@MISC{Göttingen_thequestion,

author = {Grete Hermann In Göttingen},

title = {The Question of Finitely Many Steps in Polynomial Ideal Theory ∗},

year = {}

}

In the present work, the domains in which ideals are defined are polynomial domains. An ideal will be called given if a basis of the ideal is known, and computable if a basis can be computed. This work deals with computing characteristic ideals and polynomials for a given ideal m. The computation is based on ideal theory and elimination theory as developed by E. Noether and K. Hentzelt [2,8,9] 1. I especially recommend the summary [9, §1] for the basic ideas used here. Some changes to the definitions and further corollaries will be given in §1 of this work. The computational methods below are computations in finitely many steps. The claim that a computation can be carried out in finitely many steps will mean here that an upper bound for the number of necessary operations for the computation can be specified. Thus it is not enough, for example, to suggest a procedure, for which it can be proved theoretically that it can be executed in finitely many operations, if no upper bound for the number of operations is known 2. In particular, the bounds appearing in the present work will depend only on the number n of variables, the number t of basis elements of the ideal, and the maximum degree q of these basis elements; they are independent of the coefficients of the basis elements. Using these bounds, which indicate up to what degree the variables must be considered, the problems can be reduced to problems of determinant and elementary divisor theory, which can be settled in finitely many steps by known methods. The methods provided in §§6-8, with which all of the characteristic ideals and polynomials for the ideal m can be computed, must be preceded by some preparatory theorems in §§2-5. The search for the associated prime ideals of an ideal m corresponds to, and reduces to, the simpler problem of factoring a polynomial into prime functions. Thus §2 deals with the factorization of a polynomial into prime functions. The methods used here were suggested by

many step polynomial ideal theory basis element prime function present work characteristic ideal upper bound maximum degree ideal theory elementary divisor theory necessary operation simpler problem many operation preparatory theorem ideal corresponds basic idea elimination theory polynomial domain associated prime ideal computational method

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