Generic polynomial

About: Generic polynomial is a research topic. Over the lifetime, 608 publications have been published within this topic receiving 6784 citations.

On the Construction of Galois Towers

Abstract: In this paper we study an asymptotically optimal tame tower over the field with p(2) elements introduced by Garcia-Stichtenoth. This tower is related with a modular tower, for which explicit equations were given by Elkies. We use this relation to investigate its Galois closure. Along the way, we obtain information about the structure of the Galois closure of X-0(p(n)) over X-0(p(r)), for integers 1 < r < n and prime p and the Galois closure of other modular towers (X-0(p(n)))n.

On Generic differential _{}-extensions

Abstract: Let C be an algebraically closed field with trivial derivation and let F denote the differential rational field C , with Y ij , 1 < i < n - 1, 1 ≤ j ≤ n, i < j, differentially independent indeterminates over C. We show that there is a Picard-Vessiot extension e ⊃F for a matrix equation X' = XA.(Y ij ), with differential Galois group SO n , with the property that if F is any differential field with field of constants C, then there is a Picard-Vessiot extension E D F with differential Galois group H < SO n if and only if there are fij ∈ F with A(f ij ) well defined and the equation X' = XA(fij) giving rise to the extension E ⊃ F.

On the Constructive Inverse Problem in Differential Galois Theory

Abstract: We give sufficient conditions for a differential equation to have a given semisimple group as its Galois group. For any group G with G 0 = G 1 · ··· · G r , where each G i is a simple group of type Al, Cl, Dl, E6, or E7, we construct a differential equation over C(x) having Galois group G.

On the inverse problem of Galois theory of differential fields

Abstract: 1. All fields considered here are of characteristic 0. Let F be a field, let C be an algebraically closed subfield of F. Let G be a connected algebraic group defined over C. F(G) denotes the field of all rational functions on G defined over F. If gCG then F(g) denotes the field generated by g over F. We shall say that a derivation of F(G) commutes with G*(C) if it commutes with g*, for every gEG(C), where g* denotes the automorphism of F(G) induced by the left translation by g, i.e., (g*f)(x) =f(gx), for any xCG. F denotes the Lie algebra of all derivations of F(G) that are zero on F and which commute with G*(F). If G1 is a normal subgroup of G defined over F then F(G/G1) is canonically isomorphic to a subfield of F(G); we shall identify F(G/G1) and this subfield. If R is an integral domain then (R) denotes the field of fractions of R. Every derivation d of R can be uniquely extended to a derivation of R (the extended derivation will be also denoted by d). If F1, F2 are two fields containing F as a subfield and if d1, d2 are derivations of F1, F2, respectively, such that d1i F= d2 I F and d1(F) C F then d1i d2 denotes the derivation of F1,0F F2 determined by (d1 d2)(a 0 b) -d1(a)Gb+a0d2(b), for every aGF1 and bCF2. do denotes the zero derivation of a field (it will be always clear what field we have in mind). The underlying field of an ordinary differential field 53 will be denoted by F.