In Polynomial Algbera by Values project, we are interested in computing common roots of bivariate polynomials expressed in different basis. That is when the expanded form of the polynomials are either not available or expensive to compute due to their huge sizes. For, now we are only considering bivariate polynomial equations specified in the Lagrange or Hermite interpolation bases. This situation occurs quite often in Computer Aided Geometric Design (CAGD), and there is no direct root-finding approach which proceeds without first computing the polynomials explicitly; all we know is the degree of the polynomials, and their evaluations at some nodes, including some of the derivatives. If the nodes are distinct, we are studying a root-finding problem in the Lagrange basis. Otherwise, in the Hermite basis, confluent nodes exist, and we have the evaluation of some of the derivatives of the original polynomials. The desire in this part of the thesis is to avoid the ill-conditioning caused by a change of basis, and allow the user to do all the computation in the original basis, without first converting to the monomial basis. These research projects were under the supervision of Dr. Robert M. Corless, Dr. Dhavide Aruliah, and with a close collaboration with Dr. Laureano Gonzalez-Vega. In the most recent project with Dr. Laureano Gonzalez-Vega's team at the University of Cantabria, we are developing tools and algorithms for determining the topology of a real algebraic plane curve whoes equations are known only by values at some sample points.

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