By M.-E. Alonso, E. Becker, M. F. Roy (auth.), Laureano González-Vega, Tomás Recio (eds.)
The current quantity incorporates a number of refereed papers from the MEGA-94 symposium held in Santander, Spain, in April 1994. They conceal fresh advancements within the idea and perform of computation in algebraic geometry and current new purposes in technology and engineering, rather laptop imaginative and prescient and conception of robotics. the amount can be of curiosity to researchers operating within the parts of laptop algebra and symbolic computation in addition to to mathematicians and desktop scientists drawn to having access to those subject matters.
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Extra info for Algorithms in Algebraic Geometry and Applications
2) Assume that there exists an element 9 in G j having (Cl, ... , ce) as its first i-part. Then, for i = £+ 1, ... , ... , a Ci ) = O. Proof. (1) We show ifpart. It is well-known that for a field K, any embedding of K into its algebraic closure K can be extended to an automorphism of K. The fact that a Cl ' ••• ,aCt satisfy the first t defining polynomials implies that there is a ring isomorphism T from Q( aI, ... , at) to Q( a Cl , ••• , a c , ). , T is an element of G j. Only if part is shown by the fact that every element 9 of the Galois group Gj fixes Q, that is, 0 = h(al, ...
For practical computation, improvements for this part are necessary. For factorization over algebraic number fields, several algorithms are proposed. We refer to Abott  for the history. Here, we propose a new method for the problem based on the modified Trager algorithm in . We first review Trager's algorithm presented in . Let a1, a2, ... , a£ be algebraic numbers such that the minimal polynomial of each ai is defined over Q(a1, ... ,ai-d. By applying his method directly to factorization over extension fields represented by successive extension, we obtain the following procedure.
On the other hand, we call the first one the direct approach and methods in the first one direct methods. Table-based methods ran determine Galois groups of polynomials very efficiently both in theory and in practice as far as tables for the degrees of input polynomials are known. We mention that, as a variant of table-based methods, there is another way using grouptheoretical knowledge, where the roots of an input polynomial are assigned to their approximate numerical values and functions on the roots associated with transitive subgroups in the symmetric group are computed, from which 30 H.