0, it is apparent that the derivation d = (?) + (a(x)y + b(x))(?) is not simple. Further more, Nowicki gets the following theorem, from which it is ready to decide if a derivation of the form d = (?) + (a(x)y + b(x))(?) is simple in a finite number of steps.Theorem 1.1 Let d = (?) + (a(x)y + b(x))(?) be a derivation of k[x,y] with a(x), b(x)∈k[x]. Then the following results hold.(1) If d is not simple and deg b ? deg a, then b = 0.(2) If d is not simple and deg b = deg a, then b =αa, where 0≠α∈k.(2) If deg b ? deg a and a≠0, let b = ca + r, where deg r < deg a. Then d is not simple if and only if (?) = (?) + (ay + (c' + r))(?) is not simple.In 2001, Maciejewski, Moulin-Ollagnier and Nowicki studied derivations of the form d = (?) + (y2 + a(x)y + b(x))(?) with a(x), b(x)∈k[x]. They proved that such a derivation is equivalent to the derivationâ–³_h = (?) + (y² - h(x))(?), for some h(x)∈k[x], and ifâ–³_h is not simple, thenâ–³_h has a Darboux polynomial F such that deg_y F = 1. And they gave some conditions forâ–³h to be simple.Theorem 1.2 If the degree of h(x) is odd, thenâ–³_h is simple. Theorem 1.3 If h(x) has degree 2, thenâ–³_h is equivalent toâ–³_x²-e = (?) + (y² - (x² - e))(?), andâ–³_x²-e is not simple if and only if e is an odd integer.In this thesis, we extend the results of Maciejewski, Moulin-Ollagnier and Nowicki. We first deal with derivations of the formâ–³_h = (?) + (y² - h(x))(?), where h(x) is a binomial of arbitrary degree. We present some examples of simple derivations and some criterions for simplicity of derivations. So we improve the results of Maciejewski, Moulin-Ollagnier and Nowicki from degree 2 to higher degrees.We consider the relationship between simple derivations and Darboux polynomials, and get the following conclusion.Theorem 2.1 Let h(x) = xⁿ - p. Thenâ–³_h is not simple if and only if n = 2 and p is an odd integer.Let h(x) = x^2m - px^k with k≤m - 1. Sinceâ–³_h is not simple if and only ifâ–³_h has a Darboux polynomial F such that deg_y F = 1, we first give a description of the Darboux polynomials ofâ–³_h.Lemma 2.1 Let h(x) = x^2m -px^k, k≤m-1.If derivationâ–³_h has a Darboux polynomial F such that deg_y F = 1, then the eigenvalue (?) = y±x^m.Lemma 2.2 Let h(x) = x^2m - px^k, k≤m - 1. Thenâ–³_h has a Darboux polynomial F such that deg_y F = 1 if and only if there exists a nonzero polynomial u∈k[x] such that u" - (mx^m-1 - px^k)u - 2x^mu' = 0 or u" + (mx^m-1 + px^k)u + 2x^mu' = 0.From the above two lemmas, we get the following results.Theorem 2.2 Let h(x) = x^2m - px^k, kh is simple.Theorem 2.3 Let h(x) = x^2m - px^m-1. Thenâ–³_h is not simple if and only if p≡m mod 2(m + 1) or p≡m + 2 mod 2(m + 1).We also study the simplicity of the derivationâ–³_h when h has degree 4. We consider the relationship between equivalent derivations and simplicity, and prove thatâ–³_h is simple if and only if so isâ–³_{x⁴+px²+qx+r}.Theorem 3.1 Let h(x) = x⁴ + px² + qx + r. Thenâ–³_h is not simple if and only if q is a nonzero even integer and the determinant M of the order n + 1 is zero, where n = (?)a - 1, From this theorem, we can get the following corollary immediately.Corollary 3.1 Let h(x) = x⁴ + px² + qx + r, where q = 0 or q is not an even integer. Thenâ–³_h is simple.