# A symbolic framework for general polynomial domains in theorema: applications to boundary problems

ACM Communications in Computer Algebra, Issue 177, Vol. 45, No. 3, September 2011 Abstracts of Recent Doctoral Dissertations in Computer Algebra Communicated by Jeremy Johnson Each month we are pleased to present abstracts of recent doctoral dissertations in Computer Algebra and Symbolic Computation. We encourage all recent Ph.D. graduates (and their supervisors), who have defended in the past two years, to submit their abstracts for publication in CCA. Please send abstracts to the CCA editors for consideration. Author: Loredana Tec Title: A Symbolic Framework for General Polynomial Domains in Theorema: Applications to Boundary Problems Institution: Research Institute for Symbolic Computation Thesis Advisors: Bruno Buchberger and Markus Rosenkranz In this thesis, we present a symbolic framework for working with general polynomial domains, targeted at dealing with linear boundary problems on the operator level. We consider two approaches for constructing general polynomials: the rst approach comes from universal algebra, whereas the latter employs monoid algebras. This framework is developed in Theorema, an integrated environment for doing mathematics (proving / computing / solving) in a uni ed formal frame anchored in higher-order predicate logic. A hierarchy for general polynomial domains is designed by the powerful construct of functors, a mechanism that was introduced in Theorema by B. Buchberger. In addition, an algorithmic setting is devised for polynomial reduction modulo suitable in nite systems, not covered by existing packages but needed for applications to boundary problems. In the context of symbolic boundary problems, two algebraic structures arise naturally: integrodi erential operators and polynomials. Their symbolic representation and computation involve canonical forms in certain commutative and noncommutative polynomial domains that are realized as instances of the two generic approaches. Furthermore, we establish an interesting link between integro-di erential polynomials and operators: The canonical simpli er of the former is employed in a new automated proof establishing a canonical simpli er for the latter. Various operations are implemented for applications to boundary problems, including the computation of Greenâs operators, composition and factorization of boundary problems for linear ordinary di erential equations and some simple classes of linear partial di erential equations. The implementation is completely coded in the Theorema version of higher-order predicate logic, except for some operations like solving di erential equations, which are delegated to Mathematica. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png ACM Communications in Computer Algebra Association for Computing Machinery

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