East Coast Computer Algebra Day

s of Invited Talks Subresultants revisited Joachim yon zur Gathen Fachbereich 17 Mathematik-Informatik Universit~t Paderborn Paderborn, Germany Starting in the late 1960s, Collins and Brown & Tranb invented polynomial remainder sequences (PRS) in order to apply the Euclidean algorithm to integer polynomials. Subresultants play a major role in this theory. We compare the various notions of subresultants, give a general and precise definition of PRS, and clean up some loose ends: • prove a 1971 conjecture of Brown that all results in the subresultant PRS are integer polynomials, • show an exponential lower bound on the pseudo PRS. Lastly, we show how Kronecker had, already in the 1870s, discovered many of the fundamental properties of Euclid's algorithm for polynomials. Some problems in general purpose computer algebra systems design Michael Monagan Center for Experimental and Computational Mathematics Simon Fraser University In this talk I will present three problems of interest to the computer algebra community. The first is the problem of implementing modular algorithms efficiently. Application of the Chinese remainder theorem to solve the GCD and Groebner bases problems leads to a big loss of efficiency because the data structure overhead overwhelms the cost of the modular arithmetic. The second problem is how to build a system so that all the components interact well. I will take as an example a problem of automatic differentiation from astrophysics where the function to be differentiated involves the solution of a non-linear equation. Can the CAS differentiate commands like f so lve (f--0,x--a); in a program? The third problem is a problem of trying to implement generic algorithms, efficiently. I will take as an example a linear p-adic Newton iteration. A generic version of this algorithm would work over Z mod p~ and over Fix] mod x ~ for example. Iterative solution of algebraic problems with polynomials Hans J. Stetter Technical University of Vienna Vienna, Austria In Numerical Analysis, it is standard to use an iterative solution procedure for a nonlinear problem. In Computer Algebra, one prefers exact finite manipulations which preserve the algebraic structure (like in Groebner basis computation); but often, in the end, an iterative numerical procedure can not be avoided (e.g. for zeros of a polynomial system). Furthermore, algebraic problems from Scientific Computing generally contain some "empiric" data so that their results are only defined to a limited accuracy. In this situation, an iterative approach may reduce to a few (or just one) step(s). We will at tempt to demonstrate how iterative procedures can be built upon the algebraic structure of a variety of problems for which such an approach has not been considered so far: After some discussion of zero clusters of univariate and systems of multivariate polynomials, we will mainly consider overdetermined problems like greatest common divisors, multivariate factorization, etc.; here the solution concept must be generalized to that of a pseudosolution which is an exact solution of a problem within the data tolerance neighborhood of the specified problem. Our iterative approach to the determination of pseudosolutions of such problems will prove computationally more flexible and efficient than recent "classical" approaches like [1] and [2]. [1 ] N.K. Karmarkar, Y.N. Lakshman: On Approximate GCDs of Univariate Polynomials, J. Symb.Comp. 26 (1998) 653-666 [2 ] M.A. Hitz, E. Kaltofen, Y.N. Lakshman: Efficient Algorithms for Computing the Nearest Polynomial with a Real Root and Related Problems, in: Proceed. ISSAC'99 (Ed. S.Dooley) (1999) 205-212