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 height of the Sylvester Resultant

Abstract: Let n be a positive integer. We consider the Sylvester Resultant of f and g, where f is a generic polynomial of degree 2 or 3 and g is a generic polynomial of degree n. If f is a quadratic polynomial, we find the resultant's height. If f is a cubic polynomial, we find tight asymptotics for the resultant's height.

Hopf Galois Kummer theory of formal groups

Abstract: Let /: F ?> G be an isogeny between finite n-dimensional formal groups defined over R, the valuation ring of some field extension K of Qp. Let H be the /?-Hopf algebra which arises from this isogeny. For such H, we classify Gal (H), the group of /7-Galois objects. Let Ai be the maximal ideal of R, and let P(F, K) denote the n-tuples of M under the group operation induced by F. Our main result is the construction of an isomorphism from the cokernel of P(f) to Gal (//), where P(f) is the induced map from P(F,K) to P(G,K). In geometric language Gal(//) describes the group of isomorphism classes of principal homogeneous spaces for Spec (//) over Spec (/?). Geometric methods have been used by Mazur to establish the above isomorphism, but the proof is nonconstructive. A geometric approach has also provided a formula for the cardinality of Gal (//). We give an alternative derivation of this result using formal group techniques. Introduction. Given a field K and a finite abelian group G a question in classical field theory is to determine Gal (K, G), the isomorphism classes of Galois field extensions of K with group G. When K has characteristic zero and contains a primitive nth root of unity, classical Kummer theory asserts that Gal (K, Cn) = U(K)/U(K)n where U(K) denotes the units of K and Cn is the cyclic group of order n. Furthermore, there is an explicit isomorphism from U(K)/U(K)n to Gal(/_\ Cn) given by sending u G U(K) (mod U(K)n) to K(x)/(xn - u). In their development of Hopf Galois theory Chase and Sweedier (1) provide criteria for an action (or co-action) of a cocommutative Hopf algebra on an exten sion of rings to be Galois. An /.-algebra S which admits a Galois co-action by an /.-Hopf algebra H is called an //-Galois object. Given a ring R and a cocommu tative /.-Hopf algebra //, a basic question in this theory is to determine Gal (//), the group of isomorphism classes of //-Galois objects. When K is a field and H is the dual of the group ring KG, G a finite group, Gal (//) becomes Gal (K, G). Determining Gal (//) is also of great interest because in geometric language this describes the group of isomorphism classes of principal homogeneous spaces for Spec (//) over Spec (/.). We make the conventions that all rings are commutative, all formal groups are commutative, and all Hopf algebras are abelian (commutative and cocommu tative). An /.-Hopf algebra is called finite if it is finitely generated and projective as an /.-module.

Abelsche Galoiserweiterungen von R[X]

Abstract: Our main result states that for a commutative ring R and a finite abelian group G the following conditions are equivalent: (a) Gal(R,G)=Gal (R[X],G), i.e. every commutative Galois extension of R[X]with Galois group G is extended from R. (b) The order of G is a non-zero-divisor in R/Nil(R). The proof uses lifting properties of Galois extensions over Hensel pairs and a “Milnor-type” patching theorem.

Algorithms for Extended Galois Field Generation and Calculation

Abstract: The paper aims to suggest algorithms for Extended Galois Field generation and calculation. The algorithm analysis shows that the proposed algorithm for finding primitive polynomial is faster than traditional polynomial search and when table operations in GF(p m ) are used the algorithms are faster than traditional polynomial addition and subtraction.

Polynomial systems theory applied to the analysis and design of multidimensional systems

Abstract: The use of a principal ideal domain structure for the analysis and design of multidimensional systems is discussed. As a first step it is shown that a lattice structure can be introduced for IO-relations generated by polynomial matrices in a signal space X (an Abelian group). It is assumed that the matrices take values in a polynomial ring F[p] where F is a field such that F[p] is a commutative subring of the ring of endomorphisms of X. After that it is analysed when a given F[p] acting on X can be extended to its field of fractions F(p). The conditions on the pair (F[p],X) are quite restrictive, i.e. each non-zero a(p) 2 F[p] has to be an automorphism on X before the extension is possible. However, when this condition is met, say for operators {p1,p2,...,pn 1}, a polynomial ring F[p1,p2,...,pn] acting on X can be extended to F(p1,p2,...,pn 1)[pn], resulting in a principal ideal domain structure. Hence in this case all the rigorous principles of ‘ordinary’ polynomial systems theory for the analysis and design of systems is applicable. As an example, both an observer for estimating non-measurable outputs and a stabilizing controller for a distributed parameter system are designed.