The motivation for this research originates in the practical need in science and engineering for the symbolic solution of large systems of composed polynomials. The goal is to find efficient methods for applying resultant techniques to solve systems of composed polynomials. The broader impacts are to increase the complexity of information that can be processed by scientists, engineers and by society as a whole and to thus increase productivity and cost savings. The research fosters undergraduate research (including those from groups underrepresented in the discipline) and strengthens research facilities at the principal investigator's department. Therefore it aims at motivating more students to aspire to research and engineering oriented careers. Further, it aims at increasing the percentage of graduates enrolling in graduate school. The intellectual merits for the field of symbolic computation are to advance study and efficient utilization of composition structures and resultants, which are core areas with far reaching impacts. The research approach is to apply resultant techniques to systems of equations with prevalent structure, efficiently, by utilizing the prevalent structure. The prevalent structure is composition of polynomials. Resultant techniques are methods based on a frequently utilized test for checking if systems of equations have common solutions.
The particular resultant techniques being studied are: approximating the common roots of composed polynomials by means of u-resultants, eliminating variables from skewly composed polynomials (n polynomials obtained by composing n polynomials with m polynomials where n and m are different) by means of multi-variable resultants, eliminating variables from composed ordinary differential polynomials by means of multi-variable resultants.
