Abstract

The objective of this research has been to develop methods for computing resultants of composed polynomials, efficiently, by utilizing their composition structure. By the resultant of several polynomials in several variables (one fewer variables than polynomials) we mean an irreducible polynomial in the coefficients of the polynomials that vanishes if they have 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 research comes from the following observations: Resultants of polynomials are frequently computed in many areas of science and in applications because they are fundamentally utilized in solving systems of polynomial equations. Further, polynomials arising in science and applications are often composed because humans tend to structure knowledge modularly and hierarchically. Thus, it is important to have theories and software libraries for efficiently computing 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 libraries for efficiently computing resultants of composed polynomials.

The main finding of this research is that resultants of composed polynomials can be nicely factorized, namely, they can be factorized into products of powers of the resultants of the component polynomials and of some of their parts. These factorizations can be utilized to compute resultants of composed polynomials with dramatically improved efficiency.