A computer algebra system risa/asir consists of a command asir for interactive use and several subroutine libraries which can be used as the parts of other programs. The grammar of the user language of asir is a variant of that of C and asir has a built-in dbx-like debugger. Risa's subroutine libraries include basic arithmetic subroutines, parser, evaluator and storage manager, and each of them can be used individually.This paper describes the characteristics and structure of risa system. We also show some sample programs, usage of the debugger and some timing data of fundamental calculations.

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extended abstractFor a problem how to find an extension field over which we can obtain an absolutely irreducible factor, Kaltofen gave an answer in 1983 and explicitly in I985 by employing analytic argument for showing his answer, and Chistov and Grigor'ev also gave the same answer in 1983 by algebraic arguments. Here, we give an alternative proof for Kaltofen's answer in algebraic way, independently to Chistov and Grigor'ev, and by the benefit of new way, we also give several extensions of his answer and properties of absolutely irreducible factors . We also discuss usage of our results for actual computation of absolutely irreducible factors. Here we restrict ourselves to bi-variate polynomials with integer ( or rational ) coefficients. First we state Kaltofen's answer.Kaltofen's theorem.For a bi-variate manic polynomial f(z, y) with integer coefficients, if f(~, u) is square free for an integer Q, then we can get an irreducible factor of f(z, y) over the algebraic closure of $ by factoring f(~, y) over an algebraic extension field containing a root of f(~, a).Our main results are the following:(1) We give an alternative proof of Kaltofen's theorem by using the fact that for a bi-variate polynomial f(z, y) irreducible over Q, all its absolutely irreducible factors are conjugate with each other and the set of all roots of each irreducible factor becomes an imprimitive block of the action of the Galois group of f(z, y). (2) We specialize the theorem to several coefficient domains such as fields with non zero characteristic.(3) By employing the argument used for an alternative proof, we obtain several useful properties of absolutely irreducible factors, from which we can present several quick test for absolute irreducibility of f(z, y) and information of the degree partition of f(z, y) relating to the degree partition of f(x, a), where a is an integer. (4) We apply the property, information of the degree partition, obtained in (3) to improve existing algorithms for absolute factoring. Thus, our further study on the problem in algebraic way provides several improvements on existing algorithms for factoring multi-variate polynomials. Main referencesChistov, A. L. and Grigor'ev, D. Yu. (1983), Subexponential-time solving systems of algebraic equations. I, LOMI Preprint E9-83, Leningrad. Kaltofen E. (1983), Polynomial-time reductions from multivariate to bi-and univariate integral polynomial factorization, SIAM J. Comput. 14, 469-489. Kaltofen E. (1985), Fast parallel absolute irreducible testing, J. Symbolic Computation 1, 57-67.297

Landau and Miller presented a method for determining the solvability of a manic irreducible polynomial over integers in polynomial time. In their method, a series of polynomials is constructed so that the original problem is reduced to determining the solvability of new polynomials.Here, we present an improved method for finding such a series of polynomials efficiently. More precisely, we introduce a new notion on a series of blocks in the set of all roots of the original polynomial under the action of its Galois group, and then present an efficient method for finding such a series of blocks by modifying Landau and Miller's method for finding minimal imprimitive blocks.