Resultants of Composed Polynomials

Abstract

The objective of this research has been to develop methods forcomputing resultants of composedpolynomials, efficiently, by utilizing their composition structure.By the resultant of several polynomials in several variables (one fewer variables than polynomials) we mean anirreducible polynomial in the coefficients ofthe polynomials that vanishes if theyhave a common zero.By a composed polynomial we mean the polynomial obtained from a given polynomial by replacing each variable by a polynomial.

The main motivation for this researchcomes from the following observations:Resultants of polynomialsare frequently computedin many areas of science andin applicationsbecause they are fundamentally utilized in solving systemsof polynomial equations.Further, polynomials arising in science and applicationsare often composed because humans tend to structure knowledge modularly and hierarchically.Thus, it is important to have theories and software librariesfor efficientlycomputing resultants of composed polynomials.

However,most existing mathematical theories do not adequately support composed polynomials and most algorithms as well as software libraries ignore the composition structure, thus suffering from enormous blow up in space and time.Thus, it is important to develop theories and software librariesfor efficientlycomputing resultants of composed polynomials.

The main finding of this research is thatresultants of composed polynomials can benicely factorized, namely, they can be factorized into products of powers of the resultants of the component polynomialsand of some of their parts. These factorizationscan be utilized to compute resultants of composed polynomialswith dramatically improved efficiency.